Orbital perturbations — J2 effect (oblateness), derivation of nodal precession: Foundations

This page assumes you know nothing. Every letter that appears in the parent derivation is unpacked here, in an order where each idea leans only on the ones before it. If a symbol confuses you later, it was defined here first.

1. The shape of an orbit: the ellipse itself

Before any bulge, a satellite around a round Earth traces an ellipse — a stretched circle. This is the Keplerian picture: one focus of the ellipse sits at Earth's center.

Figure — Orbital perturbations — J2 effect (oblateness), derivation of nodal precession

Why the topic needs them: the precession rate turns out to depend on both — a bigger $a$ sits farther from the bulge and feels it less; a larger $e$ swings the satellite deep near Earth where the bulge is strong.

2. Where the satellite is right now: the true anomaly $ν$

The ellipse is a track; we still need to say where on the track the satellite is.

Why the topic needs it: the bulge's pull depends on $r$ (closer = stronger, in fact as $1/ r^{3}$ for this effect). This formula is precisely what lets us later replace $r$ by $ν$ and take the orbit average — you cannot average $⟨ 1/ r^{3} ⟩$ without knowing $r (ν)$ .

3. Orienting the ellipse in 3D: $i$ , $Ω$ , $ω$

An ellipse in flat 2D is not enough — orbits live in 3D space, tilted and swivelled. Three angles pin the plane down. These are the stars of the whole topic.

Definition The three orientation angles

$i$ = inclination — the tilt of the orbit plane relative to Earth's equator. $i = 0°$ hugs the equator; $i = 90°$ goes over the poles; $i$ up to $180°$ is allowed. In the figure it is the wedge between the grey equatorial disk and the orange orbit disk.
$Ω$ = right ascension of the ascending node (Greek "Omega") — the swivel angle of the orbit plane around the polar axis. It says which way the tilted plane is turned. The "ascending node" is the point where the satellite crosses the equator going north; $Ω$ measures the angle to that crossing from a fixed reference direction in space, the vernal equinox (the direction from Earth to the Sun at the March equinox, marked $γ$ ).
$ω$ = argument of perigee (small "omega") — the rotation of the ellipse within its own plane: how far around from the node the closest point sits.

Why the topic needs them: the bulge's averaged effect leaves $a, e, i$ frozen but slowly changes $Ω$ and $ω$ . So the rate of change of $Ω$ (written $\dot{Ω}$ — see §6 for the dot) is the answer we chase.

4. How high above the equator? Geocentric latitude $ϕ$

The bulge only acts when the satellite is off the equatorial plane, so we need a symbol for "how far off."

Why the topic needs it: the extra bulge potential is written in terms of $ϕ$ ; this formula is the bridge that turns "latitude" into "where you are on the orbit," so the average can be taken.

5. Gravity of a lumpy Earth: $μ$ , $R_{E}$ , $J_{2}$ , and $P_{2}$

A perfect sphere pulls exactly toward its center. A bulging Earth does not. We need the vocabulary of that difference.

Why the topic needs them: $μ$ sets the timescale, $R_{E}$ the yardstick, $J_{2}$ the strength, and $P_{2} (sin ϕ)$ the shape of the disturbing pull. Together they build the "disturbing function" (§7) the derivation feeds on.

6. Two rate-symbols: mean motion $n$ and the dot

Formula Why the two averages come out as they do

First average — $⟨ sin^{2} u ⟩$ . Over one full loop $u$ runs through $360°$ . The function $sin^{2} u$ spends exactly as much "time" above $\frac{1}{2}$ as below it, so its average is the flat middle value: $⟨ sin^{2} u ⟩ = \frac{1}{2 π} \int_{0}^{2 π} sin^{2} u d u = \frac{1}{2} .$ The picture: a wave centred on $\frac{1}{2}$ — chop off the humps, fill the dips, you are left with $\frac{1}{2}$ .

Second average — $⟨ 1/ r^{3} ⟩$ . Here we must weight by time, and the satellite moves slower far out (near apogee) than close in. Using the orbit equation $r = p / (1 + e cos ν)$ from §2 together with the equal-areas law $r^{2} d ν = μ p d t$ , the time-average integral works out to $⟨ \frac{1}{r ^{3}} ⟩ = \frac{1}{a ^{3} ( 1 - e ^{2} ) ^{3/2}} .$ For a circle ( $e = 0$ ) this is just $1/ a^{3}$ , as it must be since $r = a$ always. The $(1 - e^{2})^{3/2}$ in the denominator is the eccentricity boost: an eccentric orbit dips deep near perigee and feels the bulge more strongly.

Why the topic needs them: the raw torque changes moment to moment, but the secular (steadily-building) drift is what matters over months. The bracket is the tool that strips away the wobble and keeps the drift.

a

measure, in one word?

The orbit's size (half the long axis).

What does

e

measure?

The orbit's shape — how stretched the ellipse is (

0

= circle).

Write

p

in terms of

a

and

e

p = a (1 - e^{2})

What angle is

ν

The true anomaly — position around the orbit from perigee, measured at Earth's center.

Write the orbit equation for

r

r = p / (1 + e cos ν)

What does

r

stand for?

The satellite's current distance from Earth's center.

What does inclination

i

describe, and what range?

The tilt of the orbit plane vs the equator, from

0°

180°

Prograde vs retrograde — what is the sign of

cos i

Prograde (

i < 90°

cos i > 0

; retrograde (

i > 90°

cos i < 0

What does

Ω

describe, and from what reference?

The swivel of the orbit plane about the polar axis, measured from the vernal equinox

γ

What is

ω

The argument of perigee — rotation of the ellipse within its own plane.

Give the formula for latitude

ϕ

sin ϕ = sin i sin u

, with

u = ω + ν

What is

μ

and its units?

μ = G M_{E}

, the gravity strength, in

m^{3} / s^{2}

What does

J_{2}

physically measure?

Earth's oblateness — how much it bulges at the equator.

Write

P_{2} (sin ϕ)

P_{2} (sin ϕ) = \frac{1}{2} (3 sin^{2} ϕ - 1)

What does

n

equal?

n = μ / a^{3}

, the mean motion.

What does an overdot mean?

Rate of change per second (e.g.

\dot{Ω}

= precession rate).

What does

⟨ sin^{2} u ⟩

average to, and why?

\frac{1}{2}

— the wave is centred on

\frac{1}{2}

What is

⟨ 1/ r^{3} ⟩

1/ [a^{3} (1 - e^{2})^{3/2}]

What is the disturbing function

R

The extra bulge potential per mass on top of point-mass gravity.

Why does a central force keep the plane fixed?

It exerts zero torque, so

L

(and the plane) is conserved.

Foundations — Orbital perturbations — J2 effect (oblateness), derivation of nodal precession

1. The shape of an orbit: the ellipse itself

2. Where the satellite is right now: the true anomaly $ν$

3. Orienting the ellipse in 3D: $i$ , $Ω$ , $ω$

4. How high above the equator? Geocentric latitude $ϕ$

5. Gravity of a lumpy Earth: $μ$ , $R_{E}$ , $J_{2}$ , and $P_{2}$

6. Two rate-symbols: mean motion $n$ and the dot

7. The engine: disturbing function $R$ , torque, and Lagrange's equations

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