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PM2017 wrote:A student at West High is angry he lost the tryouts for astronomy to PM2017. For this reason, he decides to throw PM2017's laptop into a black hole and seeing as he is actually better than PM2017 (and was unfairly disfavoured!) somehow observes that the laptop is, for some reason a perfect blackbody, and is normally blue (assume 470.00 nm). As the laptop passes through the event horizon, the student observes that the laptop seems to have a wavelength of 933.89 nm.

The student recalls from his studying that the mass of a black hole and the radius of its event horizon are related linearly. He dejectedly realizes that he can now use the wavelength data to find this relationship, but is too depressed to do so. Can you find the relationship between the mass of the black hole, in solar masses, and radius of a black hole, in kilometers?

Assume that G = 6.67e-11, c = 2.99e8, and a solar mass is 2.00e30 kg.

Hint: (Click to Show Hidden Text)

V = sqrt((2*G*M)/R)

V = sqrt((2*G*M)/R)
Since you are assuming that a linear relationship exists between mass and radius, substitute M*k for R.
V = sqrt((2*G*M)/(M*k)) = sqrt((2*G)/k)
vrec = λapp/λtrue - 1
vrec = c * (933.89nm / 470nm - 1) ~ 2.95*10^8 m/s
2.95*10^8 m/s= sqrt((2*G)/k)
vrec^2 = 8.71*10^16 m^2/s^2 = (2*G)/k
8.71*10^16 m^2/s^2 = (1.33*10-10 kg m^3 ^-2)/k
8.71*10^16  = 1.33*10^-10 m/kg / k
k = 1.33*10^-10m/kg/8.71*10^16 ~ 1.53*10^-27 m/kg
1.53*10^-27 m/kg * 1km/1000m * 2.00*10^30kg/1 Msol 
~ 3.06 km/Msol
R = 3.06 km/Msol * M

Knyte_Xjn wrote:

PM2017 wrote:A student at West High is angry he lost the tryouts for astronomy to PM2017. For this reason, he decides to throw PM2017's laptop into a black hole and seeing as he is actually better than PM2017 (and was unfairly disfavoured!) somehow observes that the laptop is, for some reason a perfect blackbody, and is normally blue (assume 470.00 nm). As the laptop passes through the event horizon, the student observes that the laptop seems to have a wavelength of 933.89 nm.

The student recalls from his studying that the mass of a black hole and the radius of its event horizon are related linearly. He dejectedly realizes that he can now use the wavelength data to find this relationship, but is too depressed to do so. Can you find the relationship between the mass of the black hole, in solar masses, and radius of a black hole, in kilometers?

Assume that G = 6.67e-11, c = 2.99e8, and a solar mass is 2.00e30 kg.

Hint: (Click to Show Hidden Text)

V = sqrt((2*G*M)/R)

Answer (Click to Show Hidden Text)

V = sqrt((2*G*M)/R)
Since you are assuming that a linear relationship exists between mass and radius, substitute M*k for R.
V = sqrt((2*G*M)/(M*k)) = sqrt((2*G)/k)
vrec = λapp/λtrue - 1
vrec = c * (933.89nm / 470nm - 1) ~ 2.95*10^8 m/s
2.95*10^8 m/s= sqrt((2*G)/k)
vrec^2 = 8.71*10^16 m^2/s^2 = (2*G)/k
8.71*10^16 m^2/s^2 = (1.33*10-10 kg m^3 ^-2)/k
8.71*10^16  = 1.33*10^-10 m/kg / k
k = 1.33*10^-10m/kg/8.71*10^16 ~ 1.53*10^-27 m/kg
1.53*10^-27 m/kg * 1km/1000m * 2.00*10^30kg/1 Msol 
~ 3.06 km/Msol
R = 3.06 km/Msol * M

PM2017 wrote:A student at West High is angry he lost the tryouts for astronomy to PM2017. For this reason, he decides to throw PM2017's laptop into a black hole and seeing as he is actually better than PM2017 (and was unfairly disfavoured!) somehow observes that the laptop is, for some reason a perfect blackbody, and is normally blue (assume 470.00 nm). As the laptop passes through the event horizon, the student observes that the laptop seems to have a wavelength of 933.89 nm.

The student recalls from his studying that the mass of a black hole and the radius of its event horizon are related linearly. He dejectedly realizes that he can now use the wavelength data to find this relationship, but is too depressed to do so. Can you find the relationship between the mass of the black hole, in solar masses, and radius of a black hole, in kilometers?

Assume that G = 6.67e-11, c = 2.99e8, and a solar mass is 2.00e30 kg.

Hint: (Click to Show Hidden Text)

V = sqrt((2*G*M)/R)

You can simplify r=2GM/c^2
2G/c^2= 1.49e-27 
1.49e-27*M= r in terms of kg and meters
Convert that in terms of solar masses and km
1.49e-27/1000*2e30 = 2.98
Radius (in km) = 2.98*Mass (in solar masses)

Name wrote:

PM2017 wrote:A student at West High is angry he lost the tryouts for astronomy to PM2017. For this reason, he decides to throw PM2017's laptop into a black hole and seeing as he is actually better than PM2017 (and was unfairly disfavoured!) somehow observes that the laptop is, for some reason a perfect blackbody, and is normally blue (assume 470.00 nm). As the laptop passes through the event horizon, the student observes that the laptop seems to have a wavelength of 933.89 nm.

