<p>I’ve been looking into this lately, and most sources I have used solve this problem numerically. I was thus wondering if there is a proper equation to solve for position without numerical, especially since the motion of planets and satellites seems to follow harmonic motion.</p>

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2023-04-14T06:32:14.257Z
The problem is Kepler's equation https://en.wikipedia.org/wiki/Kepler%27s_equation
PM 2Ring
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2023-04-14T09:11:02.737Z
@PM2Ring That's an actual answer as short as it may be...
planetmaker
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2023-04-14T20:18:17.223Z
See [this answer to *What is the analytical closed-form solution of the two-body problem to verify its numerical integration results?*](https://space.stackexchange.com/a/43237/12102) in Space Exploration SE. For Keplerian orbits you can solve for $t(\theta)$ analytically, but the inverse $\theta(t)$ needs numerical or a power series approximation.
uhoh
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2023-04-14T22:39:02.227Z
In https://space.stackexchange.com/questions/55356/how-to-find-eccentric-anomaly-by-mean-anomaly/55395#comment184997_55395 I have a short Python demo that compares different initial approximations when using Newton's method to solve Kepler's equation.
PM 2Ring
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2023-12-18T14:59:48.270Z
@uhoh ah thank you! that answers my question.
hi-bye125
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2023-12-18T22:20:39.383Z
@hi-bye125 That's great to hear! If you like you can post a short answer below, summarizing how it worked for you, and you can even accept the answer. That 1) makes an answer visible to future readers (many people and search engines don't check comments) and 2) you'll probably pick up a few upvotes and reputation points.
uhoh
<orbit><astrophysics><orbital-mechanics>
Is there an equation that can give position of a satellite around a parent body as a function of time?
hi-bye125
# Is there an equation that can give position of a satellite around a parent body as a function of time?

(Asked by: username_0 on 2023-04-14)

I’ve been looking into this lately, and most sources I have used solve this problem numerically. I was thus wondering if there is a proper equation to solve for position without numerical, especially since the motion of planets and satellites seems to follow harmonic motion.

**Tags:** orbit, astrophysics, orbital-mechanics

> **Comments:**
> **username_1**: The problem is Kepler's equation https://en.wikipedia.org/wiki/Kepler%27s_equation  
> **username_2**: @PM2Ring That's an actual answer as short as it may be...  
> **username_3**: See [this answer to *What is the analytical closed-form solution of the two-body problem to verify its numerical integration results?*](https://space.stackexchange.com/a/43237/12102) in Space Exploration SE. For Keplerian orbits you can solve for $t(\theta)$ analytically, but the inverse $\theta(t)$ needs numerical or a power series approximation.  
> **username_1**: In https://space.stackexchange.com/questions/55356/how-to-find-eccentric-anomaly-by-mean-anomaly/55395#comment184997_55395 I have a short Python demo that compares different initial approximations when using Newton's method to solve Kepler's equation.  
> **username_0**: @username_3 ah thank you! that answers my question.  
> **username_3**: @username_0 That's great to hear! If you like you can post a short answer below, summarizing how it worked for you, and you can even accept the answer. That 1) makes an answer visible to future readers (many people and search engines don't check comments) and 2) you'll probably pick up a few upvotes and reputation points.  

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