Elliptical Transfer-Intercept Orbits

Non-Hohmann Transfer Orbits that Take You Somewhere

Most of the time, when someone speaks of transfer orbits, he's referring to a special case known as a Hohmann transfer orbit. Hohmann transfer orbits have a departure occurring at one of its apsides (perihelion or aphelion) and an arrival occurring at the other apside. Thus, I could describe Hohmann transfer orbits as transfer orbits having two anchored apsides.

In this essay, I will treat a more general case of transfer orbits that have only one anchored apside, which turns out to be enough to close the equation set and permit the Keplerian elements of the transfer orbit to be found, as well as the changes of velocity required for transfer orbit insertion and, later, for matching velocity with the destination object.

In what follows, the following example problem will be used for illustration:

A spaceship is initially in Earth's orbit, but is on the opposite side of the sun from Earth. Its captain wants to enter a transfer orbit, bound for Vesta, at 12h UT on 26 June 2017. The navigator does some trial runs on a computer and discovers an elliptical transfer orbit having its aphelion at Vesta upon arrival at 4h 45m 36.036s UT on 12 June 2018. Check the navigator's work to ensure that an elliptical transfer orbit does exist for these times for departure and arrival. Show the elements of the transfer orbit and the delta-vees required for transfer orbit insertion (departure) and for matching velocity with Vesta at arrival.

Spaceship initial orbit.
a = 1.000002 AU
e = 0.016711
i = 0.0°
Ω = 0.0°
ω = 103.095°
T = JD 2454285.96

Vesta's orbital elements.
a = 2.36126914 AU
e = 0.089054753
i = 7.13518389°
Ω = 103.91484282°
ω = 149.85540185°
T = JD 2454267.1969204

Departure time,
t₁ = 12h UTC, 26 June 2017

Arrival time,
t₂ = 4h 45m 36.036s UTC, 12 June 2018

It is convenient to convert t₁ and t₂ from calendar date format to Julian date format.

Converting from Calendar Date to Julian Date

After Fliegel and van Flandern (1968).

The time zone must be Greenwich, Zulu, UT, UTC (all the same zone)

Y = the four-digit year
M = the month of the year (1=January... 12=December)
D = the day of the month
Q = the time of the day in decimal hours

A = integer [ (M−14) / 12 ]
B = integer { [ 1461 (Y + 4800 + A) ] / 4 }
C = integer { [ 367 (M − 2 − 12A) ] / 12 }
E = integer [ (Y + 4900 + A) / 100 ]
F = integer [ (3E) / 4 ]
t = B + C − F + D − 32075.5 + Q/24

Converting the time of departure, t₁, from calendar date to Julian date

t₁ = 12h UTC, 26 June 2017
Y = 2017
M = 6
D = 26
Q = 12
A = 0
B = 2489909
C = 122
E = 69
F = 51
t₁ = JD 2457931.0

Converting the time of arrival, t₂, from calendar date to Julian date

t₂ = 4h 45m 36.036s UTC, 12 June 2018
Y = 2018
M = 6
D = 12
Q = 4.76001
A = 0
B = 2490274
C = 122
E = 69
F = 51
t₂ = JD 2458281.69833375

Instead of having the initial position vectors given to us, we must calculate them by reducing the elements of the spaceship's initial orbit (around the sun) and the time of departure therefrom, t₁, in order to obtain the position vector r₁, and by reducing the elements of Vesta's orbit and the time of arrival thereto, t₂, in order to obtain the position vector r₂.

For what passes below, the Sun's gravitational parameter,

GM = 1.32712440018ᴇ20 m³ sec⁻²

The ratio of the astronomical unit to the meter,

AU = 1.495978707ᴇ11 m au⁻¹

And the

Definition of the two-dimensional arctangent function.

atn(z) = single argument arctangent function of the argument z.

Function arctan( y , x )
. if x = 0 and y greater than 0 then angle = +π/2
. if x = 0 and y = 0 then angle = 0
. if x = 0 and y less than 0 then angle = −π/2
. if x greater than 0 and y greater than 0 then angle = atn(y/x)
. if x less than 0 then angle = atn(y/x) + π
. if x greater than 0 and y less than 0 then angle = atn(y/x) + 2π
arctan = angle

Unless otherwise indicated, the coordinate system to which all unprimed vectors in this essay refer is ecliptic coordinates — heliocentric for position, and sun-relative for velocity.

Reducing Keplerian orbital elements and a time to position and velocity in heliocentric ecliptic coordinates

Find the period, P, in days.

P = (365.256898326 days) a¹·⁵

Find the mean anomaly, m, in radians.

m₀ = (t − T) / P
m = 2π [ m₀ − integer(m₀) ]

Find the eccentric anomaly, u, in radians.

The Danby first approximation for the eccentric anomaly, u, in radians.

u' = m
+ (e − e³/8 + e⁵/192) sin(m)
+ (e²/2 − e⁴/6) sin(2m)
+ (3e³/8 − 27e⁵/128) sin(3m)
+ (e⁴/3) sin(4m)

The Danby's method refinement for the eccentric anomaly.

u = u'

U = u
F₀ = U − e sin U − m
F₁ = 1 − e cos U
F₂ = e sin U
F₃ = e cos U
D₁ = −F₀ / F₁
D₂ = −F₀ / [ F₁ + D₁F₂/2 ]
D₃ = −F₀ / [ F₁ + D₁F₂/2 + D₂²F₃/6 ]
u = U + D₃
UNTIL |u−U| is less than 1ᴇ-14

The loop, just above, converges u to the correct value of the eccentric anomaly. Usually. However, when e is near one and the orbiting object is near the periapsis of its orbit, there is a chance that this loop will fail to converge. In such cases, a different root-finding method will be needed.

Find the canonical position vector of the object in its orbit at time t.

x''' = a (cos u − e)
y''' = a sin u √(1−e²)
z''' = 0

Find the true anomaly, θ. We'll use it below when we find the velocity.

θ = arctan( y''' , x''' )

Rotate the triple-prime position vector by the argument of the perihelion, ω.

x'' = x''' cos ω − y''' sin ω
y'' = x''' sin ω + y''' cos ω
z'' = z''' = 0

Rotate the double-prime position vector by the inclination, i.

x' = x''
y' = y'' cos i
z' = y'' sin i

Rotate the single-prime position vector by the longitude of the ascending node, Ω.

x = x' cos Ω − y' sin Ω
y = x' sin Ω + y' cos Ω
z = z'

The unprimed position vector [x,y,z] is the position in heliocentric ecliptic coordinates.

Find the canonical (triple-prime) heliocentric velocity vector.

k = √{ GM / [ a AU (1 − e²) ] }

k is a speed in meters per second.

Vx''' = −k sin θ
Vy''' = k (e + cos θ)
Vz''' = 0

Rotate the triple-prime velocity vector by the argument of the perihelion, ω.

Vx'' = Vx''' cos ω − Vy''' sin ω
Vy'' = Vx''' sin ω + Vy''' cos ω
Vz'' = Vz''' = 0

Rotate the double-prime velocity vector by the inclination, i.

Vx' = Vx''
Vy' = Vy'' cos i
Vz' = Vy'' sin i

Rotate the single-prime velocity vector by the longitude of the ascending node, Ω.

Vx = Vx' cos Ω − Vy' sin Ω
Vy = Vx' sin Ω + Vy' cos Ω
Vz = Vz'

The unprimed velocity vector [Vx,Vy,Vz] is the sun-relative velocity in ecliptic coordinates.

Calculate the position and velocity of the spaceship in its initial orbit at the time of departure

P = 365.257994y'' = +0.979054316Vy''' = +30019.1146
m₀ = 9.97935722x' = −0.092732158Vx'' = −30140.9504
m = 6.15348288y' = +0.979054316Vy'' = −2921.69307
u' = 6.15128508z' = 0Vx' = −30140.9504
u = 6.15128508xi = −0.092732158Vy' = −2921.69307
x''' = +0.974604719yi = +0.979054316Vz' = 0
y''' = −0.131499998zi = 0Vxi = −30140.9504
θ = 6.14906877k = 29788.8217Vyi = −2921.69307
x'' = −0.092732158Vx''' = +3983.20734Vzi = 0

Calculate the position and velocity of Vesta at the time of arrival

P = 1325.30752y'' = +0.651051227Vy''' = +20727.481
m₀ = 3.02910935x' = −2.0545179Vx'' = −6748.92645
m = 0.182899417y' = +0.646009389Vy'' = −20049.7967
u' = 0.200646945z' = +0.080867606Vx' = −6748.92645
u = 0.200648459xf = −0.13298229Vy' = −19894.5281
x''' = +2.10361404yf = −2.14957848Vz' = −2490.40168
y''' = +0.468742457zf = +0.080867606Vxf = +20933.6861
θ = 0.219245394k = 19460.2928Vyf = −1766.64767
x'' = −2.0545179Vx''' = −4232.48025Vzf = −2490.40168

We will refer to a "hypothetical" transfer orbit until we have assured ourselves that it satisfies the condition that the calculated transit time be equal, or very nearly equal, to the required transit time.

The required transit time is the amount of time that the destination object (Vesta, in our example) takes to go from where it is at t₁ to where it is at t₂. This time difference is, of course, t₂−t₁.

The calculated transit time is the amount of time, Δt, that the spaceship takes to travel, along the hypothetical transfer orbit, from where it is at t₁ to the intersection of the hypothetical transfer orbit with the orbit of the destination object.

In general, Δt will differ substantially from t₂−t₁. It is necessary that t₁ and t₂ be chosen such that Δt is nearly equal to t₂−t₁. Once we know that to be the case, we can drop the word "hypothetical," for we will have determined that the transfer orbit does, indeed, exist.

The determination of an elliptical transfer-intercept orbit from a position and time of departure and from a position and time of arrival

At time t₁ a spaceship in free orbit around the sun (i.e. there is no planet nearby) has this state vector:

xi , yi , zi , Vxi , Vyi , Vzi

At time t₂ (such that t₂>t₁) an asteroid in free orbit around the sun (i.e. there is no significant perturbing third mass) has this state vector:

xf , yf , zf , Vxf , Vyf , Vzf

We want to find out whether or not there exists a transfer orbit between the position elements of those two state vectors, such that

x₁ = xi
y₁ = yi
z₁ = zi

x₂ = xf
y₂ = yf
z₂ = zf

The subscript 1 denotes "pertaining to the transfer orbit at transfer orbit insertion" or "departure."

The subscript 2 denotes "pertaining to the transfer orbit at its intersection with the destination object's orbit" or "arrival." (Whether the destination object is actually there at that same time is a question we will answer presently.)

r₁ = √[ x₁² + y₁² + z₁² ]
r₂ = √[ x₂² + y₂² + z₂² ]
d = √[ (x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)² ]

We define the integer variable β and permit it to have only the values 1 and 2.

If β=1, an apside (perihelion or aphelion) of the transfer orbit occurs at departure.
If β=2, an apside (perihelion or aphelion) of the transfer orbit occurs at arrival.

β = either 1 or 2
φ = 3 − β
N = (−1)ᵠ

The variables β and φ will usually be subscripts. The variable N is a sign toggle factor.

m : mean anomaly
u : eccentric anomaly
θ : true anomaly

If the apside at the apsidal endpoint of the intended trajectory is the perihelion, then

mᵦ = uᵦ = θᵦ = 0

If the apside at the apsidal endpoint of the intended trajectory is the aphelion, then

mᵦ = uᵦ = θᵦ = π radians

The eccentricity of a conic section, having the sun at a focus, which includes the point of departure and the point of arrival, is found by solving, simultaneously,

The polar equation which relates the heliocentric distance with the true anomaly,

cos θ₂ − cos θ₁ = { [a (1−e²) / r₂ − 1] − [a (1−e²) / r₁ − 1] } / e

The law of cosines,

d² = r₁² + r₂² − 2 r₁ r₂ cos(θ₂−θ₁)

We can eliminate the semimajor axis by recalling that rᵦ is equal to either a(1+e), the transfer orbit aphelion, or to a(1−e), the transfer orbit perihelion. In general, the semimajor axis can always be eliminated because one or the other endpoints of the intended trajectory occurs at one or the other apside of the transfer orbit. This is why you don't need a third point on the transfer orbit to determine its elements.

