Rongzhou: /* Classic form: planetary/satellite orbit (point mass approximation) */

2026-09-24T15:04:13Z

Classic form: planetary/satellite orbit (point mass approximation)

← Older revision		Revision as of 15:04, 24 September 2026
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	Where $\mu=G(M+m)$.		Where $\mu=G(M+m)$.

			The orbit described by the equation is usually elliptical:

			[[File:Two-body-problem.ggb]]

	=== Derivation ===		=== Derivation ===

Rongzhou: Created page with "In astronomy, two body problems normally refer to the study of the gravitational interaction of two celestial objects. == Classic form: planetary/satellite orbit (point mass approximation) == Approximate a star/planet as point mass $M$, and a planet/satellite orbiting around it as point mass $m$. The orbit of the planet in the reference frame of the star satisfies the equation: {{#content:Q1785}} Where $\mu=G(M+m)$. === Derivation === Let $$ \mathbf{r}=\math..."

2026-09-24T14:41:17Z

Created page with "In astronomy, two body problems normally refer to the study of the gravitational interaction of two celestial objects. == Classic form: planetary/satellite orbit (point mass approximation) == Approximate a star/planet as point mass $M$, and a planet/satellite orbiting around it as point mass $m$. The orbit of the planet in the reference frame of the star satisfies the equation: {{#content:Q1785}} Where $\mu=G(M+m)$. === Derivation === Let $$ \mathbf{r}=\math..."

New page

In [[astronomy]], two body problems normally refer to the study of the gravitational interaction of two celestial objects.

== Classic form: planetary/satellite orbit (point mass approximation) ==

Approximate a star/planet as point mass $M$, and a planet/satellite orbiting around it as point mass $m$.

The orbit of the planet in the reference frame of the star satisfies the equation:

{{#content:Q1785}}

Where $\mu=G(M+m)$.

=== Derivation ===

Let

$$
\mathbf{r}=\mathbf{r}_m-\mathbf{r}_M
$$

be the vector from $M$ to $m$.

Let $r=|\mathbf{r}|$.

Gravity pulls $m$ toward $M$:

$$
m\ddot{\mathbf{r}}_m=-GMm\frac{\mathbf{r}}{r^3} \text{ (1)}
$$

Gravity pulls $M$ toward $m$:

$$
M\ddot{\mathbf{r}}_M=+GMm\frac{\mathbf{r}}{r^3} \text{ (2)}
$$

Therefore:

$$
\begin{cases}
\ddot{\mathbf{r}}_m=-GM\frac{\mathbf{r}}{r^3} \text{ (1)/m} \\
\ddot{\mathbf{r}}_M=+Gm\frac{\mathbf{r}}{r^3} \text{ (2)/M}
\end{cases}
$$

The relative acceleration of $m$ according to $M$, $\ddot{\mathbf{r}}$, is

$$
\ddot{\mathbf{r}}
=
\ddot{\mathbf{r}}_m-\ddot{\mathbf{r}}_M
$$

Substitute:

$$
\ddot{\mathbf{r}}
=
-GM\frac{\mathbf{r}}{r^3}
-
Gm\frac{\mathbf{r}}{r^3}
$$

$$
\boxed{
\ddot{\mathbf{r}}
=
-G(M+m)\frac{\mathbf{r}}{r^3}
}
$$

Usually, we note

$$
\boxed{
\mu=G(M+m)
}
$$

Two body problem - Revision history

Rongzhou: /* Classic form: planetary/satellite orbit (point mass approximation) */