Quasar j23483054 **
Problem: The quasar j23483054 is one of the most distant objects in the visible Universe. Use the given data to calculate its (1) zdistance, (2) recession velocity, (3) distance in megaparsecs and lightyears, (4) mass, and (5) absolute magnitude. (This problem is based on observations by Venemans et al. 2013.)
Hint:
Use relativistic formulas because of the presumed vast distance of the quasar.
Data:
 (1) 
 (2) 
 (3) 
 (4) 
 (5) 
 (6) 
 (7) 
 (8) 
Useful Equations:
: # redshift distance
: mass of black hole in solar mass units
# : # Hubble law, v is the velocity in km/s, H_{0} is the Hubble constant, and x is the distance in Mpc
Solution:
(1) To find the distance (z) to the quasar, use the formula
where restframe wavelength, and observed wavelength. For this quasar, the magnesium line at 1215.67 Å is observed at a redshifted wavelength of 9550 Å. Therefore, its distance is
 (9) 
or approximately z = 6.86.
(2) For objects at such great distances, a relativistic formula must be used to calculate the recession velocity:
 (10) 
where c is the speed of light.
 (11) 
The recession velocity of the quasar is close to the speed of light. It is near the horizon of the visible Universe.
(3) From this, it is possible to estimate its distance in Mpc, using the Hubble law:
where v is the velocity in km/s, H_{0} is the Hubble constant, and x is the distance in Mpc.
 (12) 
and in light years:
 (13) 
or about 13.5 billion light years, equivalent in lookback time to near the beginning of the Universe.
(4) To find the mass of the supermassive black hole in this quasar, a formula from De Rosa et al (2014) is used.
where for this quasar is measured at 0.94*10^{46} erg s^{1}, and FWHM (full width half maximum) broadening for the magnesium II line is found to be 5446 km s^{1}.
 (14) 
approximately equivalent to the mass of two billion Suns.
The absolute magnitude of the quasar can be found from the apparent magnitude, +24, an estimated extinction of 6 magnitudes, and the distance in parsecs, 4149 * 10^{6}, using the distance modulus formula.
 (15) 
At a distance of 10 pc, this quasar would be only a little less bright than the daytime Sun at its present distance.

References
De Rosa, G., Venemans, B., Decarli, R., Gennaro, M. Simcoe, R., Dietrich, M., Peterson, B., Walter, F., Frank, S., McMahon, R., Hewett, P., Mortlock, D., and Simpson, C. (2014). Black hole mass estimates and emissionline properties of a sample of redshift z > 6.5 quasars. APJ, 790. 145.
Venemans, B., Findlay, J., Sutherland, W., De Rosa, G., McMahon, R., Simcoe, R., GonzálezSolares, E., Kuijken, K., and Lewis, J. (2013). Discovery of three z > 6.5 quasars in the Vista Kilodegree Infrared Galaxy (Viking) Survey. Astrophysical Journal, 779, 24.
