Time Since Periapsis, Mean Anomaly, and Eccentric Anomaly

In the last chapter, we derived equations for the position of an object in a two-body system as a function of the true anomaly. However, humans don’t really think in terms of true anomaly, we think in terms of time. The only place time appeared in the last chapter was in the calculation of the orbital period.

As we’ll find in this chapter, the equations to relate the orbital position to time are transcendental for all but the circular orbit. This means we will need to use Newton’s method to solve them. Initially, we will derive separate equations each for the elliptical, parabolic, and hyperbolic orbits. Next, we will combine these using a universal variable formulation.

Time Since Periapsis

Recall the orbit equation, Eq. (113), defined in terms of the true anomaly:

\[r = \frac{h^2}{\mu} \frac{1}{1 + e\cos\nu}\]

We now want to relate the true anomaly, \(\nu\), to time. The rate of change of the true anomaly, \(\dot{\nu}\) is equal to the angular velocity of the position vector. This is exactly the azimuthal, also called the perpendicular, component of the velocity:

(185)\[v_{\perp} = r \dot{\nu} = r \frac{d\nu}{dt}\]

The \(v_{\perp}\) term in Eq. (185) makes the equation more complicated than it needs to be, so we’d like to replace it. A more convenient form of Eq. (185) is found by using the specific angular momentum to replace \(v_{\perp}\), since \(h\) is constant:

(186)\[h = r v_{\perp} = r^2\dot{\nu} \Rightarrow \frac{d\nu}{dt} = \frac{h}{r^2}\]

Substituting the orbit equation to eliminate \(r\) and separating variables, we find:

(187)\[\frac{\mu^2}{h^3}dt = \frac{d\nu}{\left(1 + e\cos\nu\right)^2}\]

Since \(\mu\) and \(h\) are constant, the left side can be directly integrated:

(188)\[\frac{\mu^2}{h^3}\int_{t_p}^{t} dt = \int_{0}^{\nu}\frac{d\nu}{\left(1 + e\cos\nu\right)^2}\]

where \(t_p\) is defined as the time since periapsis. Remember that periapsis is when \(\nu = 0\) by convention. Typically we will set \(t_p = 0\), such that:

(189)\[\frac{\mu^2}{h^3}t = \int_{0}^{\nu}\frac{d\nu}{\left(1 + e\cos\nu\right)^2}\]

The integral on the right-hand side of Eq. (189) can be found in standard tables of integrals [GradshteuinRJ07, Zwi03]. There are three forms of the equation, depending on the value of \(e\).

In [GradshteuinRJ07] (available here), the appropriate integrals are found on pages 172 and 173, No. 2.554-3 and related integrals for Eqs. (190) and (192). In [Zwi03] (available here), the appropriate integrals are found on pages 433 and 434, No. 354 and 324 for Eqs. (190) and (192). I couldn’t find the form for (191) in those references. Note that \(a = 1\) and \(b = e\) to relate the reference equations to Eq. (189).

(190)\[\int\frac{d\nu}{\left(1 + e\cos \nu\right)^2} = \frac{1}{\left(1 - e^2\right)^{3/2}}\left[2\tan^{-1}\left(\sqrt{\frac{1 - e}{1 + e}}\tan\frac{\nu}{2}\right)-\frac{e\sqrt{1 - e^2}\sin \nu}{1 + e \cos \nu}\right]\]

In Eq. (190), \(e < 1\), so it will apply for circular and elliptical orbits.

(191)\[\int\frac{d\nu}{\left(1 + e\cos \nu\right)^2} = \left(\frac{1}{2}\tan \frac{\nu}{2}+\frac{1}{6}\tan^{3}\frac{\nu}{2}\right)\]

In Eq. (191), \(e = 1\), so it will apply for parabolic trajectories.

(192)\[\int\frac{d\nu}{\left(1 + e\cos \nu\right)^2} = \frac{1}{\left(e^2 - 1\right)^{3/2}}\left[\frac{e\sqrt{e^2 - 1}\sin \nu}{1 + e\cos \nu} - \ln\left(\frac{\sqrt{e + 1} + \sqrt{e - 1}\tan\frac{\nu}{2}}{\sqrt{e + 1} - \sqrt{e - 1}\tan\frac{\nu}{2}}\right)\right]\]

In Eq. (192), \(e > 1\), so it will apply for hyperbolic trajectories.

When Eq. (189) is combined with one of Eq. (190), Eq. (191), or Eq. (191), we have a relationship where time is related to the true anomaly.

Mean and Eccentric Anomalies

As we can see, Eqs. (190), (191), and (192) are rather complicated functions of the true anomaly. To simplify handling them, we are going to define two other anomalies:

1. Mean anomaly

2. Eccentric anomaly

Note

Conceptually, it is easiest to understand the mean and eccentric anomalies in the context of an elliptical orbit. Just remember that they also apply to parabolic and hyperbolic trajectories as well.

For an ellipse, we know that the angular speed of the spacecraft is a function of the true anomaly. This is a result of Kepler’s second law, that equal areas are swept in equal times—the spacecraft must move faster near periapsis and slower near apoapsis to sweep the same area in a given time interval. Thus, the rate of change of the true anomaly is not constant!

Such a circle is said to circumscribe the ellipse.

The mean anomaly for an ellipse is defined such that it is a constant rate that gives the same period as the true anomaly. If we draw a circle that touches the ellipse at periapsis and apoapsis, then the rate of change of the mean anomaly is the rate at which a point must move around this circle to meet the spacecraft at periapsis and apoapsis.

The eccentric anomaly is defined slightly differently. Imagine the same circle that touches the ellipse at periapsis and apoapsis. Now draw a line perpendicular to the semimajor axis from the orbit to the circle, at the current true anomaly. The angle to the point defined in this way on the circle is the eccentric anomaly.

The three anomalies: true, mean, and eccentric, are shown in Fig. 47.

Once Loop Reflect

Fig. 47 The true, mean, and eccentric anomalies for an elliptical orbit. The true anomaly is shown in blue, the mean anomaly is shown in orange, and the eccentric anomaly is shown in green. The mean anomaly proceeds around the orange circle at a constant rate, while the rates of the true and eccentric anomalies vary with position. Nonetheless, they all meet at periapsis and apoapsis. Note the vertical line connecting the true and eccentric anomalies.

Solution Procedures

There are now two cases that we want to consider:

1. We know the current true anomaly and want to find the time since periapsis

2. We know the current time since periapsis and want to know the true anomaly

Each case is solved slightly differently, and the equations depend on the type of trajectory, so this will just be a general overview. Specific solutions for each type of trajectory will be in the following sections.

Given True Anomaly, Find Time Since Periapsis

1. Use the true anomaly to find the eccentric anomaly

2. Use the eccentric anomaly to find the mean anomaly

3. Use the mean anomaly to find the time since periapsis

Given Time Since Periapsis, Find True Anomaly

1. Use the time since periapsis to find the mean anomaly

2. Use the mean anomaly to find the eccentric anomaly

3. Use the eccentric anomaly to find the true anomaly