<blockquote>
<p>But where is this orbit centered?</p>
</blockquote>
<p>The true anomaly is the angle as measured from the central body between periapsis passage and the object's current location. The orbit is an ellipse with one of the two foci at the central body. This concept is central to Kepler's laws and Newtonian mechanics.</p>
<p>The eccentric anomaly is the angle as measured from the center of the ellipse between periapsis passage and the projection along the minor axis of the point in question onto the minimal bounding circle that encompasses the elliptical orbit. That minimal bounding circle is necessarily centered on the center of the orbit rather than the central body.</p>
<p>Mean anomaly is a fictitious angle. Unlike true anomaly and eccentric anomaly, mean anomaly does not indicate where the object is. Mean anomaly instead is the angular displacement that a fictitious object in a circular orbit about the central body with the same semi-major axis length as the object in question would have passed through in the time since periapsis passage.</p>
<br>
Aside: The image from wikipedia is flat-out wrong. The center is correct, but the size is incorrect. The circular orbit should intersect the ellipse, twice. It should be outside of the ellipse at periapsis and inside the ellipse at apoapsis.

CC BY-SA 4.0
2020-08-26T07:19:42.787Z
This answer does not answer. Question was: where is the fictitious orbit centered? I was not asking for definitions of the quantities. All sites say "Mean anomaly instead is the angular displacement that a fictitious object in a circular orbit about the central body", but none says where the circular orbit is centered.
jumpjack
CC BY-SA 4.0
2020-08-26T07:24:12.717Z
@jumpjack - I wrote "Mean anomaly instead is the angular displacement that a fictitious object in a circular orbit **about the central body**". I can update the answer if it isn't clear to you from what I already wrote that the central body is the center of the circular orbit.
David Hammen
CC BY-SA 4.0
2020-08-26T07:39:01.960Z
I'd write "circular orbit centered on ellipse focus". "Centered on central body" is not so helpful.
jumpjack
CC BY-SA 4.0
2020-08-26T07:44:39.323Z
@jumpjack - An ellipse has two foci. Writing "centered on the ellipse focus" is ambiguous because ellipses have two foci, so I am not changing my answer to your suggestion. The central body is in fact one of the two foci of a Keplerian ellipse. The other focus has no physical meaning.
David Hammen
CC BY-SA 4.0
2020-08-25T16:06:08.893Z
2020-08-25T16:25:45.577Z
2020-08-25T16:25:45.577Z
David Hammen
<p>Fortunately all diagrams in the question are consistent in these respects:</p>
<ul>
<li>the <a href="https://en.wikipedia.org/wiki/True_anomaly" rel="nofollow noreferrer">true anomaly</a> <span class="math-container">$\nu$</span> or <span class="math-container">$f$</span> is measured around the focus at the central body</li>
<li>the <a href="https://en.wikipedia.org/wiki/Eccentric_anomaly" rel="nofollow noreferrer">eccentric anomaly</a> <span class="math-container">$E$</span> is measured around the center of the ellipse</li>
<li>both are zero at the periapsis</li>
</ul>
<p>The mean anomaly <span class="math-container">$M$</span> is a pseudo-angular quantity useful in computing <span class="math-container">$E$</span> and <span class="math-container">$\nu$</span>, with the same zero and period but increasing linearly with time.
<span class="math-container">$M$</span> is not a geometric angle around an actual point.
There is no right place to draw it.</p>
<p>You could represent <span class="math-container">$M$</span> with a circle around the center of the ellipse, around the focus at the central body, or even around the other focus; any such choice is arbitrary.
Better yet, leave it out of the diagram to avoid suggesting a geometric meaning where there isn't one.</p>

