B Orbital parameters of stars orbiting Sagittarius A*

[jimbo007]

I was going to try and do an animation of stars orbiting Sagittarius A* but can't seem to find any useful data for it. The Wikipedia page has some data https://en.m.wikipedia.org/wiki/Sagittarius_A* and was trying to reconcile this with https://en.m.wikipedia.org/wiki/Orbital_elements since I haven't used orbital parameters/elements before. I was expecting 6 parameters but only seems to be 4 at best. Is anyone able to decrypt this data for me or point me to a location which has all the parameters?

Thanks in advance!

 

[tony873004]

http://orbitsimulator.com/gravitySimulatorCloud/mwbh.html
Here's my attempt at an animation. You can pause and scroll down through the list to view their state vectors.

 

[Janus]

I was going to try and do an animation of stars orbiting Sagittarius A* but can't seem to find any useful data for it. The Wikipedia page has some data https://en.m.wikipedia.org/wiki/Sagittarius_A* and was trying to reconcile this with https://en.m.wikipedia.org/wiki/Orbital_elements since I haven't used orbital parameters/elements before. I was expecting 6 parameters but only seems to be 4 at best. Is anyone able to decrypt this data for me or point me to a location which has all the parameters?

Thanks in advance!

Here is one of the papers from which some of the orbital parameters given were obtained, it has a more complete table of parameters.
http://iopscience.iop.org/article/1...68AD8BD224A45B380D7.c2.iopscience.cld.iop.org
(if you click on the entry in the Ref column, it will open a box with a link to the reference.)

 

[jimbo007]

Excellent thanks tony873004 that animation should it extremely easy for me to do the animation since it has the nice Cartesian coordinates. Were you able to do that animation using the orbital parameters in this thread or was it already done?

Thanks Janus that table is looking better. Looks like it provides t_0 instead of M_0, don't suppose you know how to convert from t_0 to M_0 (the mean anomaly at epoch)?

Is there any more data that exists for additional stars? 6 is a very good start, would be nice to have a few more if possible. I have seen some animations which include more stars but not sure if their data is hidden from public access or not.

 

[tony873004]

I made it years ago for my original Gravity Simulator software. The version you see is the newer "browser" version. It uses the same data. So the data in my sim is at least 6-7 years old. It was orbital elements data. I converted it to Cartesian. Here's an online calculator I made that converts between the two:
http://orbitsimulator.com/formulas/OrbitalElements.html

Since my simulation is a web page, you can view the source and get all the cartesians in one glance. Search the code for "objMass[1] ="

There's probably more recent data. The Wikipedia link doesn't give enough data to make what I made. Over the weekend I'll see if I can find more complete and current data.

edit: I just saw the table in Janus' reply. Let me know if you need help making cartesians out of that.

 

[jimbo007]

There's probably more recent data. The Wikipedia link doesn't give enough data to make what I made. Over the weekend I'll see if I can find more complete and current data.

Sounds great if you could find more complete and current data.

Yes it does look like I need help converting to Cartesian coordinates. My main trouble is with the mean anomaly which isn't provided in the table.
We have a\text{, } e\text{, } \omega\text{, } \Omega\text{, } i but no M_{0} = M(t_{0}). They have provided t_{0} but not sure if one can convert that to M_{0} with the other data provided

 

[tony873004]

M0 = 360 - (360 * (t0 - D) / T)
D is the date of your animation.
T is the period of the object's orbit.
Make sure that T, t0 and D are all in the same units.

Edit: missed a parenthesis. fixed now.

 

[jimbo007]

That's beautiful thanks tony873004. If I understand correctly, using the information below for the S1 star

we have that
<br /> M_{0} = 360 - \frac{360}{T} (t_{0}-P)<br />
using D=P
which means
M_{0} \approx 360 - \frac{360}{94.1}(2002.6 - 2016.25)

Does that sound reasonable?

 

[tony873004]

It looks good. A little background...

The 6 main orbital elements describe the size, shape, and orientation of the orbit, as well as the object's position on that orbit at a particular time.

Sometimes mean anomaly is used for position. Sometimes time since (or to) periapsis passage is used.

So for example, imagine an orbit with a period of 60 minutes. If t0 is 15 minutes, that means that 1/4 of an orbit ago it was at periapsis.
So 15/60 or 1/4 of an orbit, which is 360 degrees is 90 degrees.
360-90 = 270. So the M is 270 degrees.

