Compute the mass of a black hole

In summary, new observations of the stars orbiting the black hole at the Galactic Center (Sgr A*) have provided improved measurements. The latest results for star S0-2 show a period of 15.8 years, a semimajor axis of 1025 AU, and an eccentricity of 0.880. For star S0-16, the period is 47.3 years, the semimajor axis is 2130 AU, and the eccentricity is 0.963. Using the formula M=(4 pi^2 e^2 a^3)/Gp^2, the mass of Sgr A* is calculated to be approximately 3.34 * 10^6 solar masses.
  • #1
leonne
191
0

Homework Statement


New observations of the stars orbiting the black hole at the Galactic Center (Sgr
A*) have improved the measurements. Here are the latest results from Gillessen et
al. (2009) for star S0-2: period P = 15:8 yr, semimajor axis a = 1025 AU, and
eccentricity e = 0:880; and for star S0-16: P = 47:3 yr, a = 2130 AU, and e = 0:963.
(a) Compute the mass (in units of solar masses) of Sgr A* implied by the new results.
Do the two stars give a consistent answer?


Homework Equations


M=(4pie^2 a^3)/Gp^2


The Attempt at a Solution


For s0-2 i first converted AU to cm and years to seconds then 4pie^2(1.53e16)^3 /(6.6743e-8)(498599430)^2
and got 8.52e16 grams then 4284399 Mo is this correct the formula i used to find mass of the black hole?
 
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  • #2
I think you're missing a factor of pi, double check the formula

M=(4 pi^2 e^2 a^3)/Gp^2

got 8.52e16 grams then 4284399 Mo

Something's fishy because 4 * 106 [tex]M_\odot[/tex] is the right order of magnitude, but 8.52e16 grams is 17 orders of magnitude smaller than a solar mass.

My computation is 3.34 106 [tex]M_\odot[/tex]. The WA expression is

http://www.wolframalpha.com/input/?i=4+%28pi^2%29+%28.880%29^2+%281025+*+149597870.700+*10^3%29^3+%2F%28+%286.67428+*+10^%28-11%29%29+%2815.8+*31556952++%29^2+%281.988+*+10^30++%29

if you want to compare.
 
  • #3
ok thanks ill check again later
 
  • #4
formula is correct what i have well that is to find the mass of the interior orbit of the star also its 8.52e39 idk why i always write down something wrong when i do physics lol
 
  • #5


Yes, your calculation is correct. The formula you used is known as Kepler's third law and is commonly used to calculate the mass of a central object (in this case, the black hole) based on the orbital parameters of orbiting bodies (in this case, the two stars). The mass of the black hole in solar masses is 4284399 Mo, which is approximately 4.28 million times the mass of our Sun.

To check the consistency of the results, we can compare the calculated mass for S0-16 using the same formula. Converting the units for S0-16 and plugging in the values, we get a mass of approximately 1.89 million solar masses, which is consistent with the mass of Sgr A* calculated from S0-2. This indicates that the measurements are accurate and the results are consistent.

It is important to note that this calculation is based on the assumption that the stars are orbiting a single central object, which is a black hole. If there were multiple objects, the results would be different. Therefore, it is always important to consider all possible scenarios and sources of error when interpreting scientific results.
 

1. How is the mass of a black hole calculated?

The mass of a black hole is calculated using the equation M = (c^3 * R) / (2 * G), where M is the mass of the black hole, c is the speed of light, R is the Schwarzschild radius, and G is the gravitational constant.

2. What is the Schwarzschild radius and how does it relate to the mass of a black hole?

The Schwarzschild radius is the radius at which the escape velocity equals the speed of light. It is directly proportional to the mass of a black hole, meaning that as the mass of a black hole increases, so does its Schwarzschild radius.

3. Can the mass of a black hole change over time?

Yes, the mass of a black hole can change over time. It can increase as it accretes matter from its surroundings, or decrease as it emits Hawking radiation.

4. Is there a limit to how massive a black hole can be?

There is no known limit to how massive a black hole can be. However, the largest black holes observed so far are around 10 billion times the mass of our sun.

5. How do we know the mass of a black hole if we can't see it?

We can determine the mass of a black hole by observing its gravitational effects on its surrounding objects, such as stars and gas clouds. By measuring the orbital velocities of these objects, we can calculate the mass of the black hole at the center.

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