CC BY-SA 4.0
2020-08-26T16:51:01.013Z
After looking at Kepler equation, I think we can say that M is centered on ellipse center: M = E - e sin(E) , being E centered on ellipse center. This means that the fictitious orbit is a circle centered on ellipse center. Given that the major semiaxis is the reference parameter of the ellipse, we could assume that this would be the radius of the fictitious orbit for e=0, hence such orbit is a circle tangent to the ellipse. Hence in the end the Max-Planck-Institut für Sonnensystemforschung picture is the right one. "Centered on ellipse center" has also geometrical sense.
jumpjack
CC BY-SA 4.0
2020-08-26T18:46:10.777Z
@jumpjack M = E - e sin E expresses a numerical relation, not a geometric one. The geometric center of M is undefined. If you must define one anyway, remember that you made it up.
Mike G
CC BY-SA 4.0
2020-08-27T10:37:39.057Z
yes but what is confusing is that M = E - e sin E is a numerical relation, but for M is given a geometrical definition (the angle that IN THEORY a body would cover if it orbit was elliptic), and I have also formulas which sum up an angle centered somewhere to an angle centered (apparently) somewhere else ( https://www.aa.quae.nl/en/reken/zonpositie.html#5 ) , and to a further angle (capital $ \pi $) centered on same center of both the ellipse and the circle...
jumpjack
CC BY-SA 4.0
2020-08-27T13:48:10.047Z
@jumpjack The quae.nl article's $\Pi$ is the heliocentric longitude of the perihelion, reckoned from the ellipse focus where the Sun is. Their equation (7) for $\lambda$ assumes that the orbital inclination $i$ is negligible.
Mike G
CC BY-SA 4.0
2020-08-25T16:45:59.040Z
2020-08-26T12:20:27.577Z
2020-08-26T12:20:27.577Z
Mike G
<p>Mean longitude and mean anomaly of a body are calculated as the angle covered by the body along a fictious circular orbit, and this is ok.</p>
<p>But where is this orbit centered? Five different sources say five different things:</p>
<p><a href="https://en.wikipedia.org/wiki/Mean_anomaly" rel="nofollow noreferrer">Wikipedia</a>: circle center = ellipse focus</p>
<p><a href="https://i.sstatic.net/ogCcZ.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/ogCcZ.png" alt="Wikipedia" /></a></p>
<p><a href="http://farside.ph.utexas.edu/Books/Syntaxis/Almagest/node34.html" rel="nofollow noreferrer">University of Texas</a>: who knows?!? No ellipse at all:</p>
<p><a href="https://i.sstatic.net/gqclt.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/gqclt.png" alt="U Texas diagram, white bg" /></a></p>
<p><a href="http://www.csun.edu/%7Ehcmth017/master/node14.html" rel="nofollow noreferrer">University of California</a>: who knows? Mean anomaly mentioned but not drawn; only eclipse shown, no circle:</p>
<p><a href="https://i.sstatic.net/Q0NqA.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Q0NqA.png" alt="CSU Northridge diagram, white bg" /></a></p>
<p><a href="https://www.britannica.com/science/anomaly-astronomy#ref105658" rel="nofollow noreferrer">Britannica Enciclopedia</a>: who knows? Both circle and ellipse shown, but mean anomaly not shown:</p>
<p><a href="https://i.sstatic.net/e7Bzk.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/e7Bzk.png" alt="Britannica" /></a></p>
<p><a href="https://www2.mps.mpg.de/homes/fraenz/systems/systems3art/node15.html" rel="nofollow noreferrer">Max-Planck-Institut für Sonnensystemforschung</a>: circle centered on ellipse center</p>
<p><a href="https://i.sstatic.net/kbgHv.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/kbgHv.png" alt="Max Planck Inst. diagram, white bg" /></a></p>
<p><a href="https://www.aa.quae.nl/en/reken/zonpositie.html#5" rel="nofollow noreferrer">Theory &amp; formulas</a>: longitudes and anomalies summed up together, no pictures at all.</p>
<p>I am very confused.</p>

CC BY-SA 4.0
2020-08-25T15:13:27.013Z
Fitzpatrick@UTexas illustrates true and eccentric anomalies in a [diagram here](http://farside.ph.utexas.edu/Books/Syntaxis/Almagest/node29.html).
Mike G
CC BY-SA 4.0
2020-08-25T15:18:14.443Z
The Wikipedia illustration has scale problems, and probably should have been removed from the Mean Anomaly article years ago.
notovny
CC BY-SA 4.0
2020-08-26T07:24:17.703Z
@notovny If the image on Wikipedia is wrong, anybody (also you) can remove/fix it.
jumpjack
<orbit><coordinate><orbital-elements>
Is the "mean orbit" centered on ellipse center or ellipse focus?
jumpjack
# Is the "mean orbit" centered on ellipse center or ellipse focus?

(Asked by: username_0 on 2020-08-25)

Mean longitude and mean anomaly of a body are calculated as the angle covered by the body along a fictious circular orbit, and this is ok.

But where is this orbit centered? Five different sources say five different things:

[Wikipedia](https://en.wikipedia.org/wiki/Mean_anomaly): circle center = ellipse focus

[![Wikipedia](https://i.sstatic.net/ogCcZ.png)](https://i.sstatic.net/ogCcZ.png)

[University of Texas](http://farside.ph.utexas.edu/Books/Syntaxis/Almagest/node34.html): who knows?!? No ellipse at all:

[![U Texas diagram, white bg](https://i.sstatic.net/gqclt.png)](https://i.sstatic.net/gqclt.png)

[University of California](http://www.csun.edu/%7Ehcmth017/master/node14.html): who knows? Mean anomaly mentioned but not drawn; only eclipse shown, no circle:

[![CSU Northridge diagram, white bg](https://i.sstatic.net/Q0NqA.png)](https://i.sstatic.net/Q0NqA.png)

[Britannica Enciclopedia](https://www.britannica.com/science/anomaly-astronomy#ref105658): who knows? Both circle and ellipse shown, but mean anomaly not shown:

[![Britannica](https://i.sstatic.net/e7Bzk.png)](https://i.sstatic.net/e7Bzk.png)

[Max-Planck-Institut für Sonnensystemforschung](https://www2.mps.mpg.de/homes/fraenz/systems/systems3art/node15.html): circle centered on ellipse center

[![Max Planck Inst. diagram, white bg](https://i.sstatic.net/kbgHv.png)](https://i.sstatic.net/kbgHv.png)

[Theory & formulas](https://www.aa.quae.nl/en/reken/zonpositie.html#5): longitudes and anomalies summed up together, no pictures at all.