Notice I considered D to be 0 for simplicity. If D is after t0, you get a negative number, which is fine. You'll get an angle greater than 360, so you have to subtract off 360 to get back in the 0-360 range.

Note that they give semi-major axis in milli arcseconds. This needs to be converted to a distance in order for the conversions from orbital elements to cartesian coordinates to make sense. We need to know the distance from the Earth to the black hole to make the conversion.

from the paper

...If not specified otherwise (§ 3.3), we adopt throughout this paper R0 = 8 kpc for the Galactic center distance...

...The updated estimate of distance to the Galactic center from the S2 orbit fit is R0 = 7.62 ± 0.32 kpc...

So therefore, looking at the semi-major axis for S1 = 0.412 arcsec and converting to meters gives:
8000 parsecs x 3.08568025E+16 meters / pc * sin(0.412 / 3600) = 493075008240113 meters

As a double-check, we can plug into the period formula (P =2pi sqrt(a^3/(GM)) and convert to years.
To do this, we need to know the mass of the black hole.

...The best-fit central mass9 for an assumed distance of 8 kpc is (4.06 ± 0.38) × 10^6 (solar masses)...

...The updated estimate of distance to the Galactic center from the S2 orbit fit is R0 = 7.62 ± 0.32 kpc, resulting in a central mass value of (3.61 ± 0.32) × 10^6 (solar masses)...

...Measurements of stellar velocities and (partial) orbits have established a compelling case that this dark mass concentration is a massive black hole of about (3[PLAIN]http://cdn.iopscience.com/icons/EJ4/AJ/ucp-icons/ndash.gif4) × 10^6 (solar masses)...

I think they made a typo in that middle sentence. I think they meant the distance from Earth, not S2. That would put S2 about as far away as Earth.

The period formula is P = 2pi * sqrt(a^3/(GM))
This gives seconds. Dividing by 3.155 x 10^7 gives years.

2*pi*sqrt((8000*3.08568025E+16*sin(.412/3600))^3 / (6.67e-11*3.61e6*1.99e30))/3.155e7

gives 99.6 years. Using 4 instead of 3.61 gives 94.6 years, which is in rough agreement with the period given for S1 in the data you quoted.

You can copy and paste the above formula into Google,and play with various numbers. It outputs years.

You've now got everything properly converted for use in the calculator: http://orbitsimulator.com/formulas/OrbitalElements.html

I'm looking forward to seeing your animation! Let me know if you need help.

 

[jimbo007]

Thanks for the detailed information, you probably saved me a few weeks by mentioning that semi-major axis trap.

What I did was use the above info to calculate the initial position and velocity of the star S1 and then used Euler's method to calculate subsequent positions using 3.6 million solar masses as the mass of the black hole to calculate the new accelerations.

At least a few days from creating the animation but think I am on the right track now, will send a video once I have done something half decent but don't expect to be blown away.

Don't suppose you have found any more star data by chance?

 

[tony873004]

Footnote #26 from the Wikipedia article is the best I could find. It gives you your semi-major axis in arcseconds and in AU.
My sim has one more object than the ones listed in their table (SO-5). So it doesn't look like there's much more info to be had.

 

[jimbo007]

Unless I have miscalculated somewhere, the S13 star doesn't look to be in a relatively stable orbit. Based on that data it will fall into the black hole (in my simulation it gets flung into infinity after a few seconds). Noted that the uncertainty in the measurements for that particular star they might as well have just guessed the parameters.

At some point it comes to within 140000000000 metres of the black hole and at that distance the escape velocity is 2.75 times the speed of light which is fairly difficult to achieve. Unfortunately I don't think you have that star in your animation so can't compare with yours.

Since there is a lack of data for stars orbiting black holes and since 5 stars isn't very exciting I was thinking of doctoring up some pretend orbital elements using the Sagittarius black hole as the central body. How easy is it to mock up some orbital elements that would be stable?

 

[Janus]

Unless I have miscalculated somewhere, the S13 star doesn't look to be in a relatively stable orbit. Based on that data it will fall into the black hole (in my simulation it gets flung into infinity after a few seconds). Noted that the uncertainty in the measurements for that particular star they might as well have just guessed the parameters.