After some algebra, we get the eccentricity of the hypothetical transfer orbit.

e = 2 (cos θᵦ) rᵦ (rᵦ−rᵩ) / (rᵩ² − rᵦ² − d²)

The semimajor axis of the hypothetical transfer orbit is found from

a = rᵦ / (1 − e cos θᵦ)

The true anomaly in the hypothetical transfer orbit at the non-apsidal endpoint of the intended trajectory is found as follows:

θᵩ = θᵦ + N arccos{(rᵦ² + rᵩ² − d²) / (2rᵦrᵩ)}

The eccentric anomaly in the hypothetical transfer orbit at the non-apsidal endpoint of the intended trajectory is found as follows:

sin uᵩ = (rᵩ/a) sin θᵩ / √(1−e²)
cos uᵩ = (rᵩ/a) cos θᵩ + e
uᵩ = arctan(sin uᵩ , cos uᵩ)

The mean anomaly in the hypothetical transfer orbit at the non-apsidal endpoint of the intended trajectory is found as follows:

mᵩ = uᵩ − e sin uᵩ

The period of the hypothetical transfer orbit is

P = (365.256898326 days) a¹·⁵

The mean motion in the hypothetical transfer orbit is

μ = 2π/P

For short path trajectories (for which the arc of true anomaly going from departure to arrival is less than π radians), the calculated transit time in the hypothetical transfer orbit is

Δt = (N/μ) [mᵩ − π sin(θᵦ/2)]

Here's the test that determines whether a transfer orbit exists between heliocentric position r₁ at time t₁ and heliocentric position r₂ at time t₂. It is necessary that

Δt ≈ t₂ − t₁

And the match should be a close one, ideally a small fraction of a second. In general, this will not be the case. If the difference in the required and the calculated transit times is unacceptably large, then the spaceship pilot will have to choose either a different departure time, or a different arrival time, or both, and try again.

The procedure being demonstrated here finds elliptical transfer orbits of the short path, by which it is meant that the arc of true anomaly along the intended trajectory, from departure to arrival, is strictly less than π radians. One of the transfer orbit's apsides will occur at either the position of departure or at the position of arrival, but the other apside will not occur at all within the intended trajectory. To be complete about things, we will calculate the time of perihelion passage in the transfer orbit, whether or not the spaceship is ever there.

T = tᵦ − mᵦ/μ

The inclination of the transfer orbit is found from the cross product of the heliocentric position vectors of departure and arrival, r₁ x r₂.

Xn' = y₁ z₂ − z₁ y₂
Yn' = z₁ x₂ − x₁ z₂
Zn' = x₁ y₂ − y₁ x₂

Rn' = √[ (Xn')² + (Yn')² + (Zn')² ]

Xn = Xn' / Rn'
Yn = Yn' / Rn'
Zn = Zn' / Rn'

The vector Rn is a unit normal to the transfer orbit in the direction of the orbit's angular momentum.

i = arccos(Zn)

Having found the components of the vector normal to the transfer orbit (in the direction of the angular momentum), we now use it to find the velocity in the transfer orbit at the apsidal endpoint of the intended trajectory.

Vxᵦ'' = Yn zᵦ − Zn yᵦ
Vyᵦ'' = Zn xᵦ − Xn zᵦ
Vzᵦ'' = Xn yᵦ − Yn xᵦ

Vᵦ'' = √[ (Vxᵦ'')² + (Vyᵦ'')² + (Vzᵦ'')² ]

Vxᵦ' = Vxᵦ'' / Vᵦ''
Vyᵦ' = Vyᵦ'' / Vᵦ''
Vzᵦ' = Vzᵦ'' / Vᵦ''

Vᵦ = √[ (GM/AU) (2/rᵦ − 1/a) ]

Vxᵦ = Vᵦ Vxᵦ'
Vyᵦ = Vᵦ Vyᵦ'
Vzᵦ = Vᵦ Vzᵦ'

Now we find the angular momentum per unit mass in the transfer orbit.

hx = AU (yᵦ Vzᵦ − zᵦ Vyᵦ)
hy = AU (zᵦ Vxᵦ − xᵦ Vzᵦ)
hz = AU (xᵦ Vyᵦ − yᵦ Vxᵦ)

From here, we find the longitude of the ascending node of the transfer orbit.

Ω = arctan( hx , −hy )

Notice that hx is proportional to sin Ω, while −hy is proportional to cos Ω.

We find the argument of the perihelion of the transfer orbit as follows:

cos ω'' = (xᵦ cos Ω + yᵦ sin Ω) / rᵦ

If sin i = 0 then sin ω'' = (yᵦ cos Ω − xᵦ sin Ω) / rᵦ
If sin i ≠ 0 then sin ω'' = zᵦ / (rᵦ sin i)

ω'' = arctan( sin ω'' , cos ω'' )

ω' = ω'' − θᵦ

If ω' ≥ 0 then ω = ω'
If ω' < 0 then ω = ω' + 2π

The two known points on the hypothetical transfer orbit are

x₁ = −0.092732158 AU
y₁ = +0.979054316 AU
z₁ = 0

x₂ = −0.13298229 AU
y₂ = −2.14957848 AU
z₂ = +0.080867606 AU

The sides of the sun-departure-arrival triangle are

r₁ = 0.98343612 AU
r₂ = 2.15520567 AU
d = 3.129936551 AU

We are given that the apoapsis of the hypothetical transfer orbit occurs at the arrival position, so

β = 2
φ = 1
N = −1
m₂ = u₂ = θ₂ = π radians

The eccentricity and the semimajor axis of the hypothetical transfer orbit are

e = 0.37484849
a = 1.56759505 AU

The true, eccentric, and mean anomalies in the hypothetical transfer orbit at the non-apsidal endpoint of the intended trajectory (i.e. the departure position) are

θ₁ = 0.16062918 radians
u₁ = 0.10844236 radians
m₁ = 0.067872532 radians

The period and mean motion of the hypothetical transfer orbit are

P = 716.884602 days
μ = 0.00876457005 radians/day

The calculated and required transit times in the hypothetical transfer orbit, from departure to arrival, are

Δt = 350.698335 days
t₂−t₁ = 350.69833375 days

Subject to roundoff error, the difference is about one-tenth of a second. That's close enough. The transfer orbit exists.

The time of perihelion passage in the transfer orbit (although the spaceship is never there) is

T = JD 2457923.256033 = 18h 8m 41s UTC on 18 June 2017

Find the unit normal vector to the plane of the transfer orbit

Xn' = +0.079173779
Yn' = +0.0074990276
Zn' = +0.329531936

Rn' = 0.338992654

Xn = +0.233556030
Yn = +0.0221215048
Zn = +0.972091673

The transfer orbit's inclination to the ecliptic

i = 13.56812324°

Find the velocity in the transfer orbit at the apsidal endpoint of the intended trajectory

Vx₂'' = +2.091376254
Vy₂'' = −0.148158093
Vz₂'' = −0.499105248

V₂'' = 2.155205676

Vx₂' = +0.970383605
Vy₂' = −0.068744295
Vz₂' = −0.231581261

V₂ = 16041.367805 m/s

Vx₂ = +15566.280326 m/s
Vy₂ = −1102.752513 m/s
Vz₂ = −3714.880181 m/s

The angular momentum per unit mass

hx = +1.207943483ᴇ15 m²/s
hy = +1.144116363ᴇ14 m²/s
hz = +5.027623568ᴇ15 m²/s

The transfer orbit's longitude of the ascending node

Ω = 95.41068849°

Find the transfer orbit's argument of the perihelion

cos ω'' = −0.987126840
sin ω'' = +0.159939380
ω'' = 2.980963411 radians
ω' = −0.160629243 radians
ω = 350.79662233°

Keplerian elements of the transfer orbit

a = 1.56759505 AU
e = 0.37484849
i = 13.56812324°
Ω = 95.41068849°
ω = 350.79662233°
T = JD 2457923.256033

Now that you have the elements of the transfer orbit, you can calculate the changes-of-velocity needed for transfer orbit insertion (departure) and for matching velocity with the target asteroid at arrival.

When we reduce the elements of the transfer orbit with the time of departure, t₁, we find that the velocity of the spaceship in the transfer orbit is

Vx₁ = −34166.4329 m/s
Vy₁ = −1690.83202 m/s
Vz₁ = +8247.34992 m/s

The velocity of the spaceship in its initial orbit at t₁ was

Vxi = −30140.9504 m/s
Vyi = −2921.69307 m/s
Vzi = 0.0 m/s

So the change of velocity required at departure for transfer orbit insertion is

ΔVx₁ = −4025.4825 m/s
ΔVy₁ = +1230.8611 m/s
ΔVz₁ = +8247.3499 m/s

ΔV₁ = 9259.4983 m/s

The velocity of Vesta, when it is intercepted by the spaceship, is

Vxf = +20933.6861 m/s
Vyf = −1766.64767 m/s
Vzf = −2490.40168 m/s

When we reduce the elements of the transfer orbit with the time of arrival, t₂, we find that the velocity of the spaceship in the transfer orbit is

Vx₂ = +15566.2801 m/s
Vy₂ = −1102.75259 m/s
Vz₂ = −3714.88014 m/s

So the change of velocity required of the spaceship at arrival to match velocity with Vesta is

ΔVx₂ = +5367.4060 m/s
ΔVy₂ = −663.8951 m/s
ΔVz₂ = +1224.4785 m/s

ΔV₂ = 5545.1917 m/s

Remember that all of the unprimed vectors in this tutorial are referred to ecliptic coordinates. If you want them in celestial coordinates (so that you can use a star chart to show you the right ascension and declination in which to point the nose of your spaceship when you apply thrust), you'll still have that to do.

A check on the accuracy of the method by a numerical evolution of the state vector

As calculated from the Keplerian elements of the transfer orbit, at time t₁=JD 2457931.0, the spaceship's heliocentric state vector is

x₁ = −0.092732158 AU
y₁ = +0.979054316 AU
z₁ = 0
Vx₁ = −34166.4329 m/s
Vy₁ = −1690.83202 m/s
Vz₁ = +8247.34992 m/s

As calculated from the Keplerian elements of the transfer orbit, at time t₂ = JD 2458281.69833375, the spaceship's heliocentric state vector is

x₂ = −0.13298229 AU
y₂ = −2.14957848 AU
z₂ = +0.080867606 AU
Vx₂ = +15566.2801 m/s
Vy₂ = −1102.75259 m/s
Vz₂ = −3714.88014 m/s

The transit time of the spaceship in the transfer orbit is
t₂−t₁ = 30300336.036 sec

If we take the state vector at t₁ and numerically walk it forward in time by 1 second intervals for 30300336 seconds, we get this result.

x = −0.132983124 AU
y = −2.149579643 AU
z = +0.080867833 AU
Vx = +15566.27354 m/s
Vy = −1102.76889 m/s
Vz = −3714.87818 m/s

The relationship between apparent magnitude and radiative flux for the same spectral band

A question often asked on forums such as Quora or Yahoo Answers or Physics Forums is what's the relationship between radiative flux and apparent magnitude (for the same spectral band). The correct answer is seldom forthcoming. But it isn't hard to figure out.

