<p>Both formulae are correct. The discrepancy is because the formula from the eccentric anomaly article uses the <em>centre</em> of the ellipse as the origin, but the formula from the Kepler's equation article uses a <em>focus</em> of the ellipse (i.e, the central gravitating body, eg the Sun) as the origin. Note that <span class="math-container">$c = ae$</span> is the distance from the ellipse centre to a focus</p>

CC BY-SA 4.0
2019-12-27T17:27:05.050Z
Thank you, now it's clear. Can't upvote the answer since I don't have enough rep.
Jonas
CC BY-SA 4.0
2019-12-28T01:46:55.993Z
@Jonas I've upvoted both question and answer, we'll get you there soon!
uhoh
CC BY-SA 4.0
2021-04-12T11:46:44.173Z
See my answer [here](https://astronomy.stackexchange.com/a/43350/16685) for a diagram showing the eccentric anomaly.
PM 2Ring
CC BY-SA 4.0
2019-12-27T17:23:32.717Z
2019-12-27T17:23:32.717Z
PM 2Ring
<p>From the <a href="https://en.wikipedia.org/wiki/Eccentric_anomaly" rel="noreferrer">wiki article</a> about eccentric anomaly follows:</p>

<p><span class="math-container">$$\cos E = \frac{x}{a}$$</span>
<span class="math-container">$$\sin E = \frac{y}{b}$$</span></p>

<p>where E - eccentric anomaly, a - semi-major axis, b - semi-minor axis, P = P(x,y) a point on elleptic orbit. So if you know an E, you can find a position on an orbit:</p>

<p><span class="math-container">$$x = a \cos E$$</span>
<span class="math-container">$$y = b \sin E$$</span></p>

<p>However, from <a href="https://en.wikipedia.org/wiki/Kepler%27s_equation" rel="noreferrer">article</a> on Kepler's eqution follows:</p>

<p><span class="math-container">$$x = a (\cos E - e)$$</span>
<span class="math-container">$$y = b \sin E$$</span></p>

<p>Please, help to understand why these formaluae differ? I expect them to be the same.</p>

<orbital-mechanics><orbital-elements><eccentric-orbit>
Kepler's equation and eccentric anomaly
Jonas
# Kepler's equation and eccentric anomaly

(Asked by: username_0 on 2019-12-27)

From the [wiki article](https://en.wikipedia.org/wiki/Eccentric_anomaly) about eccentric anomaly follows:

$$\cos E = \frac{x}{a}$$
$$\sin E = \frac{y}{b}$$

where E - eccentric anomaly, a - semi-major axis, b - semi-minor axis, P = P(x,y) a point on elleptic orbit. So if you know an E, you can find a position on an orbit:

$$x = a \cos E$$
$$y = b \sin E$$

However, from [article](https://en.wikipedia.org/wiki/Kepler%27s_equation) on Kepler's eqution follows:

$$x = a (\cos E - e)$$
$$y = b \sin E$$

Please, help to understand why these formaluae differ? I expect them to be the same.

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


---

## 1 answers

### Answer by username_1 on 2019-12-27. Score: 7 | Accepted answer

Both formulae are correct. The discrepancy is because the formula from the eccentric anomaly article uses the *centre* of the ellipse as the origin, but the formula from the Kepler's equation article uses a *focus* of the ellipse (i.e, the central gravitating body, eg the Sun) as the origin. Note that $c = ae$ is the distance from the ellipse centre to a focus

> **Comments:**
> **username_0**: Thank you, now it's clear. Can't upvote the answer since I don't have enough rep.  
> **username_2**: @username_0 I've upvoted both question and answer, we'll get you there soon!  
> **username_1**: See my answer [here](https://astronomy.stackexchange.com/a/43350/16685) for a diagram showing the eccentric anomaly.  

---