At some point it comes to within 140000000000 metres of the black hole and at that distance the escape velocity is 2.75 times the speed of light which is fairly difficult to achieve. Unfortunately I don't think you have that star in your animation so can't compare with yours.

Since there is a lack of data for stars orbiting black holes and since 5 stars isn't very exciting I was thinking of doctoring up some pretend orbital elements using the Sagittarius black hole as the central body. How easy is it to mock up some orbital elements that would be stable?

If you are using ~4e6 solar masses for the black hole mass, that puts the event horizon at ~ 1.18e10 m, the minimum distance you gave above is 1.4e11 m or over 10 times the distance. This is well outside event the photon sphere and no where near where the escape velocity is even c. (it would be roughly 32% of c at that distance.)

Using the 4e6 solar mass value and the given period of 36 years, I get an semi-major axis of 2.6e14 m or ~1738 AU, which is close to the 1750 AU in the article you first linked to.
With an e of 0.395, this puts the periapis at 1.57e14 m or at over over 13,000 times the event horizon radius and at this distance, the escape velocity will be 0.88% of c.

Even the star listed just below S14 in the original article you linked to, with the same period and a much higher eccentricity of 0.974, only gets to within 6.7e12 m or 568 times further than the event horizon of the BH.
The orbital velocities at closest approaches for these stars would be 2.2e6 m/s and 1.3e7 m/s or 0.73% and 4.3% of c respectively.

 

[tony873004]

I think you have a calculation error somewhere. It's easy to make mistakes with this stuff. You're off by almost exactly 3 orders of magnitude. Perhaps you're entering meters where you should be doing km?

Like Janus said, a semi-major axis of 1750 with an ecc of 0.395 puts the star's periapsis much higher than 14... (~1 AU) number you mentioned.

An eccentricity of 0.395 on a 1750 AU orbit simply means that at apsis it is 39.5% greater than 1750, and at periapsis it is 39.5% less than 1750. So 1059 AU is the closest it should get.

Can you post your calculations? If you're using the online calculator, can you screenshot it, or give the values in each box?

 

[jimbo007]

I was hoping it would be a simple fix of adding .0 to the integer values given in the table for S13 as it is the only star with integer values but doesn't look to be the case unfortunately, especially since all other stars are producing orbits.

I am using 3.61 \times 10^6 M_{\odot} \approx 7.18029 \times 10^{36} \text{kg} as the black hole's mass
<br /> a=0.219 \text{ arcsec} =2.62095696128 \times 10^{14} \text{metres}\\<br /> e=0.395\\<br /> P=36\\<br /> t_{0} = 2006.1\\<br /> M=360-\frac{360}{P} (t_{0} - 2016.25) =1.77150919077 \text{ radians}\\<br /> \omega = 250.0 \text{ deg} = 4.363323 \text{ radians}\\<br /> \Omega = 100.0 \text{ deg} = 1.745329 \text{ radians}\\<br /> i = 11 \text{ deg} = 0.191986 \text{ radians}<br />

Eccentric anomaly:
Solve the following equation for E using Newtons method
f(E) = E - \mathit{e} \sin E - M = 0
gives
E = 2.1103871826

True anomaly:
<br /> \begin{eqnarray*}<br /> \nu &amp;=&amp; 2 \cdot \text{arctan2} (\sqrt{ 1+\mathit{e} } \sin\left( \frac{E}{2}\right),\sqrt{1-\mathit{e}}\cos\left( \frac{E}{2}\right)) \\<br /> &amp;=&amp;0.714430095777<br /> \end{eqnarray*}<br />
where
<br /> \text{arctan2}(x,y)=<br /> \left\{\begin{array}{cc}<br /> \arctan \frac{y}{x},&amp;\mbox{ if } x \gt 0\\<br /> \arctan \frac{y}{x}+\pi, &amp;\mbox{ if } y\geq 0, x &lt; 0\\<br /> \arctan \frac{y}{x}-\pi, &amp;\mbox{ if } y &lt; 0, x &lt; 0\\<br /> \frac{\pi}{2}, &amp;\mbox{ if } y &gt; 0, x = 0\\<br /> -\frac{\pi}{2}, &amp;\mbox{ if } y &lt; 0, x = 0\\<br /> \text{undefined}, &amp;\mbox{ if } y= 0, x = 0\\<br /> \end{array}<br /> \right.<br />