First, let's acknowledge that

1 parsec = 30856775814913673 meters

That'll be important down the page.

m : apparent magnitude
M : absolute magnitude
L : luminosity in watts
F : flux in watts per square meter
D : distance in parsecs
d : distance in meters

The distance modulus equation
m − M = 5 log D − 5

The inverse square law for radiative flux
F = L/(4πd²)

The magnitude scale definition
M = M☉ − 2.5 log (L/L☉)

M = M☉ − 2.5 log L + 2.5 log L☉

L = 4πd²F

M = M☉ + 2.5 log L☉ − 2.5 log(4πd²F)

M = M☉ + 2.5 log L☉ − 2.5 log(4π) − 5 log d − 2.5 log F

D = d / 30856775814913673

5 log D = 5 log(d) − 5 log(30856775814913673)

5 log D = 5 log(d) − 82.446752726110691733478528419098

m − M = 5 log D − 5

m − M = 5 log d − 87.446752726110691733478528419098

m − [M☉ + 2.5 log L☉ − 2.5 log(4π) − 5 log d − 2.5 log F] = 5 log d − 87.446752726110691733478528419098

m = M☉ + 2.5 log L☉ − 2.5 log(4π) − 2.5 log F − 87.446752726110691733478528419098

M☉ = 4.7554

L☉ = 3.827e26 watts

2.5 log(4π) = 2.7480246600552406119468651943497

m = 4.7554 + 66.457146155561248277821578807329
− 2.7480246600552406119468651943497
− 87.446752726110691733478528419098
− 2.5 log F

m = −2.5 log F − 18.982231230604684067603814806119

That's the equation that I derive, using my values for the sun's bolometric absolute magnitude, M☉, and bolometric luminosity, L☉. A possibly more authoritative source, whose name and publisher I've forgotten, gave this very similar equation, which agrees with mine to six significant figures. You should probably use his equation. Mine is just for showing how it can be found.

m = −2.5 log F − 18.982249379206

Determining the Keplerian Elements of an Elliptical Orbit from Four Observations

Presented hereafter is a method for determining a preliminary heliocentric orbit from four geocentric directions of a sun-orbiting object at four distinct times of observation. I take this method after that presented in chapter six of The Determination of Orbits by A.D. Dubyago, who credits it to

The German mathematician Karl Friedrich Gauss
The German astronomer Julius Bauschinger
The Soviet celestial mechanic Mikhail Fedorovich Subbotin

The initial data

t₁, X⊕₁, Y⊕₁, Z⊕₁, α₁, δ₁
t₂, X⊕₂, Y⊕₂, Z⊕₂, α₂, δ₂
t₃, X⊕₃, Y⊕₃, Z⊕₃, α₃, δ₃
t₄, X⊕₄, Y⊕₄, Z⊕₄, α₄, δ₄

The times of observation, tᵢ, are given in Julian date format. The vectors [X⊕ᵢ,Y⊕ᵢ,Z⊕ᵢ] are the positions of the Earth in heliocentric ecliptic coordinates, with the components being in astronomical units. The αᵢ are the geocentric right ascensions for Ceres. The δᵢ are the geocentric declinations for Ceres.

The time intervals should be about 0.5% to 1% of the object's period (estimated as 8 to 16 days for a main belt asteroid), should be near opposition with the sun, but should NOT span an apside of the object's orbit. The precision in right ascension should be 0.01 seconds or better, and the precision in declination should be 0.1 arcseconds or better.

The Earth's orbital mean motion, κ = 0.01720209895 radians/day

We will use a single value for the obliquity of the ecliptic to transform all four of the observation angles from celestial coordinates to ecliptic coordinates. First, we find the middle of the observation time window in units of 10000 years since 1 January 2000, and then we use that number to find the obliquity using the 10-degree polynomial fit of J. Laskar.

t = (t₁ + t₄)/2
T = (t − 2451545)/3652500

The obliquity in seconds of arc is

ε" = 84381.448 − 4680.93 T − 1.55 T² + 1999.25 T³ − 51.38 T⁴ − 249.67 T⁵ − 39.05 T⁶ + 7.12 T⁷ + 27.87 T⁸ + 5.79 T⁹ + 2.45 T¹⁰

The obliquity in radians is

ε = (π / 648000) ε"

Although Earth's axial tilt (i.e. the obliquity of the ecliptic) does change over time, it changes so slowly that the difference will almost always be negligible across the t₁ to t₄ time window. However, in the rare case when this isn't true, separate evaluations of the obliquity will have to be made for each time of observation.

The geocentric positions of the sun in celestial coordinates are (for i = 1 to 4)

Xᵢ = −X⊕ᵢ
Yᵢ = −Y⊕ᵢ cos ε + Z⊕ᵢ sin ε
Zᵢ = −Y⊕ᵢ sin ε − Z⊕ᵢ cos ε

The geocentric unit vectors in the direction of the target object, in celestial coordinates, are (for i = 1 to 4)

aᵢ = cos αᵢ cos δᵢ
bᵢ = sin αᵢ cos δᵢ
cᵢ = sin δᵢ

The squares of the Sun-Earth distances at times t₁ and t₄ are

R₁² = X₁² + Y₁² + Z₁²
R₄² = X₄² + Y₄² + Z₄²

The values of 2Rᵢ cos Θᵢ at times t₁ and t₄ are

2R₁ cos Θ₁ = −2 ( a₁X₁ + b₁Y₁ + c₁Z₁ )
2R₄ cos Θ₄ = −2 ( a₄X₄ + b₄Y₄ + c₄Z₄ )

where Θ is the supplementary angle to the sun-Earth-object angle.

We find some time differences:

τ₁ = κ (t₄−t₂)
τ₂ = κ (t₂−t₁)
τ₃ = κ (t₄−t₁)
τ₄ = κ (t₄−t₃)
τ₅ = κ (t₃−t₁)

We find this pair of determinants:

Φ = a₂b₄−b₂a₄
φ = a₃b₄−b₃a₄

We calculate some intermediate quantities:

A = ( a₁b₂ − b₁a₂ ) / Φ
B = ( a₂Y₁ − b₂X₁ ) / Φ
C = ( b₂X₂ − a₂Y₂ ) / Φ
D = ( a₂Y₄ − b₂X₄ ) / Φ

A' = ( a₁b₃ − b₁a₃ ) / φ
B' = ( a₃Y₁ − b₃X₁ ) / φ
C' = ( b₃X₃ − a₃Y₃ ) / φ
D' = ( a₃Y₄ − b₃X₄ ) / φ

And then more intermediate quantities:

E = τ₁/τ₂
F = (4/3) τ₁τ₃
G = AE
H = F(A−G)
I = 4Aτ₁²
K = E (B + C) + C + D
L = F (B − C + D − K)
M = 4 (Bτ₁² + τ₁τ₂C)

E' = τ₄/τ₅
F' = (4/3) τ₄τ₃
G' = A'E'
H' = F'(A'−G')
I' = 4A'τ₄²
K' = E' (B' + C') + C' + D'
L' = F' (B' − C' + D' − K')
M' = 4 (B'τ₄² + τ₄τ₅C')

Make initial guesses for the sun-object distance r₁ at time t₁ and for the sun-object distance r₄ at time t₄. For main belt asteroids, a reasonable initial guess for both times is 2.75 AU. Then use the loop below to converge, by successive approximations, to the true sun-object distances, r, and for the Earth-object distances, ρ, distances at times t₁° and t₄° (i.e. the times for the first and fourth observations, corrected for the speed of light travel time).

O = 9.999e+99
N = r₁ + r₄

while |N−O|/N > 1ᴇ-11 do
ξ = (r₁ + r₄)⁻³
η = (r₄ − r₁) / (r₁ + r₄)
P = G + ξH + ηξI
Q = K + ξL + ηξM
P' = G' + ξH' + ηξI'
Q' = K' + ξL' + ηξM'
ρ₁ = (Q'−Q)/(P−P')
ρ₄ = Pρ₁ + Q
r₁ = √[R₁² + (2R₁cos Θ₁)ρ₁ + ρ₁²]
r₄ = √[R₄² + (2R₄cos Θ₄)ρ₄ + ρ₄²]
O = N
N = r₁ + r₄

The positions of the object at times t₁ and t₄ in geocentric celestial coordinates are

x₁ = a₁ρ₁ − X₁
y₁ = b₁ρ₁ − Y₁
z₁ = c₁ρ₁ − Z₁

x₄ = a₄ρ₄ − X₄
y₄ = b₄ρ₄ − Y₄
z₄ = c₄ρ₄ − Z₄

The reciprocal of the speed of light
ç = 0.00577551833 days/AU

The sun's gravitational parameter
μ = 1.32712440018ᴇ20 m³ sec⁻²

The conversion factor from AU to meters is
U = 1.495978707ᴇ11

The conversion factor from AU/day to m/sec is
β = 1731456.8368

Correcting times of observation for planetary abberation.

t₁° = t₁ − çρ₁
t₄° = t₄ − çρ₄

The nominal time associated with the forthcoming state vector is

t₀ = ½ (t₁° + t₄°)

The nominal heliocentric distance of the object at time t₀ is

r₀ = ½ (r₁ + r₄)

Find the heliocentric position vector [x',y',z'] for the object at time t₀ in celestial coordinates.

x" = ½ (x₁ + x₄)
y" = ½ (y₁ + y₄)
z" = ½ (z₁ + z₄)
r" = √[(x")²+(y")²+(z")²]
x' = (r₀/r") U x"
y' = (r₀/r") U y"
z' = (r₀/r") U z"

Find the sun-relative velocity vector for the object at time t₀ in celestial coordinates.

S = √[ (x₄ − x'/U)² + (y₄ − y'/U)² + (z₄ − z'/U)² ]
s = √[ (x'/U − x₁)² + (y'/U − y₁)² + (z'/U − z₁)² ]
Ψ = S + s
ψ = √[(x₄−x₁)² + (y₄−y₁)² + (z₄−z₁)²]
Vx' = β (Ψ/ψ) (x₄−x₁) / (t₄°−t₁°)
Vy' = β (Ψ/ψ) (y₄−y₁) / (t₄°−t₁°)
Vz' = β (Ψ/ψ) (z₄−z₁) / (t₄°−t₁°)

The object's position in heliocentric ecliptic coordinates

x₀ = x'
y₀ = y' cos ε + z' sin ε
z₀ = −y' sin ε + z' cos ε

The object's sun-relative velocity in ecliptic coordinates

Vx₀ = Vx'
Vy₀ = Vy' cos ε + Vz' sin ε
Vz₀ = −Vy' sin ε + Vz' cos ε

The object's speed relative to the sun

V₀ = √[(Vx₀)² + (Vy₀)² + (Vz₀)²]

The semimajor axis of the object's orbit, in AU

a = (2/r₀ − V₀²/μ)⁻¹ / U

The angular momentum per unit mass in the object's orbit

hx = y₀ Vz₀ − z₀ Vy₀
hy = z₀ Vx₀ − x₀ Vz₀
hz = x₀ Vy₀ − y₀ Vx₀

h = √[(hx)² + (hy)² + (hz)²]

The eccentricity of the object's orbit

e = √[1 − h²/(aμU)]

The inclination of the object's orbit

i = arccos(hz/h)

The longitude of the ascending node of the object's orbit

Ω' = arctan(−hx/hy)
if hy>0 then Ω = Ω' + π
If hy<0 and hx<0 then Ω = Ω' + 2π

The true anomaly at time t₀

sin θ₀ = h ( x₀ Vx₀ + y₀ Vy₀ + z₀ Vz₀ ) / (r₀μ)
cos θ₀ = h²/(r₀μ) − 1
θ₀' = arctan( sin θ₀ / cos θ₀ )
If cos θ₀ < 0 then θ₀ = θ₀' + π
If cos θ₀ > 0 and sin θ₀ < 0 then θ₀ = θ₀' + 2π

The sum of the true anomaly at time t₀ and the argument of the perihelion of the object's orbit

sin(θ₀+ω) = z₀ / (r₀ sin i)
cos(θ₀+ω) = ( x₀ cos Ω + y₀ sin Ω ) / r₀
(θ₀+ω)' = arctan[ sin(θ₀+ω) / cos(θ₀+ω) ]
If cos(θ₀+ω) < 0 then θ₀ = θ₀' + π
If cos(θ₀+ω) > 0 and sin(θ₀+ω) < 0 then (θ₀+ω) = (θ₀+ω)' + 2π

The argument of the perihelion of the object's orbit

ω' = (θ₀+ω) − θ₀
if ω'<0 then ω=ω'+2π else ω=ω'

The eccentric anomaly of the object at time t₀

cos u₀ = 1 − r₀/(aU)
sin u₀ = (x₀ Vx₀ + y₀ Vy₀ + z₀ Vz₀) / √(aμU)
u₀' = arctan( sin u₀ / cos u₀ )
If cos u₀ < 0 then u₀ = u₀' + π
If cos u₀ > 0 and sin u₀ < 0 then u₀ = u₀' + 2π

The mean anomaly of the object at time t₀

M₀ = u₀ − e sin u₀

The period of the object's orbit in days

P = 365.256898326 a¹·⁵

The object's time of perihelion passage

T = t₀ − PM₀/(2π)

Example problem. Find the orbit of Ceres.

