<p>The orbital elements are on wikipedia:</p>

<p><span class="math-container">$$e=0.884\ a=0.125'',\ i=134^\circ,\, \Omega=228^\circ$$</span></p>

<p>(At an assumed distance of 8kpc, <span class="math-container">$0.125'' = 1000au$</span>)</p>

<p>It is the inclination that means that the black hole is not at the focus of the projected ellipse.</p>

<p>Imagine a circular orbit, with a central body.  If viewed from a distant but high inclination, the orbit is projected as an ellipse, with the central body at the centre of the ellipse (not the focus) </p>

<p>A distant projection of an ellipse is another ellipse, but the projection of the focus is not the focus of the projection.</p>

<p>Below you can see another example.  The ellipse is in the grey plane, with foci at A and B.  But the apparent ellipse has a completely different axis, and the focus is nowhere near the axis.  These image have been generate with geogebra and no perspective has been used.</p>

<p><a href="https://i.sstatic.net/9WpCj.png" rel="noreferrer"><img src="https://i.sstatic.net/9WpCj.png" alt="enter image description here"></a></p>

<p><a href="https://i.sstatic.net/vIQT4.png" rel="noreferrer"><img src="https://i.sstatic.net/vIQT4.png" alt="enter image description here"></a></p>

CC BY-SA 4.0
2020-04-16T10:30:11.447Z
I've put a couple of images, from geogebra (it doesn't do any fancy camera at finite distance) to show how the focus and axis don't commute with projection. I haven't tried to get the exact same perspective as S2, but it illustrates the principle
James K
CC BY-SA 4.0
2020-04-16T11:46:06.717Z
This is excellent, *thank you!*
uhoh
CC BY-SA 4.0
2020-04-16T12:13:54.743Z
Ah, I've often wondered whether about distant projections of ellipses are still ellipses. Thanks.
notovny
CC BY-SA 4.0
2020-04-16T12:30:55.980Z
Yes, I think you can show that an affine projection of a quadratic form is still quadratic.
James K
CC BY-SA 4.0
2020-04-16T09:59:20.177Z
2020-04-16T10:27:08.630Z
2020-04-16T10:27:08.630Z
James K
<p>Phys.org's <a href="https://phys.org/news/2020-04-eso-telescope-star-supermassive-black.html" rel="noreferrer">Very Large Telescope sees star dance around supermassive black hole, proves Einstein right</a> links to several ESO videos, including <a href="https://youtu.be/wstMu4ZndM8" rel="noreferrer">The star S2 makes a close approach to the black hole at the centre of the Milky Way</a>.</p>

<p>In the screen shot below, the location of Sgr A* is nowhere near a focus of the elliptical orbit.</p>

<p>In addition to the excellent accepted answer I have written <a href="https://space.stackexchange.com/a/26588/12102">this answer</a> to <em>Why does Earth not appear to be at the focus of TESS' elliptical orbit in this video?</em> and so I know that looks can be deceiving depending on one's perspective.</p>

<p>But I still don't <em>really understand</em> why the offset can look so bad here. For the video did they put the observer (the "camera") at a fairly close distance and thereby distort the perspective, or even if you choose the distance to be infinite can elliptical orbits appear to not have their central body at their focus, or at least along their axis?  </p>

<p>Is it possible to state the orbital elements of S2 in such a way and in some coordinate system so that I can plot them in 3D and then try to reproduce this effect?</p>

<p><a href="https://i.sstatic.net/Gs4DX.png" rel="noreferrer"><img src="https://i.sstatic.net/Gs4DX.png" alt="enter image description here"></a></p>

<milky-way><photography><orbital-mechanics><sagittarius-a>
Why does the closest approach of star S2 to Sgr A* not appear to be near the focus of its elliptical orbit?
uhoh
# Why does the closest approach of star S2 to Sgr A* not appear to be near the focus of its elliptical orbit?

(Asked by: username_0 on 2020-04-16)

Phys.org's [Very Large Telescope sees star dance around supermassive black hole, proves Einstein right](https://phys.org/news/2020-04-eso-telescope-star-supermassive-black.html) links to several ESO videos, including [The star S2 makes a close approach to the black hole at the centre of the Milky Way](https://youtu.be/wstMu4ZndM8).

In the screen shot below, the location of Sgr A\* is nowhere near a focus of the elliptical orbit.

In addition to the excellent accepted answer I have written [this answer](https://space.stackexchange.com/a/26588/12102) to *Why does Earth not appear to be at the focus of TESS' elliptical orbit in this video?* and so I know that looks can be deceiving depending on one's perspective.

But I still don't *really understand* why the offset can look so bad here. For the video did they put the observer (the "camera") at a fairly close distance and thereby distort the perspective, or even if you choose the distance to be infinite can elliptical orbits appear to not have their central body at their focus, or at least along their axis?

Is it possible to state the orbital elements of S2 in such a way and in some coordinate system so that I can plot them in 3D and then try to reproduce this effect?

[![enter image description here](https://i.sstatic.net/Gs4DX.png)](https://i.sstatic.net/Gs4DX.png)

**Tags:** milky-way, photography, orbital-mechanics, sagittarius-a

---

## 1 answers

### Answer by username_1 on 2020-04-16. Score: 24

The orbital elements are on wikipedia:

$$e=0.884\ a=0.125'',\ i=134^\circ,\, \Omega=228^\circ$$

(At an assumed distance of 8kpc, $0.125'' = 1000au$)

It is the inclination that means that the black hole is not at the focus of the projected ellipse.

Imagine a circular orbit, with a central body. If viewed from a distant but high inclination, the orbit is projected as an ellipse, with the central body at the centre of the ellipse (not the focus)

A distant projection of an ellipse is another ellipse, but the projection of the focus is not the focus of the projection.

Below you can see another example. The ellipse is in the grey plane, with foci at A and B. But the apparent ellipse has a completely different axis, and the focus is nowhere near the axis. These image have been generate with geogebra and no perspective has been used.

[![enter image description here](https://i.sstatic.net/9WpCj.png)](https://i.sstatic.net/9WpCj.png)

[![enter image description here](https://i.sstatic.net/vIQT4.png)](https://i.sstatic.net/vIQT4.png)

> **Comments:**
> **username_1**: I've put a couple of images, from geogebra (it doesn't do any fancy camera at finite distance) to show how the focus and axis don't commute with projection. I haven't tried to get the exact same perspective as S2, but it illustrates the principle  
> **username_0**: This is excellent, *thank you!*  
> **username_2**: Ah, I've often wondered whether about distant projections of ellipses are still ellipses. Thanks.  
> **username_1**: Yes, I think you can show that an affine projection of a quadratic form is still quadratic.  

---

