If you aren't dealing with some sort of perturbation of the orbit (from atmosphere, oblateness of the Earth, the satellite thrusting), and are not dealing with extreme timescales, and are only doing a two-body Keplerian/Newtonian simulation, the equator of the earth keeps the same orientation, as does the orbital plane of the satellite.
In that situation, you can completely ignore the rotation of the Earth as it has no effect on any of the standard orbital parameters.
| Parameter |
Symbol |
| Semi-major axis of the orbit |
$a$ |
| Orbital Eccentricity |
$e$ |
| Argument of Perigee of orbit |
$\omega$ |
| Earth's Gravitational Parameter |
$\mu$ |
| True anomaly of the satellite's current position |
$f_t$ |
Calculations below will use radians for all angular measurements.
1. Find the True Anomaly of the Ascending Node, $f_{\Omega}$.
The Argument of Perigee $\omega$ is simply the angle, measured in the direction of travel around the orbit, from the ascending node, through the center of the body being orbited, to the periapsis. As a result:
$$f_{\Omega} = 2\pi - \omega$$
2. Calculate the Eccentric Anomaly $E$ for both true anomalies
The Eccentric anomaly is a quasi-angular parameter used in Kepler's equations to convert true anomaly angle into the Mean anomaly. In this direction, the calculation uses closed-form equations.
| $$0 <= f < \pi$$ |
$$\pi <= f < 2\pi$$ |
| $$E = \arccos\left(\frac{e + \cos(f)}{e \cos(f) + 1}\right)$$ |
$$E = 2\pi - \arccos\left(\frac{e + \cos(f)}{e \cos(f) + 1}\right)$$ |
3. Calculate the Mean Anomaly $M$ for both eccentric anomalies.
Mean anomaly is another quasi-angular parameter which represents the angle swept out by a hypothetical object in a circular orbit with the same semi-major axis since periapsis passage.
$$M = E - e \sin E $$
4. Calculate the Mean Motion $n$ for the orbit.
Mean motion is the average angular velocity of an orbit with the semi-major axis $a$.
$$n = \sqrt{\frac{\mu}{a^3}}$$
5 Using the difference between the Mean Anomalies, calculate time since ascending node, $t$
If the mean anomaly of the satellite is $M_s$ and the mean anomaly of the ascending node is $M_{\omega}$:
| $$M_s >= M_{\omega}$$ |
$$M_s< M_{\omega}$$ |
| $$t = \frac{M_s - M_{\omega}}{n}$$ |
$$t = \frac{ 2\pi + M_s - M_{\omega}}{n}$$ |