Observation #1
t₁ = JD 2457204.625
X⊕₁ = +0.155228396 AU
Y⊕₁ = −1.004732775 AU
Z⊕₁ = +0.00003295786 AU
α₁ = 20h 46m 57.02s
δ₁ = −27°41′33.9″

Observation #2
t₂ = JD 2457214.625
X⊕₂ = +0.319493277 AU
Y⊕₂ = −0.965116604 AU
Z⊕₂ = +0.0000311269 AU
α₂ = 20h 39m 57.10s
δ₂ = −28°47′21.5″

Observation #3
t₃ = JD 2457224.625
X⊕₃ = +0.4747795623 AU
Y⊕₃ = −0.8983801739 AU
Z⊕₃ = +0.00002841127 AU
α₃ = 20h 31m 22.81s
δ₃ = −29°49′22.7″

Observation #4
t₄ = JD 2457234.625
X⊕₄ = +0.616702829 AU
Y⊕₄ = −0.8063620175 AU
Z⊕₄ = +0.00002486325 AU
α₄ = 20h 22m 06.57s
δ₄ = −30°41′57.3″

Converting the observation angles to radians:

α₁ = 5.44084723
δ₁ = −0.483329666

α₂ = 5.41030979
δ₂ = −0.502468171

α₃ = 5.37290956
δ₃ = −0.520509058

α₄ = 5.33245865
δ₄ = −0.53580299

Continuing through the procedure,

t = 2457219.63
T = 0.001553628
ε = 0.409057547 radians

X₁ = −0.155228396
Y₁ = +0.921851498
Z₁ = +0.399597005

X₂ = −0.319493277
Y₂ = +0.885503087
Z₂ = +0.383841559

X₃ = −0.474779562
Y₃ = +0.824271594
Z₃ = +0.357299982

X₄ = −0.616702829
Y₄ = +0.739843884
Z₄ = +0.320703493

a₁ = +0.589463371
b₁ = −0.660726081
c₁ = −0.464730008

a₂ = +0.563195268
b₂ = −0.671477524
c₂ = −0.4815901

a₃ = +0.532276132
b₃ = −0.685093499
c₃ = −0.497321844

a₄ = +0.49965704
b₄ = −0.69978587
c₄ = −0.510531662

R₁² = 1.03358381
R₄² = 1.03054208

2R₁ cos Θ₁ = 1.772595
2R₄ cos Θ₄ = 1.97920299

τ₁ = 0.344041979
τ₂ = 0.17202099
τ₃ = 0.516062969
τ₄ = 0.17202099
τ₅ = 0.344041979

Φ = −0.058607619
φ = −0.030167527

A = +0.404275125
B = −7.08013785
C = +4.84883362
D = −0.043927505

A' = +1.72864027
B' = −12.7399767
C' = +3.76138574
D' = +0.951283093

E = +2
F = +0.236729767
G = +0.80855025
H = −0.095703956
I = +0.191407912
K = +0.342297675
L = −2.91537363
M = −2.20429551

E' = +0.5
F' = +0.118364883
G' = +0.864320133
H' = +0.102305152
I' = +0.204610303
K' = +0.223373365
L' = −1.86702289
M' = −0.617533885

r₁ = 2.75 (initial guess)
r₄ = 2.75 (initial guess)

1st approximation
ρ₁ = 1.97723208
ρ₄ = 1.9223289
r₁ = 2.90652064
r₄ = 2.92071388

2nd approximation
ρ₁ = 2.001149
ρ₄ = 1.94460337
r₁ = 2.93008666
r₄ = 2.94292188

3rd approximation
ρ₁ = 2.00417695
ρ₄ = 1.94741724
r₁ = 2.93307059
r₄ = 2.94572742

4th approximation
ρ₁ = 2.00455348
ρ₄ = 1.94776713
r₁ = 2.93344165
r₄ = 2.94607627

5th approximation
ρ₁ = 2.0046002
ρ₄ = 1.94781054
r₁ = 2.93348769
r₄ = 2.94611956

6th approximation
ρ₁ = 2.00460599
ρ₄ = 1.94781593
r₁ = 2.9334934
r₄ = 2.94612492

7th approximation
ρ₁ = 2.00460671
ρ₄ = 1.9478166
r₁ = 2.93349411
r₄ = 2.94612559

8th approximation
ρ₁ = 2.0046068
ρ₄ = 1.94781668
r₁ = 2.9334942
r₄ = 2.94612567

9th approximation
ρ₁ = 2.00460681
ρ₄ = 1.94781669
r₁ = 2.93349421
r₄ = 2.94612568

10th approximation (final, converged)
ρ₁ = 2.00460681
ρ₄ = 1.94781669
r₁ = 2.93349421
r₄ = 2.94612568

HEC positions in AU at t₁ & t₄

x₁ = +1.33687069
y₁ = −2.2463475
z₁ = −1.33119794

x₄ = +1.58994315
y₄ = −2.10289848
z₄ = −1.31512559

The reciprocal of the speed of light
ç = 0.00577551833 days/AU

Aberration corrections to time

çρ₁ = 0.011577643 days
t₁° = 2457204.61

çρ₄ = 0.011249651 days
t₄° = 2457234.61

Epoch of state vector
t₀ = 2457219.61 JD

The object's state vector in heliocentric ecliptic coordinates

x₀ = +1.46520344 AU
y₀ = −2.52458426 AU
z₀ = −0.349479243 AU
Vx₀ = +14610.4367 m/s
Vy₀ = +7967.42879 m/s
Vz₀ = −2442.63758 m/s

The object's distance from the sun at t₀
r₀ = 2.93980995 AU

Sun-relative speed
V₀ = 16819.9661 m/s

The sun's gravitational parameter
μ = 1.32712440018ᴇ20 m³ sec⁻²

The semimajor axis of the object's orbit
a = 2.76694735 AU

The angular momentum per unit mass in the object's orbit

hx = +1.3390648ᴇ15 m²/sec
hy = −2.2844842ᴇ14 m²/sec
hz = +7.26435028ᴇ15 m²/sec

h = 7.39026848ᴇ15 m²/sec

The eccentricity of the object's orbit
e = 0.076026341

The inclination of the object's orbit
i = 10.5918141°

The longitude of the ascending node of the object's orbit
Ω = 80.3183813°

The true anomaly at time t₀
θ₀ = 147.669798°

The sum of the true anomaly at time t₀ and the argument of the perihelion of the object's orbit
(θ₀+ω) = 220.296384°

The argument of the perihelion of the object's orbit
ω = 72.6265867°

The eccentric anomaly of the object at time t₀
u₀ = 145.259666°

The mean anomaly of the object at time t₀
M₀ = 142.777370°

The period of the object's orbit
P = 1681.12408 days

Times of perihelion passage

T₀ = 2456552.87
T₁ = 2458234.01 = T₀+P

A summary of the calculation results and a comparison with JPL's numbers

Orbital elements (as calculated)

a = 2.76694735 AU
e = 0.076026341
i = 10.5918141°
Ω = 80.3183813°
ω = 72.6265868°
T₀ = JD 2456552.87
T₁ = JD 2458234.01

Elements for Ceres from JPL Horizons for JD 2457219.5 (16 July 2015)

a = 2.76800849 AU
e = 0.075773402
i = 10.5922177°
Ω = 80.3268351°
ω = 72.6626266°
T = JD 2456552.64

More comments censored by YouTube

Censored Comment 1.

21:40. No, whites do not make up 76.3% of the population. You've added together whites and mestizos ("Hispanics") and have presented the percentage sum of the two races. Whites alone are only about 60% of the US population. Mestizos are about 16% of the US population. The US government frequently lumps these two groups together in order to create the appearance that whites commit more crimes, both in total and per capita, than they really do. In jurisdictions where the crimes of mestizos and the crimes of whites are separately tracked, mestizos show a per capita perpetration rate that is about three times higher than that of whites. Generally speaking, though it does depend considerably on which crime is involved, blacks have a per capita perpetration rate that is as far above that of mestizos as is that of mestizos above that of whites. As a rule of thumb, the per capita crime rate for mestizos is usually close to the geometric average between that of blacks and that of whites.

And, yes, there really is a Marxist "deep state" animus against white people. How can you tell? It shows in the fact that mestizos are treated as a group distinct from whites when they are the victims of crime, whereas mestizos are packed into the "white" offender category when they are the perpetrators of crimes.

Debunking White Privilege & Addressing Income Disparities in the U.S.
by Data Driven Conclusions

Censored Comment 2.

Even non-CRT diversity and inclusion training would conflict with quality and with efficiency in production for the market. The reason why is that races differ with each other in a general and distributed way, in the way that the book The Bell Curve (Richard Herrnstein and Charles Murray) strives to explain. Of course, of course, you can find exceptions. Of course, of course, you can find anecdotes that go contrary to the general trend. But, in the long run, the reason for Affirmative Action, for hiring quotas for certain minorities, for political correctness in choosing who does what and who gets promoted, is racial differences.

You can even put the math behind it in writing.

While it is true that someone’s race doesn’t determine his individual IQ, it does determine the probability for a randomly chosen member of a race having an average IQ being at, or above, a specified value.

The normal distribution that most closely matches the IQ distribution of white male US citizens is 103.08±14.54 (Jensen & Reynolds, "Sex Differences on the WISC-R," Personality & Individual Differences, volume 4, number 2, pp. 223-226, 1983).

The normal distribution that most closely matches the IQ distribution of US-resident mulattoes (usually called "blacks" or "African-Americans") is 85.0±13.0 (a typical finding of studies since 1950).

A good approximation of the fraction, f, of a race having an average IQ of x̄ and a standard deviation in IQ of σ, which is above the minimum IQ of μ, can be found as follows:

f(μ) = [σ√(2π)]⁻¹ ∫(μ,∞) exp{ −[(x−x̄)/σ]²/2 } dx

Taking advantage of the normal distribution's symmetry, we make it more easily integrable.

f(μ) = ½ − [σ√(2π)]⁻¹ ∫(x̄,μ) exp{ −[(x−x̄)/σ]²/2 } dx

You can avoid integrating the probability density function if you have a handy error function to call.

f(μ) = 1 − ½ { 1 + erf [(μ−x̄)/(σ√2)] }

Let us suppose that an employer wants to hire workers for a job that, in his opinion, requires a minimum IQ of 130 for satisfactory performance. He lives in an area that is demographically typical for the United States, where whites outnumber blacks by a ratio of five.

The fraction of whites who are qualified for the job on the basis of IQ is

f( μ=130.0, x̄=103.08, σ=14.54 ) = 0.0320528311

The fraction of blacks who are qualified for the job on the basis of IQ is

f( μ=130.0, x̄=85, σ=13 ) = 0.0002685491

If the population of whites and of blacks were equal in size, then the ratio of mentally qualified whites to mentally qualified blacks would be 119.355755.

Since whites outnumber blacks in the area where the employer's business is, by a ratio of five, the actual ratio of mentally qualified whites to mentally qualified blacks is 596.778775.

If the employer needs fewer than 100 new employees, it could very easily turn out that he will hire no blacks at all, even if he uses no racism whatsoever in selecting his hirelings. In fact, of the occasions in which this scenario plays out, and exactly 100 new workers are hired, the employer will have hired...

100 whites and 0 blacks on 84.5% of occasions
99 whites and 1 black on 14.2% of occasions
98 whites and 2 blacks on 1.2% of occasions
97− whites and 3+ blacks on 0.1% of occasions

Because the United States is a First World country in which most of the best jobs are mentally challenging jobs, purely free-market hiring practices will exclude a demographically disproportionately high fraction of low-average-IQ races from those jobs. This is normal and natural. It is the only way by which a country can remain competitive internationally, especially with countries that don't engage in politically correct tampering with the free market for labor. A focus on merit is a good thing. It is in conflict with diversity and inclusion. Therefore, diversity and inclusion are bad ideas, and to focus on them, instead of on merit, is harmful.

The Staging Area for Antifa/BLM in Portland, Oregon

In Portland, Oregon, there are several blocks near the intersection of Hawthorne bridge and Burnside bridge, both of which span the Willamette River, that have no permanent buildings, where Antifa/BLM rioters have been squatting for several months. They live in tents and under tarps that are mostly blue in color.

The city government has set aside these areas for their use and has provided these leftist troublemakers with services, such as portable toilets and privacy screening fences.

The enemy anarchists include many people who have become addicted to illegal narcotics and who are controlled through their addiction by their leaders, who supply the drugs. The leaders, however, are paid for their "work" by rich subversives, like the Jewish billionaire George Soros.

The rioters use their tent-town as a staging area. It is where they go to sleep during the day, so that they can emerge upon the streets refreshed and ready for another night of arson, vandalism, assault, robbery, and murder. Some of their number stand watch, using slingshots to shoot rocks at passers-by to discourage investigation.

