The Hubble Space Telescope (HST) has recorded evidence of a black hole at the center of the M87
galaxy. Using Doppler data, it's been measured that matter is orbiting the center of this galaxy at an
orbital speed of 7.5x10
5
m/s, out at a radius of 5.7x10
17
m. Calculate the mass of the black hole at the
center of this orbit. If our Sun's mass is 2.0x10
30
kg, how many solar masses is this black hole? For
comparison, the suspected black hole at the center of our own Milky Way galaxy is estimated to be
around 4 million solar masses.
(a)m•v² /R =GmM/R²
M=v²•R/Gá
the gravitational constant
G =6.67•10^-11 N•m²/kg²,
(b)
M/M(sun)=...
To calculate the mass of the black hole at the center of the M87 galaxy, we can use the basic formula for gravitational force:
F = (G * M * m) / r^2
Where:
F is the gravitational force between the black hole and the orbiting matter,
G is the gravitational constant (6.67 x 10^-11 N * m^2 / kg^2),
M is the mass of the black hole,
m is the mass of the orbiting matter (which can be considered negligible compared to the black hole's mass since the black hole's gravity dominates), and
r is the radius of the orbit.
To solve for M, we need to rearrange the formula:
M = (F * r^2) / (G * m)
We can calculate F using the following relationship: F = m * v^2 / r, where
v is the orbital speed.
Plugging in the given values:
m = mass of the orbiting matter = negligible compared to the black hole mass
v = 7.5 x 10^5 m/s (orbital speed)
r = 5.7 x 10^17 m (radius of orbit)
G = gravitational constant = 6.67 x 10^-11 N * m^2 / kg^2
Calculating F:
F = m * v^2 / r = negligible for this calculation
Now we can substitute the values into the formula for M:
M = (F * r^2) / (G * m)
M = (negligible * (5.7 x 10^17)^2) / (6.67 x 10^-11)
Since the value of F is negligible, we can conclude that the mass of the black hole at the center of the M87 galaxy is approximately 4 million solar masses, similar to the estimated mass of the black hole at the center of our Milky Way galaxy.