# Sgr. A* - Mathematical calculations

Hey guys

First real thread besides my introductory thread.

I'm a Danish student in what is somewhat equivalent to my junior/senior year in the american school system. That was a small attempt at school system conversion.

I'm writing a project about the black hole Sgr. A* in the centre of our galaxy. I'll try to explain my situation the best I can:

The project: As mentioned I'm writing a school project. The grade is pretty influential on my average grade, so I'm interested in writing a good project. I'm already past half of my writing time.
Based on coordinates given in a seperate document, I've plotted the ellipse of star S2's movement around Sgr. A* which is assumed to be a black hole at the centre of the Milky Way. I've corrected for the tilt, so that I could calculate the true semi-major axis and I am currently doing a few other procedures in order to get the period, so I may apply Kepler's 3rd law in order to calculate the mass of the black hole.

My issue: I'm writing a few paragraphs on the properties of a black hole. I've included a little about the Schwarzchild radius, gravitational lensing, the theory of general relativity, the three externally observable properties which are the mass, angular momentum and the charge of the black hole in accordance with the No-hair hypothesis.

Since I'm writing in BOTH mathematics and physics, I'd like to perhaps include a few formulas and/or calculations that show understanding of what I've only described with words (the above-mentioned). The only mathematical calculation I've done for this section, is calculating the Schwarzchild radius of the Earth, which is very basic. My maths is limited to what I consider basic calculus understanding and my physics knowledge is also limited compared to a few of the guys on this site, that obviously understand physics and astronomy in great depth.

By the way I also plan on writing about the accretion disc, perhaps the jets and the singularity - I still have to fully grasp those parts yet through a bit of reading.

Does anyone have (relatively simple) calculations or formulas, that I could apply to my section about black holes generally, that would demonstrate a higher degree of understanding? Links to sources are also very much appreciated.

Much appreciated!

Alexander

## Answers and Replies

Matterwave
Science Advisor
Gold Member
This seems like quite a broad subject. The center of our Milky Way is not only a black hole, but a super massive black hole, with mass ~few million solar masses (which I assume you will get soon after using Kepler's law). You might want to calculate the Schwarzschild radius for such a massive black hole. Another interesting calculation would be to look at the tidal gravity at the event horizon using a Newtonian approximation (a GR calculation might be a bit math intensive) and seeing if, for such a massive black hole, one would have problems crossing the event horizon.

Hey Matterwave,

Thank you very much for your reply.

The subject seems broad indeed, but I've been given a set of research questions for this project, that confines and narrows down the task at hand.
I've already been given information on the SMBH in Sgr A*, so its mass and other properties are known to me from the research documentation of R. Schrödel et al called Stellar dynamics in the central arcsecond of our galaxy. I'm just supposed to try to recreate the calculations of the mass of the black hole, that they've calculated and compare my calculated mass with their results. That is the main aim - there are several other tasks as well, which are more mathematically related to the definition of ellipses, their conic definition, their parametrical definition and other properties.

I'll try to ask my teacher about the Newtonian approximation or look it up. I'm not familiar with it, but it seems comprehendable for me.

Again, thank you very much for taking your time to reply - much appreciated!

PS: I apologize for my grammatical errors - English is not my main language. But they may also be a product of slightly fast typing.

Matterwave
Science Advisor
Gold Member
By "Newtonian approximation" I mean just use Newton's law of gravitation to find the tidal forces, not Einstein's General Relativity.

stevebd1
Gold Member
Here are some old library entries that have been turned into regular posts which provide an introduction to various equations-

Radius of a black hole
What is a photon sphere
Black hole thermodynamics
What is irreducible mass
What is frame dragging

Here's a paper/presentation that provides a decent summary of various equations-
Solutions of Einstein’s Equations & Black Holes

Here's a paper that measures the spin of sgr A*-
Measuring spin of a supermassive black hole at the Galactic centre

For more in-depth information, I would suggest 'Compact Objects in Astrophysics' by Max Camenzind which covers black holes and accretion disks.

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Hey stevebd1,

Thank you very much for your answer. I'll take a look at the links and books you've recommended. A lot of it seems very relevant and useful. I appreciate that.

Alexander