The lesson the MSM never learns. FIRST, there was Nicholas Sandmann. NOW, there is Kyle Rittenhouse

Back to the Kyle Rittenhouse case. It has been revealed that the gun that 17-year-old Kyle used to defend, first, a shop owner's property from arson and, later, himself from attack by the rioting arsonists, belonged to a friend of his who lived in Wisconsin. Contrary to what you might have been hearing from the lying leftist mainstream media, that gun never crossed state lines.

Here's a possible repeat of the lesson that the MSM should have learned, but didn't learn, from their lies about Nicholas Sandmann. Sandmann is the teenager who the media falsely accused of racially disrespecting a native American, resulting in death threats, massive harassment, and unjust penalties against himself. When the truth came out, Sandmann began suing the media institutions who had lied about him and had refused to retract those lies with a suitable apology. And he has won probably millions of dollars in out-of-court settlements from MSM corporations.

Now here we go again. Probably, after he is exonerated from his politically motivated first-degree murder charges, Kyle Rittenhouse will have standing to sue the media corporations for defamation. And I hope that he wins, too.

Comment on Race, IQ, and Socio-Economic Status Censored from YouTube

@hydrolito That's right. But cultures differ, and our culture is one of the more technologically advanced cultures. Nearly all of our most highly paid jobs — technical, clerical, legal — are jobs that require a high IQ to be competitive in the labor market. Because blacks have an IQ distribution that is downwardly offset from that of whites by about 15 points, they will (in general) find themselves out-competed by whites for the best jobs, and the result will be a generally lower socio-economic status. Just as we can see around us today.

Now, in an attempt to give black people a boost up, we have Affirmative Action laws, racial hiring quotas, and Equal Opportunity requirements that burden employers with less qualified workers than they might otherwise have. And, despite all of these remedial efforts, blacks still lag behind. What the Marxists tell you about "systemic racism" is false, but it is even falser than you might think. It is blacks who have the privileges from the system. But their inborn racial disadvantages with respect to living in a technically advanced culture are so large that these privileges haven't been enough to compensate.

The Chinese Communist Party and the US military-industrial complex are BOTH guilty for coronavirus

It now seems that the Chinese government was partly correct to blame the novel coronavirus on the United States' military-industrial complex. The earliest work on an artificially altered coronavirus took place in US academic and research institutions, such as Harvard, Emory, and Chapel Hill. That research was off-shored to China when officials in the NIH began asking questions about possible breeches of the Biological Weapons Convention.

The NIH was not initially insistent that the coronavirus research by US institutions be curtailed. Their first opposition to it appears to be a "cover-your-ass" kind of objection, made for show, but failing to threaten consequences for failure to comply with a pause order. The NIH letter to those institutions said that a pause, while desired, would be voluntary on their part. Later, though, the heat would ramp up, and that's when the researched was outsourced to China.

Of course, the Chinese government was happy to take the contract for the continuing development of a genetically modified coronavirus. The US government subsidized the research that took place at the Wuhan laboratory, though the Chinese did the work and reaped at least as much potential military benefit as their US sponsor did.

The primary motivation for a pandemic grade of coronavirus was probably economic, rather than military. The incentive was enabling the present elites in both China and the United States to milk their respective populations of money (via medical fees relating to a vaccine), and to justify governmental actions that facilitated increased control of the populations by the elites.

The release of a novel coronavirus to the world was, I think, an accident. The nCov19 virus isn't much use as a biological weapon. Apparently, the Chinese had chosen to raise the virus' infectiousness first, leaving the fine-tuning of its host targeting and the severity of its pathology for later. The released virus was thus in some intermediate form and was incomplete and inadequate as a bioweapon.

Divine Heritage, Chapter 3, Sections 1 through 5

Chapter 3

Brookstone School
Columbus, Georgia
Late 2044

I was jogging to the college campus the next day when Vanessa Emory's white Mercedes caught up with me, horn beeping. There wasn't much traffic. I stopped. So did she, leaning over to roll down the passenger side window.

"Get in here," she told me.

I did. In a few seconds we were both heading down the street toward Brookstone College.

"I heard last night's newscast," said Ms. Emory. "We don't want a repeat of what happened Monday."

"I'm not especially worried about gang members," I said.

Ms. Emory nodded. Which was strange because I knew why I could be confident, but how could she?

"Your safety is a concern, nonetheless. However, that is only part of it." She turned right at the intersection, just past where I'd been attacked. "Another consideration is that Brookstone might become liable for any injuries you inflict on the upstanding gentlemen of the Krack gang during any future repetition of those curbside negotiations that you had with them yesterday."

I recalled that one of Brookstone's deans was my legal guardian at the moment. I nodded.

"I expect you've already figured out the rest," she said.

I thought that I had.

"I might not avoid legal consequences next time," I said.

"That's part of it. The police granted you favor because you're a preteen who has never been in a fight before, whereas those punks you beat up have lengthy criminal records for assault, robbery, drug dealing, and violations of the gun laws. But if you get into more fights, the police will notice that one name keeps popping up regularly in police reports. Yours. And then you might be presumed to be at fault, even if you are never the one to instigate violence."

I knew about the fallacy. Whether they are police officers, judges, or administrators, the majority of people in authority have difficulty distinguishing between the cause of problems and the focus of problems. That confusion is what enables much of that destructive phenomenon known as "office politics."

It happens among students in grade school, too. If several kids don't like a certain other kid, they each will contrive to have a problem with him, and report it to the teachers or to the principal. The school officials don't know that the complaints are orchestrated by conspiracy, and they incorrectly assume that they just have this one problem kid to deal with. And, most of the time, the conspiracy achieves its purpose. But Ms. Emory had hinted that there was more.

"What did I miss?"

"You should have guessed. Owing to the stupidity of the media for mentioning your name on television, the Kracks know who you are and, approximately, where you live. And they have a history of vendetta."

I knew I should have killed them. I could have. No one would have suspected me of doing the deed. And because I did not, the girls in my dorm were in danger. As if she were reading my mind, Ms. Emory spoke.

"Not even you can be in two places at once. We will speak again after your classes are finished."


Dr. Roper had set a fast pace, as Ms. Emory predicted he would. Already he was treating higher-order derivatives and their meaning during the first half of class, and presenting different methods for integration during the latter half. He'd given homework, some of which tried to confuse his students about which order of derivative to set equal to zero in order to find a local extreme of the next lower order. Other problems involved integration, which would have been devilishly convoluted for someone without experience in knowing when to use a trigonometric substitution, when to integrate by parts, and when to have a peek in The CRC Handbook of Standard Mathematical Tables and then reverse-engineer the logic behind an integral identity.

So far, my experience had enabled me to surf the class without having to exert myself much. I'd earned the gratitude of a few students one day by dropping by a study hall frequented by math and science majors, and correcting a few of my fellow college freshmen who had neglected to transform the differential dx to its new space, f(u) du, after making a substitution.

Yes, Brookstone College considered me a freshman, even though Brookstone GS called me a sixth-grader.

Dr. Roper had assigned a homework problem in which we were to find the analytic solution to an indefinite integral. The integral looked difficult, but it was not. You started with a trigonometric substitution, x equals the tangent of u, and you worked out the trigonometry until you obtained the transformed integral in its simplest form. Then you used integration by parts, grouped terms, applied a couple of trig relations, and did some factoring. But in the study room, when my older classmates asked me for help, I went to the blackboard and wrote:

∫ [ (7x Arctan x) / (1+x²)² ] dx

(A miracle occurs here.)

= (7/4) [ x + (x²−1) Arctan x ] / (x²+1) + K

The amusement that greeted my abbreviated demonstration was loud enough to bring Ms. van Peenen, a math teacher of Dutch extraction who treasured peace and quiet, out of her office to tell us students to hush. Then I had to get to my next class. I heard voices tapering in decrescendo behind me.

"How'd she do it?"

"Solved it in her head on the way down the hall."


"I know. I can barely chew gum and walk."

Physics 101 wasn't nearly as challenging. It was almost like high school physics, with a little calculus thrown in. Our hardest homework problem so far had been to derive the formula by which one would calculate the horizontal range of a projectile on a flat, airless world having a gravity field that did not vary with altitude, as a function of its initial velocity. I turned in the ridiculously easy homework assignments and tried to hide my boredom. When would the really good stuff begin?

I'd kept thinking about Vanessa Emory's words "not even you," as if she were Lois Lane reminding Superman that even he had his limitations, through my calculus and physics classes. Since first meeting her on the bus in Atlanta, Ms. Emory had shown an interest in me that had been very unusual for such a highly placed executive, not to mention someone whose aging father owned most of the school I attended. At first, I'd thought that she was shepherding me because my famous IQ-test results made me a feather in Brookstone School's cap, but lately I'd begun thinking that her interest was even greater than that could account for.


After class, I was coming down the stairs to the exit of the building that was nearest the street that led back to the grade school campus. Ms. Emory was waiting near the exit, and apparently had been waiting the whole time I was in class. Thinking about that gave me the creeps. She was nice, and she had been very helpful to me, but just what the hell was going on here? A vice president of Brookstone School wouldn't wait two hours in the corridor of a class building just to play chauffeur to a student, no matter how promising the student might be.

"Are you all done?"

"With class? Yes. Where are we going now?"

Vanessa Emory smiled. She knew I'd guessed that she had something in mind.

"To my house," she said. "There are some things I want to tell you in private. I've called Dean Klang and told him that you're with me, and that you'll be late getting back to your dorm."

So we got into her car, and she drove to a large, impressive house a short distance outside Columbus in Muscogee County. We went inside and, first thing, we had a snack. Then we went into her living room and sat down.

"This is one of my father's homes, though he isn't staying here at the moment," she said. "Do you like it?"

"It's nice," I admitted, looking around at the drapes, the furniture, the carpet, the lighting fixtures. The polished and carved mahogany paneling that faced the walls. The jade statuettes. The antique clock that showed the correct time. "Expensive looking."

"Quite expensive. My father's taste in home furnishings runs to the high end of quality. Now let me get to the reason I asked you to come here."

Vanessa Emory held out her hand. The indoor lights went out. The heavy drapes were drawn against the sun, so, lacking indoor lighting, it had become somewhat dark in the living room. How had she turned the lights off?

"No, I don't have a remote control device," she said. "I turned off the lights by willing the circuit open. It's a divine power that I have, similar to your power to alter the rate at which you experience time."

That startled me. No one else had ever guessed what I could do with time.

"Now watch this," she said, as she made a ball of light appear above her palm. It grew in brightness until the room was as well illuminated as it had been by the electric lights. "About twenty thousand years ago, the gods and goddesses about whom the Greeks would, much later, tell in their legends, actually lived. They built a civilization of which there remain only a few traces, now mostly buried in parts of Europe. If they'd endured, they might have explored space and colonized the moons and planets of our solar system."

Vanessa Emory smiled sadly.

"But they did not endure."

"Why not?" I asked.

"The ancient divines made the mistake that all of the higher races since have made. They married, or informally consorted with, lesser men and women. The Greek legends recollect this failing in the stories about Zeus, or Jove, in which he frequently took mortal lovers and had children by them. In truth, it wasn't just one god who did that. Nearly all of them started doing it. For some reason, race-mixing became popular among the divines of long ago. And after only a few centuries, a half-breed race of demigods arose, and the race of pure gods died out."

"Tragic," I said.

"Yes, it was. Of course, if that hadn't happened, we wouldn't be here. I am a demi-goddess. And so are you."

I considered that. It certainly explained the facts as I knew them.

"Why us?" I asked. "Why aren't powers like ours more common among, um, white people?"

"Ah. You've guessed more than I thought," said Ms. Emory. "Yes, the white race is a degenerate form of the race of demigods. Mortals had outnumbered the gods by a very large ratio. Perhaps by a thousand to one. And the first generation of demigods continued the race-mixing ways of their fully divine parents. So the god-genes became ever more dilute as the generations continued to roll by. Eventually, the only special benefit the white race had from the divine side of their family tree was a slightly higher average intelligence than other humanoid races."

"The Asians have a higher average IQ than than whites do," I pointed out.

Vanessa Emory dismissed her ball of light and turned the electric lights back on with another wave of her hand.

"The Asians," she said, "got their advantages from racial admixture with white people." She sat on a soft chair that faced where I was sitting on her sofa. "Do you know what population pressure is?"

