<p>I need to calculate the mean anomaly of an elliptic orbit at a specific point in time.<br> I found two different formulas for <span class="math-container">$M(t)$</span> and I'd like to know how they correlate and if they result in different ouputs.<br><br>
The first formular is:<br>
<span class="math-container">$M(t) = \frac{2 \: \pi}{T}\:(t-t_o)$</span><br>
and its taken from <a href="https://doi.org/10.1007/978-3-662-54716-8" rel="nofollow noreferrer">this</a> book.</p>
<p>The second formular is:<br>
<span class="math-container">$M(t) = M_0 + \Delta t \sqrt{\frac{\mu}{a^3}}$</span><br>
taken from <a href="https://downloads.rene-schwarz.com/download/M001-Keplerian_Orbit_Elements_to_Cartesian_State_Vectors.pdf" rel="nofollow noreferrer">this</a> document.<br></p>
<p>Inputs:<br>
<span class="math-container">$T$</span>... Orbital Period<br>
<span class="math-container">$t_o$</span>... Starting point in time (epoch)<br>
<span class="math-container">$t$</span>... Point in time<br>
<span class="math-container">$\Delta t$</span>... Elasped time: <span class="math-container">$t-t_0$</span><br>
<span class="math-container">$\mu$</span>... Standard gravitational parameter <span class="math-container">$\mu=GM$</span><br>
<span class="math-container">$a$</span>... Semi-major axis<br>
<span class="math-container">$M_0$</span>... Mean anomaly at epoch</p>

<orbital-mechanics><orbital-elements>
How do different formulas for calculating the mean anomaly of an elliptic orbit correlate?
Laila
# How do different formulas for calculating the mean anomaly of an elliptic orbit correlate?

(Asked by: username_0 on 2020-10-03)

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](https://doi.org/10.1007/978-3-662-54716-8) book.

The second formular is:  
$M(t) = M\_0 + \Delta t \sqrt{\frac{\mu}{a^3}}$  
taken from [this](https://downloads.rene-schwarz.com/download/M001-Keplerian_Orbit_Elements_to_Cartesian_State_Vectors.pdf) 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

**Tags:** orbital-mechanics, orbital-elements

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