# The Orbital Plane

## Contents

# 4 The Orbital Plane@

- Info
**A Change of Perspective:**The solution of the*general*2-body problem is developed several different ways utilizing the now known special properties admitted by the system.- Description
Switch into the orbital frame and rewrite the constants of motion in polar coordinates.

## 4.1 Reorienting the coordinate system@

Knowing that the motion **must** be 2-dimensional, we can choose a more
convenient reference frame using a coordinate rotation. Let this rotation
be denoted \(\mathbf{R} \in \mathrm{SO}(3)\) such that

Although they look different, \((x, y, z)\) and \((X, Y, Z)\) both represent the **same** object.

It’s very convenient to utilize the conservation of angular momentum and
the Laplace vector to define the precise action of
\(\mathbf{R}\) on the two coordinate systems.
**We choose to orient the** \(z\) **axis in the direction of the angular momentum**
**and the** \(x\) **axis in the direction of the Laplace vector.**

This rotation is **not** time-dependent — that is, \(\mathbf{R}\) is
a constant matrix representing a rotation between the two coordinate systems.

The standard form of the equations of motion is consequently *unchanged*.

## 4.2 Switching into polar coordinates@

By choosing this new coordinate system, we have reduced the problem to determining just 2 unknown quantities — the parameterizations of polar coordinates,

where \(r = |\mathbf{r}|\) as usual and \(\theta\) is the angle made between the reduced mass particle and the \(x\) axis.

Important

The relative motion of the two bodies (\(\mathbf{r}\))
*in the orbital plane* is fully determined by the set of polar coordinates
\((r, \theta)\). **The 2-body problem has been reduced from determining all 6 variables to only 2.**

### 4.2.1 Developing new equations of motion@

With polar coordinates being used to describe position, it’s extremely convenient to change our reference frame again by switching into the (noninertial) cylindrical coordinate system.

This transformation is especially useful as the expression for the relative position simplifies dramatically to

From this, we need to calculate the **inertial** velocity and acceleration before continuing.
These expressions are given by

Simply inserting these back into the standard form of the 2-body problem gives us a new set of differential equations in terms of \(r\) and \(\theta\).

### 4.2.2 Rewriting the conserved quantities@

#### 4.2.2.1 Angular momentum \(h(r,\dot{\theta})\)@

The angular momentum \(\mathbf{h}\) expressed in polar coordinates is

Since \(\mathbf{h} = h \mathbf{e}_z\) too by construction of \(\mathbf{R}\) (where \(h = |\mathbf{h}|\) is a constant of motion), the conservation of angular momentum provides

This simple equation proves to be **very** useful, and we’re even able to get
some information out of it (since we know that \(h \geqslant 0\) and \(r > 0\)).

Important

**The trajectory always goes in the counterclockwise direction!**(\(\dot{\theta} \geqslant 0\))A

**collision**will occur if there’s no angular momentum.Proof

Set \(\dot{\theta} = 0\). Then \(\ddot{r} < 0\). This behavior eventually drives \(r \to 0\) (called a

*collision*) in**finite time.**

#### 4.2.2.2 Laplace vector \(A(r,\theta,\dot{r},\dot{\theta})\)@

The Laplace vector \(\mathbf{A}\) expressed in polar coordinates is

(The last equality holds by construction of \(\mathbf{R}\).)

To summarize, we get two scalar equations from the invariance of the Laplace vector.

Important

**Combined with angular momentum, we can get expressions of** \(r(\theta)\) **and** \(\dot{r}(\theta)\) **very quickly!**

#### 4.2.2.3 Energy \(\mathcal{E}(r,\theta,\dot{r},\dot{\theta})\)@

The energy \(\mathcal{E}\) expressed in polar coordinates is

Important

Paired with angular momentum, this expression “integrates” the \(\ddot{r}\) equation of motion with integration constant \(2\mathcal{E}\).

Similarly, this relation provides another expression for \(\dot{\theta}\) in addition to that from the conservation of angular momentum using the knowledge that \(\dot{\theta} > 0\) for interesting motion.