I nodded. "It's when there are too many people living in a territory that can't grow enough food for everyone to eat."

"Food or some other necessary resource," said Ms. Emory. "But usually it is food. Well then. About fifteen thousand years ago, after the gods were gone and the demigods had grown few, white tribal groups began wandering from their original homelands in Europe and in northern Asia. They went in all directions. Those who went into central Asia met a new humanoid race, which we refer to as the Yoyoi. The Yoyoi were the original Asian race, a primitive race having an average intelligence inferior to that of the invading whites. And here is where the white race repeated the error that caused the extinction of the ancient gods."

"Whites married, or informally consorted with, the Yoyoi and made a new hybrid race," I guessed.

"Exactly so," said Ms. Emory. "And that new hybrid race, over the course of time, became the modern race that, today, we call 'Asians.' To the extent that they have beauty and mental ability, they got it from our race. It certainly was never present in the original Yoyois."

"But their average IQ is higher than the average IQ for white people," I said. "If dilution lessened our godlike attributes, then surely the further dilution with the Yoyoi would have lessened them the more."

"Before the dilution had spread far, the wisest of the Yoyoi-Aryan hybrids became politically ascendant over the others and determined that their culture would practice a form of eugenics aimed at cultivating two traits. One of them was intelligence. The other was respect for authority. As the centuries passed, those in leadership positions weren't always the wisest or smartest Asians, and yet the Asians kept bowing to them anyway, simply because they were the civil authorities."

"The two traits sometimes get in each others' way."

"Yes. Meanwhile, in Europe, white people were far more divided, more rambunctious, more tempestuous, more prone to rebellion. The lack of discipline made a unification of white civilization late to reappear. However, it had its own eugenic effect on the race. One of the effects was a broadening of the normal distribution for white intelligence. Or, in statistician's terms, the standard deviation rose. With the passing of time, there was a reduction of the hump in the middle of the bell curve and a rise in the percentage of whites found at the extremes."

"So white people have higher percentages of both idiots and geniuses than the Asians do," I said, following the logic. "And a smaller percentage of mediocrities."

"And that's an advantage," said Vanessa Emory. "Can you tell me why?"

"Of course," I said. It was obvious. "Those who do the most challenging tasks, and advance the sum of human knowledge, are always those in the high extreme of the distribution of intelligence. When it comes to pushing the envelope, the mediocrities count no more than the retards do. So flattening the distribution and squeezing equal percentages in both directions increases the percentage of the race that can make significant scientific achievements and contributions to culture."

"Yes," said Ms. Emory. "That's quite a good summary. And that is why the white race, rather than the Asian race, produced the world's first technical civilization. The gods never tinkered with electronics or with nuclear physics because they didn't need to. Their innate abilities were much, much greater than those of their demigod offspring were."

Ms. Emory continued. "But also, in eugenic terms, there's another advantage to a larger standard deviation. It makes culling to improve the race with respect to the trait having the flattened distribution more rapidly effective."

She'd answered my question about why demigods were no longer common. But there remained the other side of the coin.

"If the god-genes became more dilute with time, then why do any demigods or demi-goddeses exist today at all?"

"How do you feel about Adolf Hitler?" she asked.

"I think he was a man who underestimated his opponent, and lost a war because of that miscalculation."

"Well, so much is true. However, the Führer had plans to improve his nation genetically. He ordered one of his senior deputies, Heinrich Himmler, to start a program of human breeding. Its purpose was to promote human biological virtues among the German people. Biological virtues in general, that is. But along the way, somewhere, Himmler discovered that there was a bit of truth to the legends of the ancient Greeks, and he began to focus his program on recovering the god-genes as a special task of the Lebensborn project. Whether Hitler himself knew about it is unclear. Himmler didn't always fully account for his doings."

"So the Nazis brought back the god-genes?"

"To some extent, they did. Himmler barely knew that he was on to something when Germany lost the war. Most people believed that the Lebensborn project ended when Germany fell to the Allies. But it continued in secret. Himmler gave the task to SS officers who escaped to Argentina. The project was very quietly expanded to include white people in Australia, then in America, in the United Kingdom, and, finally, back in Europe once again. I am of the fourth generation of the project. You are of the fifth."

"How do you know?" But I'd guessed the answer before my question was completely asked.

"The internet has made genealogical research a rather simple matter," said Ms. Emory. "I looked up your family tree. You have Lebensborn ancestors on both your mother's and your father's sides."

"Do your parents have talents like yours?"

"My father doesn't. My mother is dead. But, no, she didn't. Or, I should say, not as far as I know. Apparently, the genes that enable some manifestation of divine power only rarely occur to the necessary extent, or line up in the proper way, even among those who carry them. At the molecular level, they're just alleles. Many of them are probably recessives."

My father and my mother had been introduced to each other by their own parents, by my two sets of grandparents. They didn't just happen to hook up at school and start dating. It fit right into what Ms. Emory was telling me. It was even possible that my parents were part of the Lebensborn project and still didn't realize it.

"How many of us are there?"

"Demi-divines? Not many," she said. "I doubt that there have been as many as ten alive at any one time. Lebensborn has had only a little success in bringing back the gods, but it's a start."

"Why did you tell me this?" I asked.

"To give you an idea of the importance of your staying alive, of not taking unnecessary risks with yourself. And to encourage you, when the time comes, to have children. Many."

"Did you?"

A pained expression crossed Vanessa Emory's face.

"Five," she said. "And all of them were killed before they could grow up."


"Well," said Vanessa Emory. "That's what I think. My two daughters were killed in a hit-and-run automobile accident, and the police never found the driver. My oldest son fell to his death from an apartment building rooftop, and the police never found out who had pushed him. In fact, they said it appeared to have been an accidental fall, though I suspect that they simply wanted to close the case. Another died of food poisoning. The last, a boy about your age, was killed by blacks during a flash riot. They swarmed the streets, attacking any white person they saw. They saw him, and so he died."

"I suppose, then, that I ought to marry a demigod once I come of age. It would be the best way to concentrate the god-genes."

"It would," agreed Vanessa Emory. "But there is a problem. There are, at present, no living male demi-divines. You'll have to find the best mortal man you can, the man who is physically and mentally the most perfect, because such a man is likely to have an above-average concentration of the god-genes."

"Such a man is likely to be already claimed."

"Then you cheat," said Vanessa Emory. "You don't have to marry him. You only need to get his genes combined with yours in a baby. Once you've done that, you need trouble neither him nor his wife evermore."

She looked at the antique clock on the marble fireplace mantle.

"I must get you back to your dorm," she said with a curious smile. "Norman Klang might be wondering what I'm doing with you."

"Giving me a stern safety lecture," I suggested.


We left the house and got back into her Mercedes.


My track class was held alongside a general PE class, from which I was exempted because training for a sport, such as track, was considered to be an acceptable equivalent. The event we were training for at the moment was the 400-meter relay race. I was one of four girls who had to run 100 meters with a baton, which I was to hand off on the run to the next girl, unless I were the last girl, in which case I carried it across the finish line.

The girls in the PE class were doing their exercises in the middle of the football field. They were subdued today because yesterday the coach had scolded them for chanting, while doing their push-ups, "We must! We must! We must increase our bust! The bigger the better. The better the bigger. The boys are depending on us!" Repeat. Coach Braun thought it was immodest. I thought it was just girls will be girls.

The coach was pleased with me because I was the fastest girl on the team. I'd been careful not to let my speed-up get out of hand. Warp two was plenty speedy enough. Cheating? Of course it wasn't cheating. Cheating was something like using drugs, such as amphetamines or steroids. Being a demi-goddess who could bend time wasn't against the rules at all.

I heard Coach Braun call "On your mark." The girls crouched into their starting positions. "Get set." Looking on from the other side of the track, I saw six butts rise several inches each, as muscles tensed in twelve thighs. "Go!"

Off they went. Coming around the first curve, the half-dozen girls in the first relay caught up with those in the second, who snatched their batons and took them on around the track. Beth Griffin was the girl with the baton I was to take. I started off as she got close, let her catch me. I went on double-time as I took the baton. My team had been in last place due to a near-fumble of our baton on the first hand-off. I remedied that to the extent of tying with the leading team at the third hand-off. Our final relay girl, Tamara Cook, crossed the finish line in second place.

Oh well.

I jogged over to the finish line on normal time rate. Coach Braun kept shooting quick glances at me. He was holding his stopwatch in his right hand. Suddenly, it occurred to me that I might have run my hundred meters faster than I should have.

The relay race is a team sport, and it isn't polite to ask one's coach about one's individual performance. It would sound too much as if one were trying to take a bigger share of the credit for winning or avoid some of the blame for losing. And that isn't good sportsmanship. So I jogged on by the coach to the bleachers, while the next twenty-four girls took their places around the track for their event.

I swear that my hearing is getting better. Or, I should say, when I want my hearing to improve, it does. When Coach Fuller came out to the track, Coach Braun had a quiet word with her. I tuned in.

"Keep an eye on Brenda Jones," he said. "I timed her in the relay. Perhaps I made a mistake with the stopwatch. But if I didn't, then she ran that hundred meters in ten point two seconds."

"Not possible," said Coach Fuller. "That would be a world's record for the women's 100-meter sprint, and it would be almost a record for the men's."

"As I said, I might have made a mistake with the stopwatch. Just keep an eye on her. Even without timing her, I can see that she's easily the fastest girl we've ever had at Brookstone. And..." I looked away just in time. "I have the feeling she's holding back. As strange as that sounds."

"Well, all right. I'll observe her for you while you're at Glisson Camp. But if she's that fast, then she's from another planet."

"I've been training boys and girls for a long time," said Coach Braun, who was in his sixties. "I can usually tell when a student isn't giving an event his best effort. And while it would be utterly ridiculous for me to find fault with Brenda's excellent event times, it does seem to me that she isn't trying as hard as she can." Coach Braun gave his class over to Coach Fuller and headed off to wherever it was he had to go.


All right, then. My IQ had made me famous. And now I'd probably become doubly so, since, now that my coach had discovered some of my speed, I could hardly avoid becoming Brookstone's track star. I'd only meant to make up for the sloppiness of two of my teammates in the relay race. I hadn't intended to earn the "fastest woman in the world" title. Maybe I could convince the coach that he'd made a mistake with that darned stopwatch.

The answer turned out to be simple. I reduced the speed-up to about fifty percent above normal. That way I could still go as fast as I needed to, while looking as if I were trying harder, which I was.


I came back to the dorm room after my morning classes feeling very bad. My hips felt as if they were in a vice that was trying to pinch them until my legs fell off. I also felt vaguely nauseous, and I figured that if ever there were a day to cut my afternoon classes, this was it. As I opened the door, I saw that Ruby Pierce had a guest. LaChandra Stints.

As you've probably figured out, blacks make me uneasy. I've never had a good relationship with a black person, and I've found them generally to be a bunch of beggars, cheaters, blame-shifters, and chip-on-the-shoulder makers of many excuses for their own shortcomings. Or else violent predators. So I wasn't expecting LaChandra Stints to be quite a charming girl, as eloquent yet concise in speech as a demi-goddess.

One who wasn't starting her first menses.

Yes, that was what was happening. I hadn't been asleep when Mrs. Joiner had explained to the girls in her elementary school biology class about what to expect. But I hadn't known that I was going to be entertaining when it happened.

"Brenda," said Ruby. "This is LaChandra Stints. She's on the school debate team, and she's really smart, like you. When she found out you were my roommate, she asked if she could come to our room to meet y—"

By this time, I had my suitcase down from the shelf over the closet, had taken the Advil and one of the pads, and was heading back out the door toward the wing bathroom.

"Pleased to meet you. Be back in a minute. I have an emergency to take care of."

Okay, so it was an awkward first meeting. But what was I supposed to do?

As it turned out, I wasn't bleeding yet. But I could feel that it wouldn't be long. The pain had gotten worse. I dragged myself back from the bathroom and into the room, where I lay down on my bed and groaned.

"Should I come back later?" asked LaChandra.

"I can talk," I said. "I just don't want to move, and I'm cutting my classes for the rest of the day."

"I don't suppose you'll be setting the track on fire today, huh?" said Ruby.

"Uh-uh. Is it always this bad?" I hoped not.

"No," said LaChandra. "But the first time often is. Your body isn't accustomed to the prostaglandins, which is why you're having severe cramps."

I nodded, then wished I hadn't. I had gotten a headache, too.

