Third-body perturbations

WHAT is a third-body perturbation?

WHY "difference"? Newton's law gives absolute accelerations in an inertial frame. But we track the satellite relative to Earth. Earth is free-falling toward the Moon just like the satellite is. In a free-falling frame, the common part of the acceleration cancels — only the gradient (the variation of the Moon's field across the Earth–satellite gap) survives. This is exactly why astronauts feel weightless in orbit: they and their capsule fall together.

HOW to derive it from first principles

Let (all vectors from an inertial origin):

• $\vec r_s$ = satellite position, $\vec r_E$ = Earth position, $\vec r_3$ = third body position.
• $\mu = GM_E$, and the third body has mass $m_3$ with $\mu_3 = Gm_3$.

Step 1 — Absolute accelerations (Newton).

Satellite feels Earth + third body: $\ddot{\vec r}_s = -\mu\,\frac{\vec r_s-\vec r_E}{|\vec r_s-\vec r_E|^3} + \mu_3\,\frac{\vec r_3-\vec r_s}{|\vec r_3-\vec r_s|^3}$

Why this step? Each mass produces $-\,GM\,\hat r / r^2$ pointing toward it; write both pulls on the satellite.

Earth feels the third body (we care about Earth's own motion): $\ddot{\vec r}_E = \mu_3\,\frac{\vec r_3-\vec r_E}{|\vec r_3-\vec r_E|^3}$

Why? The Moon tugs the whole Earth, accelerating our observation frame. We must subtract this.

Step 2 — Go to Earth-centered relative coordinates.

Define $\vec r = \vec r_s - \vec r_E$ (satellite w.r.t. Earth) and $\vec d = \vec r_3 - \vec r_E$ (third body w.r.t. Earth). Subtract: $\ddot{\vec r} = \ddot{\vec r}_s - \ddot{\vec r}_E = -\mu\,\frac{\vec r}{r^3} + \underbrace{\mu_3\!\left(\frac{\vec r_3-\vec r_s}{|\vec r_3-\vec r_s|^3} - \frac{\vec r_3-\vec r_E}{|\vec r_3-\vec r_E|^3}\right)}_{\text{perturbation }\vec a_{3b}}$

Why this step? The first term is the familiar Keplerian pull. The bracket is the third-body perturbing acceleration — literally pull-on-satellite minus pull-on-Earth.

Step 3 — Clean form. With $\vec s = \vec r_3 - \vec r_s = \vec d - \vec r$:

The tidal approximation (the useful 80/20 core)

Almost always $r \ll d$ (satellite–Earth distance ≪ Earth–Moon/Sun distance). Expand to first order in $r/d$.

Step 1 — Expand $|\vec d - \vec r|^{-3}$. Write $|\vec d-\vec r|^2 = d^2 - 2\vec d\cdot\vec r + r^2$. Let $\hat d = \vec d/d$: $|\vec d-\vec r|^{-3} = d^{-3}\left(1 - 2\frac{\vec d\cdot\vec r}{d^2} + \frac{r^2}{d^2}\right)^{-3/2}\approx d^{-3}\left(1 + 3\frac{\vec d\cdot\vec r}{d^2}\right)$

Why? Binomial $(1+x)^{-3/2}\approx 1-\tfrac32 x$; keep terms linear in $r/d$.

Step 2 — Substitute and drop $O((r/d)^2)$: $\vec a_{3b} \approx \mu_3\left[\frac{(\vec d-\vec r)}{d^3}\Big(1+3\frac{\vec d\cdot\vec r}{d^2}\Big) - \frac{\vec d}{d^3}\right]\approx \frac{\mu_3}{d^3}\Big(3(\hat d\cdot\vec r)\,\hat d - \vec r\Big)$

Key scaling insight: the disturbance $\propto \mu_3/d^3$, not $\mu_3/d^2$. So the Sun (huge $\mu_3$ but far) and the Moon (small $\mu_3$ but near) compete. Plug numbers: the Moon's tidal effect on Earth-orbit satellites is about twice the Sun's — despite the Sun being far more massive.

Worked examples

Common mistakes (steel-manned)

Flashcards

Connections

• Two-body problem — the unperturbed baseline we subtract from.
• Tidal forces — same $3(\hat d\cdot\vec r)\hat d-\vec r$ gradient physics.
• J2 perturbation and oblateness — the other main perturbation, dominant at low altitude.
• Gauss and Lagrange planetary equations — how $\vec a_{3b}$ turns into slow drifts in $i,\Omega,e$.
• GEO stationkeeping — where luni-solar N–S drift forces regular burns.
• Restricted three-body problem — full treatment when $r/d$ is not small.

Concept Map

Hinglish (regional understanding)