Distance to black hole:
<br /> \begin{eqnarray*}<br /> r_{c} &amp;=&amp; a(1-\mathit{e} \cos E) \\<br /> &amp;=&amp; 3.15286729 \times 10^{14} \text{metres}<br /> \end{eqnarray*}<br />

Calculate x-coordinate and y-coordinate in random frame (dont think z is necessary for this purpose)
<br /> \begin{eqnarray*}<br /> x_{rand} &amp;=&amp; r_{c} \cos \nu \\<br /> &amp;=&amp; 2.38188643 \times 10^{14}<br /> \end{eqnarray*}<br />
<br /> \begin{eqnarray*}<br /> y_{rand} &amp;=&amp; r_{c} \sin\nu \\<br /> &amp;=&amp; 2.06571760 \times 10^{14}<br /> \end{eqnarray*}<br />

Transform to a different random frame treating the black hole at the origin.
<br /> \begin{eqnarray*}<br /> x &amp;=&amp; x_{rand}(\cos(\omega)\cos(\Omega) - \sin(\omega)cos(i)\sin(\Omega)) - y_{rand}(\sin(\omega)\cos(\Omega)+\cos(\omega)\cos(i)\sin(\Omega)) \\<br /> &amp;=&amp;-5.61288464 \times 10^{14}<br /> \end{eqnarray*}<br />

I think I will stop there as there is a million places the calculation went wrong and chances are they happened in one of the above steps

 

[jimbo007]

I just dug up an old dusty book from the garage and found it gave the following for S1
<br /> \mathit{a} = 20.5 \text{ mpc} \\<br /> \mathit{e} = 0.496 \\<br /> \mathit{i} = 120.82 \text{ deg} \\<br /> \Omega = 341.61 \text{ deg} \\<br /> \omega = 115.3 \text{ deg} \\<br /> P = 132 \text{ years}<br />
The trouble with this data (besides being different from the values given for S1 in post 8) is we don't have M or t0 so how would one calculate M for this different set of data?

 

[tony873004]

They might simply be describing the size, shape and orientation of the orbit without regard as to where in this orbit the star is.

On you previous post, I'm not familiar with that method, but even trying to follow it, I get different answers than you, specifically with E and x.
See here. This will let you play with the numbers. Press "Run" under the code.
http://orbitsimulator.com/code/tdunn/code01.html?sag.txt

 

[jimbo007]

I got those instructions from:
https://downloads.rene-schwarz.com/...Orbit_Elements_to_Cartesian_State_Vectors.pdf

Thanks for the new calculator I will play with the numbers and try to see what's going on.

For the new data can I just set M=0 for all stars?

 

[tony873004]

Yes. You won't be positioning the star along the orbit properly without M or t0, but your orbit will be fine.

 

[tony873004]

I made a mistake above with E. I'll fix it in a minute.

on second thought, that's too much algebra for me tonight to isolate E. from that equation!

 

[Janus]

By gleaning data from a couple of the references listed in the OP and feeding them into the orbit simulator I have on my computer (Gravsim), I created the following Sim of a number of the stars orbiting Sagittarius A. Unfortunately, the Sim has no way to generate videos directly, so I had to resort to recording it from my monitor with a digital camera and saved the recording to youtube.(so excuse the poor quality)
It starts with the motions of the stars(highly sped up of course), and then finishes by tracing out the orbits.

 

[tony873004]

That's pretty good for a camcorder aimed at a screen!
I'm surprised you didn't get a flickering strobe effect.
Unless you go full screen, the res is pretty good.
Did you use this one: http://www.grav-sim.com ?
You can download Camtasia to make screen recordings without the camcorder. It's about $200 but they let you use it for free for 30 days.
There's other free ones out there, but I never had much luck with them. I broke down and spent the $200.

 

[Janus]

That's pretty good for a camcorder aimed at a screen!
I'm surprised you didn't get a flickering strobe effect.
Unless you go full screen, the res is pretty good.
Did you use this one: http://www.grav-sim.com ?
You can download Camtasia to make screen recordings without the camcorder. It's about $200 but they let you use it for free for 30 days.
There's other free ones out there, but I never had much luck with them. I broke down and spent the $200.