When her genius IQ was discovered by the media, about two years ago, LaChandra Stints had been held up as a symbol of black equality with whites. Um, no. I take it back. The media had strongly implied that she proved that blacks were smarter than whites. When my IQ proved to be much higher still, nothing was said that implied white mental superiority. Not on television or in the press, anyway. As I thought about it, I seemed to remember that following both her burst of fame and mine, there had been an increase in televised images of blacks using computers and electronic gadgets in incidental spots of television programming and in advertisements.

Still, LaChandra was the real thing. I could tell that much as she spoke. Really intelligent people have a precision of diction that less intelligent people can't reach. It comes partly from having a larger vocabulary, and partly from being able to think ahead while speaking. That's not always a reliable guide to someone's IQ, though, as it can be faked by an actor who has rehearsed his lines. That's why most television personalities aren't really as smart as they sound. But LaChandra wasn't reading from a script.

The only class that she and I had in common was the History of the American Revolution. I'd noticed her there, but it was a large class, and I had always been rushed afterward to eat and then run to the college campus for my classes in calculus and physics. I'd never stopped to speak with her. I was raised in Atlanta, which had been a racial pressure-cooker since before I was born. Blacks prey on whites there with theft, assault, and rape, and the authorities mostly let them get away with it. Atlanta's whites, for their part, pretend (in between getting robbed, beaten, or raped) that there's nothing at all wrong with the city, except for a little of the "random" crime that happens everywhere. So you should understand that I hadn't been in a hurry to acquaint myself with LaChandra.

Why was a ninth-grader taking a sixth-grade class in history? Because she'd missed it at her old school, and she had to take it here because it was part of the core curriculum at Brookstone. Not even I had been allowed to skip any of it. Brookstone allowed me to take college science and math courses, but it wasn't going to let me out of the usual sixth-grade science and math courses, even though I obviously already know the subject matter. Rules don't always make sense. Or maybe it would be better to say that the actual purpose of a rule isn't always what its makers say it is.

LaChandra and I exchanged pleasantries while I tried to deal with my first-ever full-blown attack of menstrual cramps. She probably sensed that it wasn't a good time for a lengthy comparison of life experiences, so she left after a few more minutes. She lived in another dorm. Ruby saw her to the wing exit of Mathews Hall, then returned to the room.

"Biology never was my favorite subject," I said. "Now I know why."

Ruby made sympathetic noises. We spoke a while about LaChandra, whom Ruby seemed to consider my mental equal. I knew better, but I understood that Ruby might not see what I could. A higher mind can sort rank among lower ones much more accurately than the reverse. Although my contact with her had been brief, I could estimate that her IQ was perhaps about 140, making LaChandra the equal of Sarah Wiesman. But not mine.

Let me try to convey how rare it is for a black to have a genius IQ.

Although the distribution of intelligence among the members of a race isn't perfectly normal, the normal distribution does make a good approximation of the population distribution when only one race is present. Or, conversely, one of the ways you can tell that more than one race is in a population is if its IQ spread deviates significantly from a normal distribution, such as by being bimodal or skewed. The reason that the normal distribution makes a good approximation for the distribution of a characteristic like intelligence is related to the way genetic inheritance works. I'll save that discussion for another time.

The fraction, f, of a race having an average IQ of x̄ and a standard deviation in IQ of σ, which is above the minimum IQ of μ, is found from

f(μ) = ½ − [σ√(2π)]⁻¹ ∫(x̄,μ) exp{ −[(x−x̄)/σ]²/2 } dx

You can avoid integrating the probability density function if you have a handy error function to call.

f(μ) = 1 − ½ { 1 + erf [(μ-x̄)/(σ√2)] }

In the equation, x̄ is the average IQ for the race, and σ is its racial standard deviation in IQ. For white US-residents in the year 2040, those numbers were x̄=103 and σ=16.4. For black US-residents, x̄=85 and σ=12.4. The minimum IQ required by Brookstone School for student enrollment is 130. If you set μ=130, you find that the fractions of whites and of blacks who are eligible to attend this school are 0.0498467387 and 0.0001422428469, respectively. In other words, one white student in twenty has what it takes to get into this school, but only one black student in 7030 does.

Since there are about equal numbers of whites and blacks in this part of the United States (Georgia and Alabama), there should be about 350 white students for each black student here. In fact, the ratio is more lopsided than that, since the total number of students in grades six through twelve is about 1400, and there's only one black student in those grades who actually does meet the customary minimum IQ requirement, namely LaChandra Stints. Where are the three other blacks who should qualify? If I had to guess, I'd say they were attending a less expensive school with which their parents are more comfortable.

There are also two other black students at Brookstone who didn't meet the IQ requirement, but were allowed to enroll on a sports wavier. One of them is a player on the Brookstone high school football team, at which sport, I was told, he is quite good. The other wavier had been granted to the fastest sprinter on the boy's track team.

But, as far as I know, there aren't any white students at all who received a wavier of the minimum IQ requirement to enroll at Brookstone, no matter what their athletic merits might be, and that tells the tale as far as I'm concerned. Brookstone School had recently become infected with political correctness. Possibly the installment of a new corporate president had had something to do with that. As yet the consequential troubles were negligible. But that would not continue. It was the first leak in the levee, the trickle that might easily become a flood. Political correctness would eventually destroy Brookstone School, as it has destroyed so many other institutions, unless the entrance standards were strictly enforced, once again.

Divine Heritage, Chapter 1, Sections 1 and 2
Divine Heritage, Chapter 1, Sections 3 through 6
Divine Heritage, Chapter 1, Sections 7 through 10
Divine Heritage, Chapter 1, Sections 11 through 13
Divine Heritage, Chapter 2, Sections 1 through 3
Divine Heritage, Chapter 2, Sections 4 through 6
Divine Heritage, Chapter 2, Sections 7 through 9
Divine Heritage, Chapter 2, Sections 10 through 12
Divine Heritage, Chapter 3, Sections 1 through 5

Divine Heritage, Chapter 2, Sections 10 through 12

After returning with Ms. Emory from the college campus, I bought my sixth grade textbooks at the bookstore on the grade school campus. The total (with sales tax) came to just short of $700, which, I'm sure you'll agree, is plenty to pay for three lousy textbooks. It left me with hardly any money at all left in my account at Brookstone Bank. No doubt my father could wire more money into it, but it wouldn't do to ask. Explicitly, that is. I might not be as slick as Sarah Weisman, but I knew that the art of getting money out of one's father consists mostly of making him think that giving it to me had been his idea.

I was back in my dorm. It had been a wearing day, even with Ms. Emory's help. I was on eBay looking for my two college textbooks, but nobody was selling those particular titles just now. Ruby was watching over my shoulder as I turned the browser to Amazon. I typed the ISBN for my calculus book into the search window and watched the list of offers come up.

"Eight hundred dollars?" asked Ruby.

"It was almost twelve hundred in the college bookstore," I said, making sure that I was looking at the latest edition of the textbook. Another part of the college textbook scam involved the publishers constantly making trivial changes to the books and republishing them as a new edition, after which all of the professors would regard the previous edition as obsolete.

But I didn't even have eight hundred dollars, so I looked for used books. There was, I discovered, a paperback version of the book. While used hardcover copies were selling for around $500, the used paperback copies began at—

"Forty-nine cents." I laughed.

"Get that one!" Ruby urged.

"No," I said. "See the quality description. It's rated as 'acceptable,' which really means 'not acceptable.' Likewise 'good' means 'okay in a pinch,' and 'very good' really means 'acceptable.' I'm looking a little further down the list."

The store selling the first book was My Grandma's Goodies. Another book, selling for fifty cents, was rated at 'good,' and it was being offered by Goodwill Industries of Central Florida. Then came a listing by Belltower Books, at 'very good' condition, for ninety-nine cents. The next offer was from Alibris, a name I recognized, for a book in 'good' condition.

"I think I'll get the book offered by Belltower Books," I said. "They offer expedited shipping, too, which I'll take because I need to have the book as soon as possible."

"Still a bargain, considering the price tag on a new book at the bookstore."

I agreed. But there was another problem.

"What is my shipping address here?"

"Oh. If you don't have a mail box at the Student Union yet, then you get your mail in care of Norman Klang, Mathews Hall, Brookstone School GSC, Columbus, Georgia, three one nine oh four."

I typed that into Amazon as my shipping address, right below my name. I'd had the account with Amazon already, so it already had my credit card number. I found a similarly sweet deal on a copy of my physics textbook. Then I clicked on the check-out button and paid for my books, selecting the expedited shipping option, which cost me more than the books themselves had.

"Und now ve vait," said Ruby with a mock German accent.

I began writing an email to my parental units.

Dear Dad and Mom,

I'm writing now to give you my email address and to tell you my status. I've moved into Mathews Hall on Brookstone's grade school campus. I'm sharing room #107 with a very nice girl named Ruby Pierce. She's a year older than I am, is in seventh grade, and attended Brookstone last year. She's showing me the lay of the land, so to speak. You can send me packages in care of Norman Klang, Mathews Hall, Brookstone School GSC, Columbus GA 31904.

You targeted my funding very accurately, Dad. I paid my tuition, my housing fee, my cafeteria ticket, and I have bought all of my books. I should say, though, that the college textbooks cost rather more than they did when you were going to school, and I had to order used copies from various vendors through Amazon online. But they will arrive within a few days. It shouldn't be a problem. I bought the three required textbooks for the sixth grade courses at the GSC bookstore for $700. This quarter, those courses will be History of the American Revolution, Algebra 1, and English Composition 1.

I would have gotten out of taking the classes that I could already teach if it were permitted. But they don't let you CLEP the core curriculum here. However, they do permit me to take college courses in addition to the sixth grade ones, and I've been accepted by the college faculty for Physics 101 and for an 'honors' course in calculus that combines differential and integral calculus, and analytic geometry, into a single five-credit hour course.

I've also enrolled in some sort of track-and-field endeavor, though I'm not certain yet of the details. One of Brookstone's executives appears to have taken a personal interest in me, and she has acted on several occasions as my patron, opening doors for me that might otherwise have remained shut. I owe Ms. Vanessa Emory a great deal. I only wish that I knew why she's been such an avid champion for me.

Though expenses have left me broke, I'm in no immediate need of money.
I sent the email.

"Will that work?" asked Ruby, who had read what I wrote.

"It will work once," I said, grinning.

Classes would begin tomorrow. I had all of my sixth grade lessons in the morning, all in the same building here at GSC. Algebra (8:05-9:00), English (9:05-10:00), History (10:05-11:00) with five minutes slack between classes. I'd eat in the cafeteria from 11:30 to noon. Then I'd run from GSC to the college from 12:30 to 12:45. Two miles in fifteen minutes shouldn't be a problem for me. My calculus class began at one in the afternoon, followed by physics (2:15-3:15). At 3:30, I'd run back to GSC and join the track team out by the football field, where practice began at 4 o'clock. With all that running, my wind should be pretty darn good by the end of the quarter.

"That's a very heavy schedule," said Ruby. "You're going to run two miles, twice a day, carrying your books, and begin training for the track team the moment you finish that second two-mile run?"

Put that way, the schedule did look a bit difficult. I'd forgotten about having to carry my books.

"I'd better get a backpack," I said, and called up Amazon again.

"You'd better grow wings," said Ruby.


As I'd expected, algebra was boring, easy, and a sure "four" in my GPA basket, provided that I could stay awake long enough to take the midterm and the final exam. English composition was more iffy, since the judgment of the teacher had more play in assigning grades. I'd have to learn the teacher to some extent in order to ace that class. But I'd done the same with Mrs. Fergus at Morningside, and I'd no doubt that I could do it with Mr. Wilson at Brookstone.

It was history that presented difficulties. All those doings of the political figures of the American Revolution, the ideas of the philosophers behind the politicians, the adventures of the military leaders in front of the politicians, names, dates, quotes. Bleah. I wondered whether it would be wise for me to disagree with some of the ideas of the Founding Fathers during class, or in an essay for class.

Yes, there are notions, popular with the revolutionary luminaries, that I would dispute. One of them was put forth by Thomas Jefferson, an otherwise sensible fellow who became fond of the silly idea that the common man represented a reservoir of wisdom that would nudge the country back into its true course, if it were to stray from it. Which is nonsense. Common folk are no such resource, and their votes constitute no such restoring force. You don't get wisdom by summing mediocrities, and most people throughout all the ages have been mediocrities.