The student recalls from his studying that the mass of a black hole and the radius of its event horizon are related linearly. He dejectedly realizes that he can now use the wavelength data to find this relationship, but is too depressed to do so. Can you find the relationship between the mass of the black hole, in solar masses, and radius of a black hole, in kilometers?

Assume that G = 6.67e-11, c = 2.99e8, and a solar mass is 2.00e30 kg.

Hint: (Click to Show Hidden Text)

V = sqrt((2*G*M)/R)

Assuming you don't have to use the given values/equations

Alternative solution: (Click to Show Hidden Text)

You can simplify r=2GM/c^2
2G/c^2= 1.49e-27 
1.49e-27*M= r in terms of kg and meters
Convert that in terms of solar masses and km
1.49e-27/1000*2e30 = 2.98
Radius (in km) = 2.98*Mass (in solar masses)

PM2017 wrote:

Knyte_Xjn wrote:

PM2017 wrote:A student at West High is angry he lost the tryouts for astronomy to PM2017. For this reason, he decides to throw PM2017's laptop into a black hole and seeing as he is actually better than PM2017 (and was unfairly disfavoured!) somehow observes that the laptop is, for some reason a perfect blackbody, and is normally blue (assume 470.00 nm). As the laptop passes through the event horizon, the student observes that the laptop seems to have a wavelength of 933.89 nm.

The student recalls from his studying that the mass of a black hole and the radius of its event horizon are related linearly. He dejectedly realizes that he can now use the wavelength data to find this relationship, but is too depressed to do so. Can you find the relationship between the mass of the black hole, in solar masses, and radius of a black hole, in kilometers?

Assume that G = 6.67e-11, c = 2.99e8, and a solar mass is 2.00e30 kg.

Hint: (Click to Show Hidden Text)

V = sqrt((2*G*M)/R)

Answer (Click to Show Hidden Text)

V = sqrt((2*G*M)/R)
Since you are assuming that a linear relationship exists between mass and radius, substitute M*k for R.
V = sqrt((2*G*M)/(M*k)) = sqrt((2*G)/k)
vrec = λapp/λtrue - 1
vrec = c * (933.89nm / 470nm - 1) ~ 2.95*10^8 m/s
2.95*10^8 m/s= sqrt((2*G)/k)
vrec^2 = 8.71*10^16 m^2/s^2 = (2*G)/k
8.71*10^16 m^2/s^2 = (1.33*10-10 kg m^3 ^-2)/k
8.71*10^16  = 1.33*10^-10 m/kg / k
k = 1.33*10^-10m/kg/8.71*10^16 ~ 1.53*10^-27 m/kg
1.53*10^-27 m/kg * 1km/1000m * 2.00*10^30kg/1 Msol 
~ 3.06 km/Msol
R = 3.06 km/Msol * M

Your turn!

Knyte_Xjn wrote:
Extremely sorry for the extremely late post!
1. (a) What is the minimum speed required for a celestial object to free itself from the gravitational attraction of a massive object?
(b) You observe a planet orbiting a massive star with a mass of 74 Msol. Calculate the value mentioned in 1a for this planet to derail from its orbit and move away from the star's gravitational influence given that the radius of the orbit is 3.4 AU.
(c) An object given ________(answer to 1a) will continually move away from the other object, its speed __________ approaching _________. (fill in the blanks)

1. Escape velocity
2. sqrt(2GM/r) is sqrt(2*6.67e-11*1.48e32kg/5.1e11m) = sqrt(3.87e10)=196754 m/s
3. a object given [u]escape velocity[/u] will continually move away from the other object, its speed [u]gradually[/u] approaching [u]0[/u]

Name wrote:

Knyte_Xjn wrote:
Extremely sorry for the extremely late post!
1. (a) What is the minimum speed required for a celestial object to free itself from the gravitational attraction of a massive object?
(b) You observe a planet orbiting a massive star with a mass of 74 Msol. Calculate the value mentioned in 1a for this planet to derail from its orbit and move away from the star's gravitational influence given that the radius of the orbit is 3.4 AU.
(c) An object given ________(answer to 1a) will continually move away from the other object, its speed __________ approaching _________. (fill in the blanks)

Answer (Click to Show Hidden Text)

1. Escape velocity
2. sqrt(2GM/r) is sqrt(2*6.67e-11*1.48e32kg/5.1e11m) = sqrt(3.87e10)=196754 m/s
3. a object given [u]escape velocity[/u] will continually move away from the other object, its speed [u]gradually[/u] approaching [u]0[/u]

Name wrote:1. A star with the same radius and twice the temperature of our sun is found. What is its luminosity (in solar luminosity)?
2. The parallax of a type 1a supernova was found to be .05 milliarcsec. What its distance (in parsecs)?
3. What is the supernovas apparent magnitude?
4. The apparent magnitude of the star in part 1 and the supernova in part 2/3 is the same. What is the distance of the star?

[strikethrough]1. 8 (stefan boltzmann)
2. 20,000 (1000/parallax)
3. -2.8 (distance modulus; abs mag is -19.3)
4. really close, or 0.859 parsecs (Abs mag = 2.53, because the 2.5th root of 8 is ~2.30 4.83 - that is 2.53; apparent magnitude is -2.83)[/strikethrough]