This is the one I have: http://www.orbitsimulator.com/gravity/articles/what.html
I normally don't have much need to capture video from my computer. Most animations I due are generated frame by frame with Pov-Ray, which I can then assemble into a video. I could have done the same with this one, but it would have been a a lot of work and a lot of time rendering enough frames to show several orbits. It was just simpler to use the gravity simulator.

 

[jimbo007]

Nice camera work there Janus - you did well to extract data for 10 stars.

tony873004 I take it the calculation for E is still cactus in your code01.html calculator? I thought my E=2.11 was pretty close

 

[tony873004]

tony873004 I take it the calculation for E is still cactus in your code01.html calculator?

I fixed E. Still get a different x than you.
http://orbitsimulator.com/code/tdunn/code01.html?sag.txt

This is the one I have: http://www.orbitsimulator.com/gravity/articles/what.html

That's the one I wrote.

Unfortunately, the Sim has no way to generate videos directly

If you have the latest version you can make it take a series of screen shots which can then be assembled into an animated GIF or perhaps to YouTube to animate. (see post 47: http://www.orbitsimulator.com/cgi-bin/yabb/YaBB.pl?num=1176774875/45#45.)

 

[Janus]

Using my normal POV-Ray method, here's an animation that gives a more three dimensional perspective of the respective orbits.

 

[tony873004]

That's awesome!

 

[jimbo007]

Ok I think I found the problem. Looks like there were 2 issues, the first one was user error on my behalf. That x-value discrepancy we were having was because I accidentally gave you the value for S14 rather than S13. The other issue was in the jungle of + and - signs I got one of them around the wrong way for the y coordinate calculation.

I just spent 30 mins trying to upload my animation but it blew up. Will try again tomorrow before Janus upstages me with more of his fancy animations.

 

[jimbo007]

After great difficulty I managed to get something uploaded to youtube.

Below is my cubes/stars orbiting the black hole



Not quite as good as Janus's but I plan to improve in subsequent simulations

 

[Loren]

You might also want to check out Astrosynthesis V3.
I think there is a trial version, too.
I used it to create some fictional star systems for a novel I am writing and it is an interesting program that allows you to modify and create a large number of variables.
There's nothing like DIY, but it might be an interesting comparison with your work.

 

[tony873004]

Nice! I like glow of the stars.

 

[jimbo007]

Thanks for the link Loren I will check it out and yes would be to compare. I am going to try and create a more 3 dimensional perspective like Janus's and get the camera zoom around with some smoke/fog effects if I can ever figure out how to get it working. The smoke/fog is turning out to be a big CPU hog so have to try to work out some shortcuts

 

[m4r35n357]

I made it years ago for my original Gravity Simulator software. The version you see is the newer "browser" version. It uses the same data. So the data in my sim is at least 6-7 years old. It was orbital elements data. I converted it to Cartesian. Here's an online calculator I made that converts between the two:
http://orbitsimulator.com/formulas/OrbitalElements.html

Since my simulation is a web page, you can view the source and get all the cartesians in one glance. Search the code for "objMass[1] ="

There's probably more recent data. The Wikipedia link doesn't give enough data to make what I made. Over the weekend I'll see if I can find more complete and current data.

edit: I just saw the table in Janus' reply. Let me know if you need help making cartesians out of that.

Pardon the bump, but I was recently involved in another thread (which I now can't find) where someone else was asking for just this information and it did not seem to be available. I stumbled across this thread by accident!
I have a simulator suite that does black hole orbits and an (unadvertised) Newtonian n-body simulation, and the latter takes a list of Cartesian state vectors as its input (I already have data for the Solar System from NASA's Horizons facility).
Now I can start to look at the galactic centre . . .
[EDIT 1] OK I see it is Solar System data, but I hope there is galactic centre stuff somewhere in this thread if I dig deep enough.
[EDIT 2] Now looking at your simulator, precisely how does one pause it?

 

[tony873004]

[EDIT 2] Now looking at your simulator, precisely how does one pause it?

There is a [||] button on the left of the screen.

 

[m4r35n357]

I made a mistake above with E. I'll fix it in a minute.

on second thought, that's too much algebra for me tonight to isolate E. from that equation!

Ah, the only button I didn't click on for some reason ;) Thanks, will investigate now.
[EDIT] BTW do you have mass data for the bodies? If not I'll have to see if I can look it up somewhere.

