Estimating Mass of Central Object Using S2's Orbit Motion and Kepler's 3rd Law

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In summary, the conversation revolved around using Kepler's 3rd law in solar units to find the mass of a central object based on the orbit motion of star S2. The participants were given a hint to use the arcsec relation and read the radius of the orbit from the graph, with a suggested estimate of 0.1" for the semi-major axis. There was also a discussion about the possible tilt of the orbit and whether it needed to be corrected for in order to accurately estimate the mass of the central object.
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I want to understand how one can go about estimating the mass of a central object given the a graph the orbital motion of S2, or any given star for that matter.
During one of my class lectures, we were shown a graph of star S2's orbit motion (shown at the bottom of this post) and was told to try and figure out how to find the mass of the central object using Kepler’s 3rd law in solar units, as shown below:
1678026251763.png


We were also given a hint to use the arcsec relation and read the radius of the orbit from the image, as shown below:
1678026338495.png


From my notes, it looks like we had to use the declination value of 0.1, but I still don't understand how exactly we got to that point. Does anyone happen to know why this is and how to generally use these types of graphs to estimate the mass of a central object?

1678026170065.png
 
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The "radius" in the formula is actually half the semi-major axis of the ellipse (which obviously doesn't have a constant radius). Based on the diagram, 0.1'' might be a good estimate for the semi-major axis.
 
  • #3
As @pasmith said, reading from the diagram, the semimajor axis is about 0.1". This is the θ in your second formula. It is not a declination, it is the angular size of the orbit.
 
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Related to Estimating Mass of Central Object Using S2's Orbit Motion and Kepler's 3rd Law

What is Kepler's 3rd Law and how is it used to estimate the mass of a central object?

Kepler's 3rd Law states that the square of the orbital period (T) of a planet is directly proportional to the cube of the semi-major axis (a) of its orbit, mathematically expressed as T^2 ∝ a^3. When applied to a system where a smaller object orbits a much larger central object, this law can be used to estimate the mass of the central object. By measuring the orbital period and the semi-major axis of the orbit, one can rearrange the law to solve for the mass of the central object.

What data is required to estimate the mass of the central object using S2's orbit?

To estimate the mass of the central object using S2's orbit, you need the orbital period (T) of S2 and the semi-major axis (a) of its elliptical orbit. These measurements can be obtained through long-term observations of S2's position and velocity as it orbits the central object, which in this case is the supermassive black hole at the center of our galaxy.

Why is S2's orbit particularly useful for estimating the mass of the central black hole in our galaxy?

S2's orbit is particularly useful because it is one of the closest stars to the supermassive black hole at the center of our galaxy, known as Sagittarius A*. S2 has a relatively short orbital period of about 16 years, allowing for complete orbital observations within a human lifetime. Its high velocity and close proximity to Sagittarius A* provide precise data that can be used to accurately estimate the mass of the black hole.

How do astronomers measure the semi-major axis and orbital period of S2?

Astronomers measure the semi-major axis and orbital period of S2 by tracking its position and velocity over time using high-resolution telescopes and instruments, such as the Very Large Telescope (VLT) and the Keck Observatory. By plotting S2's trajectory and fitting it to an elliptical orbit, they can determine the semi-major axis. The time it takes for S2 to complete one full orbit around Sagittarius A* gives the orbital period.

What is the estimated mass of the central object, Sagittarius A*, based on S2's orbit?

Based on observations of S2's orbit, the estimated mass of Sagittarius A*, the supermassive black hole at the center of our galaxy, is approximately 4 million times the mass of the Sun. This estimation is derived from applying Kepler's 3rd Law to the measured orbital parameters of S2.

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