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I'm looking for a general formula or procedure calculate time to reach any given true anomaly.

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    $\begingroup$ It is unclear what you are asking $\endgroup$
    – Slarty
    Aug 1 at 16:31
  • $\begingroup$ @Slarty Lets say I want to find how long it will take to get so that my true anomaly is 270 degress. $\endgroup$
    – Sam
    Aug 1 at 17:11
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    $\begingroup$ Is this what you are looking for ? The formula is what is given in the question and is not verified, I think. $\endgroup$
    – AJN
    Aug 1 at 17:26
  • $\begingroup$ @AJN it seems to be the opposite. You've linked to "How to get true anomaly from time?" and this question asks "How to get time from true anomaly?" $\endgroup$
    – uhoh
    Aug 2 at 2:14
  • $\begingroup$ @Slarty I think it was clear enough but I've modified the wording a bit. Sometimes questions don't have to be long. At this point I hope it's clear enough so that closing the question which blocks anyone from posting answers won't be necessary. $\endgroup$
    – uhoh
    Aug 2 at 3:50
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Assumptions:

  • The value you are requesting is the time-of-flight from, or to the periapsis of your specified orbit to the specified True Anomaly. We will designate this value as $t$.
  • We have the following information:
Symbol Parameter
$e$ Orbital Eccentricity $0\le e \le 1$ for an elliptical orbit, $e > 1$ for a hyperbolic orbit.
$a$ Semimajor Axis (with the convention that semimajor axis is a negative value for a hyperbolic trajectory)
$\theta$ True Anomaly in radians. The assumptions of the following calculations assume that the value is in the range $-\pi \le \theta \le \pi$
$\mu$ Standard Gravitational Parameter of the body being orbited.

I have chosen to limit this to the Elliptical and Hyperbolic cases. The math is straightforward in the circular case ($e = 0$), since the orbiting object moves at a constant speed, and the parabolic case ($e = 1$) has its own set of equations.

1. Calculate the Eccentric Anomaly($E$) or Hyperbolic Eccentric Anomaly ($F$) from True Anomaly. The Eccentric anomaly is a geometric quasi-angular parameter used to solve Kepler's equations.

Elliptical Orbit Hyperbolic Orbit
$E=\pm\cos^{-1}\left({\frac{e+\cos\theta}{1+e\cos\theta}}\right)$ $F=\pm\cosh^{-1}\left({\frac{e+\cos\theta}{1+e\cos\theta}}\right)$

Use the value of $E$ or $F$ that has the same sign as True Anomaly $\theta$. At the periapsis ($\theta = 0$) and apoapsis ($\theta = \pm \pi$) of the orbit, or if the orbit is circular, Eccentric Anomaly is equal to True Anomaly.

Keep the value in radians for the next step.

2. Calculate the Mean Anomaly ($M$) from the appropriate Eccentric Anomaly.

Mean Anomaly is a quasi-angular parameter equal to the angle swept out by a hypothetical object in a circular orbit whose radius equals the semimajor axis, in the time it would take for the object in the orbit you calculating for to get to or from periapsis.

Elliptical Orbit Hyperbolic Orbit
$M = E - e \sin(E)$ $M=e \sinh(F) -F$

Again, at the periapsis ($\theta = 0$) and apoapsis ($\theta = \pm \pi$) of the orbit, or if the orbit is circular, Mean Anomaly is equal to both True Anomaly and Eccentric Anomaly.

3. Calculate the Mean Motion, $n$

Mean motion is the angular speed of a hypothetical body in a circular orbit whose radius equals the orbit's semimajor axis, and is used to relate Mean Anomaly to the passage of time. $$n=\sqrt{\frac{\mu}{|a|^3}}$$

It is also related to the orbital period $P$, which can be calculated as: $$P=2\pi n$$

4. Calculate the time since Periapsis, $t$

For many expressions of the Keplerian orbital elements, there is a reference value called Mean Anomaly at Epoch $M_0$, which is the mean anomaly of the orbiting body at a specific reference time $t_0$, called the Epoch. The time of flight between these reference points and the current position can be expressed as:

$$t-t_0 = \frac{M-M_0}{n}$$

Since we are using the periapsis as the reference point here, $t_0 = 0$ and $M_0 = 0$, and the equation above becomes: $$t = \frac{M}{n}$$

A positive value indicates the object is ascending from periapsis, and a negative value indicates the object is descending towards periapsis. In the latter case, if the orbit is elliptical, to get time since periapsis, add the orbital period $P$ to the negative value.

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