I am very confused.

**Tags:** orbit, coordinate, orbital-elements
> **Comments:**
> **username_1**: Fitzpatrick@UTexas illustrates true and eccentric anomalies in a [diagram here](http://farside.ph.utexas.edu/Books/Syntaxis/Almagest/node29.html).  
> **username_2**: The Wikipedia illustration has scale problems, and probably should have been removed from the Mean Anomaly article years ago.  
> **username_0**: @notovny If the image on Wikipedia is wrong, anybody (also you) can remove/fix it.  

---

## 2 answers

### Answer by username_3 on 2020-08-25. Score: 5

> But where is this orbit centered?

The true anomaly is the angle as measured from the central body between periapsis passage and the object's current location. The orbit is an ellipse with one of the two foci at the central body. This concept is central to Kepler's laws and Newtonian mechanics.

The eccentric anomaly is the angle as measured from the center of the ellipse between periapsis passage and the projection along the minor axis of the point in question onto the minimal bounding circle that encompasses the elliptical orbit. That minimal bounding circle is necessarily centered on the center of the orbit rather than the central body.

Mean anomaly is a fictitious angle. Unlike true anomaly and eccentric anomaly, mean anomaly does not indicate where the object is. Mean anomaly instead is the angular displacement that a fictitious object in a circular orbit about the central body with the same semi-major axis length as the object in question would have passed through in the time since periapsis passage.

  
Aside: The image from wikipedia is flat-out wrong. The center is correct, but the size is incorrect. The circular orbit should intersect the ellipse, twice. It should be outside of the ellipse at periapsis and inside the ellipse at apoapsis.

> **Comments:**
> **username_0**: This answer does not answer. Question was: where is the fictitious orbit centered? I was not asking for definitions of the quantities. All sites say "Mean anomaly instead is the angular displacement that a fictitious object in a circular orbit about the central body", but none says where the circular orbit is centered.  
> **username_3**: @jumpjack - I wrote "Mean anomaly instead is the angular displacement that a fictitious object in a circular orbit **about the central body**". I can update the answer if it isn't clear to you from what I already wrote that the central body is the center of the circular orbit.  
> **username_0**: I'd write "circular orbit centered on ellipse focus". "Centered on central body" is not so helpful.  
> **username_3**: @jumpjack - An ellipse has two foci. Writing "centered on the ellipse focus" is ambiguous because ellipses have two foci, so I am not changing my answer to your suggestion. The central body is in fact one of the two foci of a Keplerian ellipse. The other focus has no physical meaning.  

---

### Answer by username_1 on 2020-08-25. Score: 3

Fortunately all diagrams in the question are consistent in these respects:

* the [true anomaly](https://en.wikipedia.org/wiki/True_anomaly) $\nu$ or $f$ is measured around the focus at the central body
* the [eccentric anomaly](https://en.wikipedia.org/wiki/Eccentric_anomaly) $E$ is measured around the center of the ellipse
* both are zero at the periapsis

The mean anomaly $M$ is a pseudo-angular quantity useful in computing $E$ and $\nu$, with the same zero and period but increasing linearly with time.
$M$ is not a geometric angle around an actual point.
There is no right place to draw it.

You could represent $M$ with a circle around the center of the ellipse, around the focus at the central body, or even around the other focus; any such choice is arbitrary.
Better yet, leave it out of the diagram to avoid suggesting a geometric meaning where there isn't one.

> **Comments:**
> **username_0**: After looking at Kepler equation, I think we can say that M is centered on ellipse center: M = E - e sin(E) , being E centered on ellipse center. This means that the fictitious orbit is a circle centered on ellipse center. Given that the major semiaxis is the reference parameter of the ellipse, we could assume that this would be the radius of the fictitious orbit for e=0, hence such orbit is a circle tangent to the ellipse. Hence in the end the Max-Planck-Institut für Sonnensystemforschung picture is the right one. "Centered on ellipse center" has also geometrical sense.  
> **username_1**: @jumpjack M = E - e sin E expresses a numerical relation, not a geometric one. The geometric center of M is undefined. If you must define one anyway, remember that you made it up.  
> **username_0**: yes but what is confusing is that M = E - e sin E is a numerical relation, but for M is given a geometrical definition (the angle that IN THEORY a body would cover if it orbit was elliptic), and I have also formulas which sum up an angle centered somewhere to an angle centered (apparently) somewhere else ( https://www.aa.quae.nl/en/reken/zonpositie.html#5 ) , and to a further angle (capital $ \pi $) centered on same center of both the ellipse and the circle...  
> **username_1**: @jumpjack The quae.nl article's $\Pi$ is the heliocentric longitude of the perihelion, reckoned from the ellipse focus where the Sun is. Their equation (7) for $\lambda$ assumes that the orbital inclination $i$ is negligible.  

---

