How do different formulas for calculating the mean anomaly of an elliptic orbit correlate? I need to calculate the mean anomaly of an elliptic orbit at a specific point in time. I found two different formulas for $M(t)$ and I'd like to know how they correlate and if they result in different ouputs.
The first formular is:
$M(t) = \frac{2 \: \pi}{T}\:(t-t_o)$
and its taken from this book.
The second formular is:
$M(t) = M_0 + \Delta t \sqrt{\frac{\mu}{a^3}}$
taken from this document.
Inputs:
$T$... Orbital Period
$t_o$... Starting point in time (epoch)
$t$... Point in time
$\Delta t$... Elasped time: $t-t_0$
$\mu$... Standard gravitational parameter $\mu=GM$
$a$... Semi-major axis
$M_0$... Mean anomaly at epoch
Kepler's Third Law including the constants of proportionality is:
$$GM T^2 = 4\pi^2 a^3$$
Substituting $\mu = GM$, this can be rearranged to give:
$$\frac{2\pi}{T} = \sqrt{\frac{\mu}{a^3}}$$
Which lets you rewrite your first formula as your second one and vice-versa.
Note that in the first formula the reference epoch $t_0$ is assumed to be the time of periapsis, which corresponds to the case $M_0 = 0$. Your definition of $\Delta t$ should also read $\Delta t = t - t_0$, i.e. time since the reference epoch.