 

[tony873004]

I don't have that data. I imagine you'll just have to approximate based on the color of the stars.

 

[m4r35n357]

Ah, sorry I didn't mean to post that, was just looking at the source! Yes I notice that only the BH seems to have mass. Does that mean you are doing some sort of n body Kepler analysis?

 

[m4r35n357]

Using my normal POV-Ray method, here's an animation that gives a more three dimensional perspective of the respective orbits.

Would it be possible for you to provide me a list of body masses, and a snapshot of their coordinates and velocities? I'd love to try this out.

 

[Yukterez]

the orbit simulator I have on my computer (Gravsim)

What metric does that simulator use, Newtonian or relativistic? Orbits around black holes can look much different under Einstein than they would under Newton as shown in this thread where the same initial conditions give very different results for Newton and Einstein: Black hole orbits
Since it says n-body in the description I guess it is Newtonian, but maybe it would be better to simulate the orbits as test particles in Schwarzschild metric to get the right eccentricity and perihelion shifts. But that depends on which effect is stronger: the mutial attraction of the orbiting stars reative to each other or the relativistic effect of curved spacetime generated by the central black hole (which is only the case if the orbits get close enough so that the ratio rs/r is not neglible).

 

[m4r35n357]

But that depends on which effect is stronger: the mutial attraction of the orbiting stars reative to each other or the relativistic effect of curved spacetime generated by the central black hole (which is only the case if the orbits get close enough so that the ratio rs/r is not neglible).

I think bearing in mind the very short observational period we are working with, any effects due to particles outside about $10R_s$ to $100R_s$ would be "lost in the noise", so I'm comfortable with a Newtonian approach. Not that I feel we have much choice in the matter ;)

 

[Yukterez]

I think bearing in mind the very short observational period we are working with, any effects due to particles outside about $10R_s$ to $100R_s$ would be "lost in the noise", so I'm comfortable with a Newtonian approach.

When the observed velocity at the perihelion at 10rs is for example 0.2236068c there would be a notable difference between the Newtonian and the relativistic orbit:

So if the closest perihelion were at that distance it might be better to neglect the mutual attraction of the orbiting stars and threat them as test particles, but therefore take the relativistic metric of the black hole into account, but if the closest perihelion were at 100rs it's surely better to stay with Newton.

Not that I feel we have much choice in the matter

We could simulate it in Schwarzschild metric (if the simulations we already have aren't already)

 

[Yukterez]

I see at Wikipedia that the lowest orbit has a semimajor axis of around 1000 Au while the rs of the black hole is only 0.1 Au. In that case there is no need for a relativistic simulation and Newton does the job.

 

[Janus]

I see at Wikipedia that the lowest orbit has a semimajor axis of around 1000 Au while the rs of the black hole is only 0.1 Au. In that case there is no need for a relativistic simulation and Newton does the job.

With the listed eccentricity, I get a precession of apsides of ~ 0.17 degrees per orbit. With a 14.53 year orbit, this works out to ~30500 years for the apsides to rotate a full 360 degrees. ( Compare this to the 43 seconds of arc per century precession for Mercury, which would take ~3,000,000 years to complete a full rotation.)

 

[m4r35n357]

When the observed velocity at the perihelion at 10rs is for example 0.2236068c there would be a notable difference between the Newtonian and the relativistic orbit:

Agreed, but my point was we don't have enough actual data to check our predictions accurately against (order of 15 year orbits).

So if the closest perihelion were at that distance it might be better to neglect the mutual attraction of the orbiting stars and threat them as test particles, but therefore take the relativistic metric of the black hole into account, but if the closest perihelion were at 100rs it's surely better to stay with Newton.We could simulate it in Schwarzschild metric (if the simulations we already have aren't already)

Yes, I suppose we could use the potential (with extra term due to GR) in a n-body simulation. We would also need to consider interactions between stars that pass nearby each other around the perihelion (we'd need to use that potential for all the stars), so I'd call that a modified Newtonian analysis really.
Then what if the black hole is spinning?

 

[jimbo007]

Interesting animations by Yukterez, unfortunately it's a bit beyond me to calculate relativistic orbits at the moment

Below is my 2nd attempt at a simulation using more stars