- 3652 - Calculating Milky Way’s blackhole. Black Holes are both simple and complex. We can calculate their mass, radius, lifetime, energy consumption using simple algebra. At the same time, their immense gravity causes space to bend, lengths to shorten, time to slow and mass to increase.
--------------------- 3652 - Calculating Milky Way’s Black Hole
- The theory of relativity comes into play big time. This review will make the simple calculations and you just have to realize things are more complicated than that.
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- Let’s start with a simple calculation for the mass of the Milky Way Galaxy. We observe the Large Megellanic Cloud which is a smaller galaxy orbiting the Milky Way in our Local Group of galaxies. The distance from the center of the Milky Way to the Megellanic Cloud is 160,000 lightyears. The orbital velocity of the Cloud is 300 kilometers / second.
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- The Megellanic Cloud is in an elliptical orbit but we can approximate things by assuming it is a circular orbit and the circumference of the orbit is 2* pi* radius, (2pi*r)
Since we assumed the orbit to be circular we can assume the orbital speed to be constant at 300 km/sec.
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-------------------- Velocity = distance / time, and the time is the period of the orbit.
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-------------------- The period of orbit = distance / velocity.
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-------------------- The period of orbit = 2*pi*radius / velocity.
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- Using Kepler’s and Newton’s laws of motion we know that the period of orbit squared equals the radius of orbit cubed. ( period^2 = radius^3). To get this into units we measure the formula becomes:-
---------------- period^2 = 4*pi^2 * radius^3 / Gravitational Constant * Mass
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--------------- 4*pi^2* radius^2 / velocity^2 = 4*pi^2 * radius^3 / Gravitational Constant * Mass
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---------------- Mass = radius * velocity^2 / Gravitational Constant
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- Using this formula we can calculate the mass of the Milky Way, given the Gravitational Constant = 6.67 * 10^-11 meters^3/ kiogram*seconds^2:
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---------------- Mass = 160,000 lightyears * (300 km/sec)^2 / 6.67 * 10^-11 meters^3/ kiogram*seconds^2
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----------------- Mass of the Milky Way = 2.042 *10^42 kilograms
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---------------- The mass of the Sun = 1.9891*10^30 kilograms
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---------------- Mass of the Milky Way = 10^12 Solar Mass
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- Therefore, the mass of the Milky Way could contain 1 trillion stars the size of our Sun. 1,000,000,000,000 solar mass. Or, 1,000,000,000,000 stars. But, how much of this mass is located in the Black Hole in the center or the Milky Way?
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- Only recently have we been able to see stars near the center of the Milky Way because the interstellar dust is blocking our view. Using infrared telescopes astronomers can see stars orbiting the center of the Milky Way. And, over the last 20 years they have measured several orbits.
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- One star was in circular orbit at a radius of 20 light days traveling at 1,000 km/second. Using this same formula we can calculate the mass of the Black Hole:
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---------------- Mass = radius * velocity^2 / Gravitational Constant
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---------------- Mass of Black Hole = 20 lightdays * (1,000 km/sec)^2 / 6.67 * 10^-11 meters^3/ kiogram*seconds^2
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----------------- Mass of the Black Hole = 7.77*10^38 kilograms
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---------------- The mass of the Sun = 1.9891*10^30 kilograms
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---------------- Mass of the Black Hole = 3.91*10^8 Solar Mass
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- The Black Hole at the center of the Milky Way has a mass of 391 million solar mass. That hardly puts a dent in the 1,000,000 million solar mass of the galaxy.
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- The Black Hole at the center of the Milky Way is inactive for the most part. It has very modest X-ray and Gamma ray emissions compared to other active galaxies. This means that our Black Hole is not supporting a massive rotating accretion disk at this time in its evolution.
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- X-ray flares have been observe from time to time indicating to astronomers that material is falling into the Black Hole. When things fall in to a Black Hole 10% to 40% of its matter gets radiated away before it disappears beyond the Hole’s event horizon.
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- The faster the Hole rotates the longer the material radiates, up to 40%. The slower the rotation, like expected with our Milky Way Black Hole the lesser the time for the material to radiate. We will therefore use 10% mass to energy radiation for our calculations.
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- The apparent brightness radiation is equal to the actual luminosity / 4*pi*radius^2. Astronomers us this formula to estimate the luminosity of the flares to be 10^40 watts in power. Watts is energy radiation per second. And, 10% of the mass is consumed in radiation.
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----------------- Therefore mass = 10 * Energy / speed of light^2. ( M = 10*E/c^2)
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---------------- Mass = 1.1 * 10^24 kilograms / second.
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--------------- Mass = 3.5 * 10^31 kilograms / year.
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---------------- Mass = 17 Solar Mass / year.
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- The Milky Way Black Hole is consuming 17 Solar Mass per year. If it was consuming this amount over its entire life time or 10 billion years then it would have consumed 170,000,000,000 Solar Mass up to this time. That is 170 billion stars the size of our Sun that have been consumed assuming it had a constant appetite.
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- Now that we have calculated the mass of the Black Hole we can easily calculate the radius of its Event Horizon.
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------------- Radius of Black Hole = 2 * G * Mass / c^2
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------------- Radius of Black Hole = 2 * 6.67 * 10^-11 meters^3/ kiogram*seconds^2* 7.77*10^38 kilograms / (3*10^8 m/s)^2
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--------------- Radius of Black Hole = 11.52 * 10^11 meters
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--------------- One Astronomical Unit = 1.496 * 10^11 meters
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--------------- Radius of Black Hole = 7.7 Astronomical Units, an AU is the distance from the Earth to the Sun, about 93 million miles. The Black Hole is only about 15 AU across.
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- Also with the mass of the Black Hole we can calculate the lifetime of the Black Hole:
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----------- Lifetime = 10,240 * pi^2 * G^2 * Mass^3 / h * c^4
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----------- Lifetime = 1.01065 * 10^5 * (6.67 * 10^-11 meters^3/ kilogram*seconds^2)^2 * ( 7.77 * 10^38 kilograms)^3 / (6.63 * 10^-34 kg*m^2/sec )* (3*10^8 m/sec)^4
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----------- Lifetime = 39.28*10^99 seconds
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------------ Lifetime = 12.45*10^92 years
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- Not to worry, this Black Hole in the middle of our galaxy is going to live for a long time.
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- So, with simple algebra we calculated the mass, radius, energy consumption, and lifetime of the Black Hole at the center of our galaxy. Go Figure?
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August 16, 2022 Calculating Milky Way’s Black Hole 3652
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--------------------- --- Tuesday, August 16, 2022 ---------------------------
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