# The Orbit Equation¶

Now, we’ll return to the equation of relative motion, Eq. (33), repeated here for reference:

Our goal is to be able to integrate this equation to find a scalar equation. An analytical equation will be more accurate than the numerical techniques we used earlier, and a scalar equation is easier to work with than a vector one.

We start by taking the cross product of this equation with the angular momentum, \(\vector{h}\):

To make this easy to integrate, we want both sides to be \(d/dt(\ldots)\). Let’s try to replace the left-hand side of the equation. Since \(\ddot{\vector{r}} = d/dt\left(\dot{\vector{r}}\right)\), lets try to pull a \(d/dt\) out. By the product rule of differentiation, we find:

But the angular momentum is constant, so its derivative \(\dot{\vector{h}} = \vector{0}\) and the second term in Eq. (111) is zero. Therefore:

So, the left-hand side of the formula is in terms of \(d/dt(\ldots)\). Let’s get the right-hand side into the same form so we can simplify. After a bunch of algebra, we find:

Substituting everything together, we find:

Rearranging:

This equation can be integrated to find:

where \(\vector{B}\) is called the **Laplace vector** and is the constant of integration. The Laplace vector has the same dimensions as \(\mu\), so we can transform it into a dimensionless number by dividing the equation by \(\mu\):

where \(\vector{e} = \vector{B}/\mu\) and is called the **eccentricity vector**. Since \(\vector{B}\) lies in the orbital plane, \(\vector{e}\) also lies in the orbital plane. The line along \(\vector{e}\) is called the **apse line**. These coordinates are shown in Fig. 29.

We now want to transform Eq. (112) to be in terms of \(r\) and \(\nu\), which is called the **true anomaly**, defined as the angle from the apse line to the \(m_2\). This will result in a scalar equation, which is easier to work with than the vector equation.

To obtain a scalar equation, we take the dot product of Eq. (112) with \(\vector{r}\). After some algebra, we end up at:

where \(e = \mag{\vector{e}}\) is called the **eccentricity**. Eq. (113) is called the **orbit equation** and it defines the path of \(m_2\) around \(m_1\), relative to \(m_1\). In this equation, \(h\), \(e\), and \(\mu\) are all constant. Since \(e\) is the magnitude of \(\vector{e}\), it is strictly positive, \(e \geq 0\).

Important

Put a big star next to Eq. (113). We are going to use it for the rest of the course!

The orbit equation describes **conic sections**, meaning that all orbits are one of four types, as shown in Fig. 30. The particular type of orbit is determined by the magnitude of the eccentricity:

Circles: \(e = 0\)

Ellipses: \(0 < e < 1\)

Parabolas: \(e = 1\)

Hyperbolas \(e > 1\)

We are going to handle each of these in turn in a few sections. In the meantime, we are going to do a little bit of work directly with the orbit equation.