Democracy is a stupid idea for the simple reason that the wisest people are always outvoted. It really is possible for millions of people, each voting in accordance with their own interests, to drive their national vehicle off the cliff of hard reality, so that they and their country die.

Imagine that you took apart two old-fashioned pocket watches and scattered their parts across a pair of tables. To one of the tables, you invited a hundred people, randomly picked off the street, and told them to vote democratically on how to put the pieces back together again. To the other table, you invited a watch-maker. At which table would a working watch most likely be reassembled first?

However, there's a come-back argument. For a system of government other than democracy, who chooses the leader? That is, who ensures that a statesman is invited to assemble policy at the national table, and not some blowhard politician whose only talent is talking magnificently about himself?

No, not the common people. They aren't wise and are no proper judges of wisdom in others. If you leave the choice of leadership to them, they'll pick blowhard politicians almost every time. That would be true even if blowhard politicians and wise statesmen occurred among the candidates for high office in equal numbers. Of course, the real situation is even worse, since for every wise statesman who comes along, there are about a thousand blowhard politicians.

I'd say that war would determine which countries were the best ruled, with victory going to the more wisely led countries most of the time. People would sooner or later learn their lesson regarding the pursuit of power by those wannabe leaders who are ambitious but unworthy. Or, rather, the people who survived would learn that lesson.

From a divine point of view, it isn't all that important how many countries don't learn it in time, and fall as a consequence. From a cosmic perspective, it isn't important how many people are enslaved or exterminated. What matters is that natural selection would tend to preserve those countries that did learn rapidly enough, and the arrangements that those countries had made for the marriage of wisdom and power would be preserved along with them.

I could speculate about what those arrangements would be, but I would only be guessing. But that's why liberals are foolish to sneer at tradition. Traditional mores and culture are usually well-culled adaptations for the people among whom they evolved. What even the greatest minds would be hard put to contrive through planning, nature brings forth by the processes of natural selection. Including war.

For anyone interested in betting with the odds on his own survival and that of his country, I'd give this advice: if you want to be on the side that wins in the long run, you must first recognize that what decides struggles is power and the skill with which it is put to use.

On the other hand, I doubted that Mr. Ham was another Socrates, and so it probably wouldn't be wise for me to assert my opinions against those of Thomas Jefferson in Mr. Ham's history class.

On the third day of class, my books and the backpack arrived from Amazon. When I entered the dorm lobby, Donna Lane, who was acting as a receptionist for Mathews Hall, waved me over and gave them to me. That was five days ago. It was Monday again, and I was finding out that what you can do easily for one day isn't so easy when you must do it day after day after day.

After history, I headed back to my dorm room, took the sixth grade books out of my backpack, and put the college textbooks and my calculator in. Then, leaving the backpack, I walked to the cafeteria and ate whatever they were serving that didn't wiggle by itself. I returned to the dorm room and picked up my backpack, put it on, went out of the dorm through the wing exit. And started running.

I'm glad that I'd gotten a small pack that had a chest strap. Otherwise that thing would have bounced too much. It was just big enough for two textbooks, a thin notebook, a calculator, and some mechanical pencils. I ran along at about warp factor two, or twice normal speed. It felt no more strenuous than jogging, but my strides were longer, as they were for a run without the speed-up. I'd become so accustomed to running with an altered time rate that adjusting my step was now reflex, and I no longer made embarrassingly high leaps unless I wanted to.

How fast was I running? Oh, maybe about fifteen miles per hour. Fast enough to get me to the classroom before Dr. Roper closed the door, but slowly enough that anyone watching me would think me merely an excellent distance runner in good training. I'd already made this trip ten times, five times each way, running along the sidewalk. I drew looks, but not many, so I know that I must look like a normal running girl, wearing a backpack.

I was approaching an intersection where I'd have to make a right turn, when, from an alley between two buildings came several members of a black gang, obviously interested in me.

Well, I might be late to class, but I had to do my civic duty.

I ran past, dodging them, straight into that alley. The black youths came running after me in pursuit, thinking they had me now. It was a blind alley, dead-ending at the wall of a third building, with no exit except the one I'd entered by. I went to warp four and jumped over the blacks, clearing their reaching hands by about eighteen inches, and landing between them and the exit. Then I turned to fight. One of the blacks reached for me. I snapped his arm at the elbow and threw him into the wall on the left. I punched the second on his flat nose and saw blood fly out of his broad nostrils. I kicked the third in the groin so hard that he was punted though the air. I left the fourth with a dislocated jaw and the fifth with some broken ribs.

Job done, I adjusted my backpack's straps, left the alley, and resumed my run. The delay didn't even make me late for class. I got past the classroom door with several minutes to spare.

In the first day of class, Dr. Roper had gone through the theory behind derivatives, or "how much one thing changes when you change something else." And he explained that the derivative of a function is the slope of the line which is tangent to the function. The next day, he taught us about Riemann sums and gave out homework assignments. The day after that, he spoke of limits in general and limits of Riemann sums in particular, followed by another homework assignment. The fourth day, we were introduced to the geometrical idea of integration. You know, those tall, skinny rectangles that fit between the independent variable's axis and the curve of the function?

Yesterday, we got into the rules for differentiating and integrating polynomials. In class, I'd said, "So, they're each others' inverse operations," as if I were catching on. Ha! Dr. Roper was impressed by what appeared to be the quickness of my deduction. Okay, I schmoozed for brownie points, and I got them. But I did discover the inverse relationship between differentiation and integration for myself. I'd just done it a year earlier, and nobody saw me do it then.

So far, my calculus class had not yet caught up with what I'd known about calculus last April, when I worked out part of Mrs. John's homework problem at Morningside.

Physics 101, with Dr. Linder, wasn't even that hard. The only difference between college physics and high school physics is that in college the textbook doesn't pre-digest the differential equations for you. However, they are all easy differential equations that have variables separable and are easily integrated to provide the functions you'd see in the high school textbooks. I knew that more difficult math was ahead, but it didn't look as if I'd get to the heavy stuff in the 100 series of physics courses.

Still, credit is credit. I wished that getting that credit left me with more time to explore the college library and dig for stuff that I didn't already know.


"In other news tonight, the police investigation into the beating of some youths has revealed that their assailant was an 11-year-old Brookstone student named Brendalyn Jones. Yes, apparently a little girl beat five members of the Krack gang so badly that all of them were sent to Columbus Memorial Hospital for serious to critical injuries. Police say that the girl, who came forward the moment her classes at Brookstone's college campus were over, testified that the youths attempted to ambush her as she was going to class."

"Mark," said the news anchorman. "Why is an 11-year-old girl going to the Brookstone College campus? She seems a little young to be enrolled there."

"Peter," said the reporter. "She is young. But Brendalyn Jones is the girl who made headlines across America a few weeks ago after tests revealed that her IQ is somewhere above 200. She has been given permission to take college level classes, and she must run each day from Brookstone GSC to Brookstone College because she's too young to drive a car."

"And how is it that a little girl beat up so many gang members?"

"The police aren't too sure of that," said the reporter. "Apparently, Miss Jones has had martial arts training of some kind."

"Will the youths recover from their injuries?"

"Several of them will be in the hospital for a while. One of them has three broken ribs. Another has had to have his lower jaw put back into place. A third has a burst left testicle and a bruised right—"

"My word, Mark, that is one tough little girl."

"Peter, she told the police that it was their good fortune that she hadn't killed any of them. She said that she had no choice other than to use nearly lethal force because the odds were five-to-one against her."

"Well," said the anchorman. "I think we can all understand that. Turning to events at the state capitol..."

The girls in the dorm lobby were clustered around the TV. Watching the local news was a habit with us girls because it told us where the gang activity in Columbus was the thickest, so we could avoid those areas. But never had the lobby been so quiet, with attention so fixed on the six o'clock news as it was on that Tuesday evening. I was sitting on the broad central rise of a furry, brown, all-the-way-around couch which half filled the dorm lobby, my legs crossed, reading about the ride of Paul Revere (and the similar rides of William Dawes and Samuel Prescott) from my history book on the rise before me, when the eyes of thirty other girls turned my way.

It was so comical that I couldn't help laughing. What was even funnier is that most of them started asking whether I was all right, even though there I was, sitting on the couch rise, reading a book.

"Nobody better mess with my roommate," said Ruby, acting fierce. "Or they'll have to deal with me!"

Giggles and guffaws broke out among the girls at that. Well, at least the shock-and-awe mood that the newscast had created was broken. The other girls congratulated me on my victory and related horror stories about girls who had been raped or beaten by gang members. Which meant black gang members, but they were careful not to name the race of the perpetrators. It wasn't the first time I'd noticed that people were unwilling to discuss the socially significant differences between the races in a candid manner. Although it seemed such a simple thing to do, nearly everyone had been strongly conditioned to avoid it.

Although I wasn't familiar with the techniques of military brainwashing, I didn't think that psychological conditioning of any sort could be stronger than that which had instilled within so many people a reluctance to talk honestly about racial differences.

An eleventh grade girl named Patricia Greenwood excused herself, saying that she had to go study in a less noisy place. I suspected that she wanted to be the one to bear the first gossip about my fight downtown because, on her way out of the lobby exit to the east wing, she told me "It's nice to see the good people win for a change!" Perhaps she meant the white people. It had been a while since we were winners, hadn't it?

About a hundred years.

Survival is the greatest school, and Death is its best teacher. But no living thing graduates, ever. The beneficiaries of nature's lessons aren't individuals, but races, which endure so long as they pass the tests and which prosper by how high they score. The white race had gotten itself into trouble partly by being too generous, and, in its generosity, making itself vulnerable. Other races had been quick to take advantage of that vulnerability. They infiltrated white countries, seeking out key positions of control, of supervision, of decision-making, and of power, which, once they had them, they used them to benefit their own people at the expense of white people. Non-whites of every stripe had become favored above whites, and, being favored, they received rewards even when there were white people who deserved them more.

My fight could, conceivably, have landed me in jail. Or it could have imposed on my parents a legal obligation to pay fines and the hospital expenses of those gang members. The reason that didn't happen was that I'm only eleven years old—a little kid—and a girl, and have no prior record of mischief of any sort. All of those things added up to a degree of favor that outweighed the favor that those gang members had just for being black, once their prior records for trouble-making had been subtracted. Or, rather, my favor exceeded my opponents' favor this time. If I had to fight again, especially against blacks, that earlier fight would weigh against me, even if I were as justified next time as I'd been before.

To be sure, I could have outrun those blacks. I hadn't admitted that to the police because, in their opinion, it would have put blameworthiness upon me. Why didn't I just outrun them? Because the next little girl who happened to walk down that street, past that alley, would not have had my advantages. She'd have been robbed, beaten, raped, and probably murdered by those black youths. It was morally necessary that I deprive those gang members of their ability to harm someone who actually was an ordinary child, someone like whom I only appeared to be.

And there was one other reason as well. There is no idea more obscene than that decent people should be expected to give ground or right-of-way to vile predators. Good should roar so that evil trembles, not the other way around.

Those five black youths would recover. But would they reform? Had I taught them a lesson that would change their predatory behavior? No. They'd return to their previous lives, maybe a little more cautious than they were before. But sooner or later they would attack another innocent victim.

The conviction was growing in me that I'd made a mistake by not killing them. It was a mistake that I'd made before, in my old neighborhood in Druid Hills, when I had defeated four teenage blacks who had attacked me. By letting them live, I'd made the violent deaths of some number of other people probable. Though it hadn't occurred to me at the time, I'd chosen between the lives of those gang members and the lives of whomever it was they would someday murder, and I'd chosen wrongly.

And now that my fighting skill was known, I wouldn't be excluded from suspicion if gang members began turning up dead. I could no longer afford to do what was right. My moral weakness, which had made me reluctant to kill when killing was proper, had cost me that much.

Divine Heritage, Chapter 1, Sections 1 and 2
Divine Heritage, Chapter 1, Sections 3 through 6
Divine Heritage, Chapter 1, Sections 7 through 10
Divine Heritage, Chapter 1, Sections 11 through 13
Divine Heritage, Chapter 2, Sections 1 through 3
Divine Heritage, Chapter 2, Sections 4 through 6
Divine Heritage, Chapter 2, Sections 7 through 9
Divine Heritage, Chapter 2, Sections 10 through 12
Divine Heritage, Chapter 3, Sections 1 through 5