1. What is the radius of the event horizon for an object the mass of our Sun? Using the equation for black hole event horizon radius as a function of mass 2 * G * M radius = ------------- c^2 2 * 6.67x10^(-11) * 2x10^(30) = ------------------------------- ( 3x10^(8) )^2 = 3000 m approx 2. Take a look at the figure shown below (you might want to print it out). It shows three spectra taken of the galaxy M84. The top panel is the spectrum of light coming from just below the center of the galaxy, the middle panel is the spectrum at the very center, and the bottom panel is the spectrum of light coming from just above the center. The strongest line is emitted by ionized nitrogen atoms. In the lab, we measure its wavelength to be 6583 Angstroms. a. Use the wavelength of the peak of the emission line in the middle panel to calculate the average speed of the entire galaxy. Is it coming towards us, or away from us? The observed wavelength in the center of galaxy is about 6608 Angstroms, so shift in wavelength = 6608 - 6583 = 25 Angstroms This is caused by the Doppler shift, so we can calculate the speed of the entire galaxy as shift in wavelength v = ------------------- * c rest wavelength 25 Angstroms = ------------------- * c 6583 Angstroms = about 1,100 km/s b. Use the shift in peak wavelength above and below the center of the galaxy to estimate the speed with which gas is orbiting the central black hole in this galaxy. Above the center (R = +0.05 arcsec), the peak wavelength is about 6600 Angstroms; below the center, the peak wavelength is about 6614 Angstroms. The difference from the central wavelength is shift = (6614 - 6608) = about 6 Angstroms below center = (6608 - 6600) = about 8 Angstroms below center avg shift is about 7 Angstroms from value at center So the Doppler shift tells us that above or below center, the gas must be moving at about 7 Angstroms rotational speed = -------------- * c 6608 Angstroms = about 320 km/s c. If the gas is moving in a circular orbit around the center of the black hole, and we are viewing the orbit edge-on, we can use a bit of physics to derive the mass of the black hole. In the case of this galaxy, we think that the gas is moving in an orbit roughly 4 pc in radius. Use this orbital radius, plus your estimate for the speed of the gas, to estimate the mass of the black hole in kg. If gas orbits an object of mass M at distance R, and moves with orbital speed v, then v^2 * R M = --------- G ( 320x10^3 m/s)^2 * (4 x 3.1x10^16 m) = ----------------------------------------- 6.67x10^(-11) N*m^2/kg^2 = 2 x 10^(38) kg d. Convert your value to solar masses. How many stars like the Sun would it take to equal the mass of the black hole at the center of this galaxy? One solar mass is about 2x10^(30) kg, so this central black hole has a mass of 2 x 10^(38) kg black hole = ------------------- 2 x 10^(30) kg/sun = about 100,000,000 suns