Elliptic Orbits

Prepared by: Noah Leigh, Ilanthiraiyan Sivaganamoorthy, and Angadh Nanjangud

In this lecture we aim to cover the following topics:

1. The Eccentricity Vector

2. The Equation of the Orbit

3. Elliptic Orbits

4. Vis-Viva Equation

The Eccentricity Vector

Deriving The Eccentricity Vector

The Eccentricity vector \(\mathbf{e}\) helps us describe the shape of an orbit and its direction. We can find it by taking the cross product of the velocity and the specific angular momentum:

1. Such as:

   (75)\[\ddot{\mathbf{r}} = - \frac{\mu}{r^3} \mathbf{r}\]

   Taking the cross product with the specific angular momentum, \(\mathbf{h}\):

   \[ \ddot{\mathbf{r}}\times\mathbf{h} = - \frac{\mu}{r^3} \mathbf{r}\times\mathbf{h} \]

   Using the rule for \(\mathbf{a}\times\mathbf{b}\times{\mathbf{c}}=(\mathbf{c}\cdot\mathbf{a})\mathbf{b}-(\mathbf{b}\cdot\mathbf{a})\mathbf{c}\) with \(\mathbf{h}=\mathbf{r}\cdot\dot{\mathbf{r}}\) :

   \[ -\frac{\mu}{r^3}\left[ (\dot{\mathbf{r}}\cdot\mathbf{r})\mathbf{r}-(\mathbf{r}\cdot\mathbf{r})\dot{\mathbf{r}} \right]\rightarrow-\frac{\mu}{r^3}(\dot{r}r\mathbf{r}-r^2\dot{\mathbf{r}})\rightarrow-\frac{\mu}{r^2}(\dot{r}\mathbf{r}-r\dot{\mathbf{r}}) \]
   \[\Rightarrow \frac{d}{dt}\left(\mu\frac{\mathbf{r}}{r}\right)\]

   This gives us:

   \[ \Rightarrow \dot{\mathbf{r}}\times\mathbf{h} - \mu\frac{\mathbf{r}}{r} = constant \]

   Which can be rewritten as:

   (76)\[\mathbf{e}=\frac{\dot{\mathbf{r}}\times\mathbf{h}}{\mu}-\frac{\mathbf{r}}{r}\]

   This is our equation for the eccentricity vector!

2. The first integrals of motion:

   • \(\mathbf{e}\) is a constant vector, implying both constant direction and magnitude.

   • \(\mathbf{e}\) lies in the plane of motion and is fixed.

   Let’s define:

   (77)\[e = \frac{\mathbf{e}}{\mu}\]

   where \(e\) is the eccentricity.

Properties

• \(\mathbf{e}\) lies in the plane of motion and (since it is constant) has a fixed magnitude and direction. \(\Rightarrow \hat{i}\equiv\mathbf{e}\), so we can use the eccentricity as one of the axes.

• The eccentricity, \(e=|\mathbf{e}| \geq 0\), determines the shape of the orbit:

  • \(e = 0\): Circular orbit

  • \(0 < e < 1\): Elliptic orbit

  • \(e = 1\): Parabolic orbit

  • \(e > 1\): Hyperbolic orbit

The Equation of the Orbit

If we look at the dot product of \(\mathbf{e}\cdot\mathbf{r}\) we know that: \(\mathbf{r}\cdot\mathbf{e}=recos(\theta)\) and if we recall \(\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})=(\mathbf{a}\times\mathbf{b})\cdot\mathbf{c}\) then:

\[\frac{\mathbf{r}\cdot(\dot{\mathbf{r}}\times\dot{\mathbf{h}})}{\mu}-\frac{\mathbf{r}\cdot\mathbf{r}}{r}=\frac{(\mathbf{r}\times\dot{\mathbf{r}})\cdot\mathbf{h}}{\mu} - \frac{r^2}{r} = \frac{h^2}{\mu}-r\]

So then this equivalence implies:

(78)\[r = \frac{\frac{h^2}{\mu}}{1+ecos\left(\theta\right)}\]

\(\theta\) is called the true anomaly.

This is another way of expressing Kepler’s 1st Law - the orbit of a planet is an ellipse with the sun at one of the two focii. This is an Equation of the Orbit.

Note: \(\theta\) can also be expressed as \(v\) or \(f\)

r describes the shape of the orbits as conic sections through the varying eccentricities shown before but it is important to note that:

• Operative orbits are always circular or elliptic. As these keep the satellite closest to the planet of interest for as long as possible.

• Hyperbolic orbits are maneuvers used on interplanetary missions to reduce propellant use by using the gravity of a body to reach another target body.

Elliptic Orbits

Fig. 1 An ellipse demostrating the different properties of an elliptical orbit, all shown in different colours.

a = semi-major axis
b = semi-minor axis
p = semi-latus rectum
\(\textcolor{#646464}{\mathbf{e}}\) is aligned with the apsides and point towards the perigee

Key Parameters

1. semi-major axis \(a\): This is the axis of the ellipse that has the longest diameter.

2. semi-minor axis \(b\): This is the axis of the ellipse with the shortest diameter.

3. Semi-latus rectum \(p\): This is the distance from the foci (orbitted body) to the orbit measured perpendicular to the semi-major axis.

   (79)\[p = a(1 - e^2)\]

4. Perigee (\(r_p\)): The lowest point in an orbit/the point closest to the orbitted body.

   (80)\[r_p = a(1 - e)\]

5. Apogee (\(r_a\)): The highest point in an orbit/the point furthest from the orbitted body.

   (81)\[r_a = a(1 + e)\]

Determination of the Orbit

1. Given focus \(\mathbf{r_o}\) and \(\mathbf{v_o}\), compute \(\mathbf{h}\) and \(\mathbf{e}\).

2. Compute the semi-latus rectum, \(p = \frac{h^2}{\mu}\)

3. Compute the semi-major axis,\(a\), using \(p\), (79): \(a=\frac{p}{1-e^2}\)

4. Compute the apsides points (most extreme points in the orbit) using (80) and (81) .

Vis-Viva Equation

The Vis-Viva Equation is used to find the velocity at any given r when a maneuver is planned.

The Equation:
The total energy of the orbit is the sum of the kinetic energy and the potential energies of the orbit.

\[ E= \frac 1 2 v^2-\frac{\mu}{r} = \frac 1 2 \dot{r^2} + \frac 1 2 \frac{h^2}{r^2} - \frac{\mu}{r} = \frac 1 2 \dot{r^2} + \phi_{eff} \]

The sum of the potential energies is the effective potential energy, \(\phi_{eff}\):

(82)\[\phi_{eff} = \frac 1 2 \frac{h^2}{r^2}-\frac{\mu}{r}\]

At perigee and apogee the velocity = 0 so \(E_{r_p}=\phi_{eff}(r_p)\) so we can derive an equation for the specific orbital energy as a function of \(a\).

(83)\[E=-\frac{\mu}{2a}\]

\(\Rightarrow\) orbits with the same semi-major axis have the same orbital energy.

This means that the Vis-Viva Equation is, using equations (82) and (83):

(84)\[E = \frac 1 2 v^2 - \frac{\mu}{r}= -\frac{\mu}{2a}\]

So the velocity of an orbitting body can be found whether you have the position or the semi-major axis of its orbit.

Conclusion

• The eccentricity vector is a fundamental constant vector in orbital mechanics.

• The orbit equation describes different conic sections based on the value of the eccentricity \(e\).

• The Vis Viva equation (84) is a crucial tool in mission planning for calculating velocities at various points in an orbit.