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Quasar j2348-3054 **


Problem: The quasar j2348-3054 is one of the most distant objects in the visible Universe. Use the given data to calculate its (1) z-distance, (2) recession velocity, (3) distance in megaparsecs and light-years, (4) mass, and (5) absolute magnitude. (This problem is based on observations by Venemans et al. 2013.)



Use relativistic formulas because of the presumed vast distance of the quasar.



lambda[0] := 1215.67*`Å`



lambda[obs] := 9550*`Å`



FWHM[line] := 5446*Unit('km')/Unit('s')



Unit('pc') := 3.26*Unit('ly')



A := 6



m := 24



H[0] = 70*Unit('km')/(Unit('s')*Unit('Mpc'))

H[0] = 70*Units:-Unit('kg')/(Units:-Unit('s')*Units:-Unit('Mpc'))


`λL`[lambda] = .94*10^46*Unit('erg')/Unit('s')

`λL`[lambda] = 0.9400000000e46*Units:-Unit('erg')/Units:-Unit('s')



Useful Equations:

NULL:  #  redshift distance

NULLz = sqrt((c+v)/(c-v))-1

NULL:  mass of black hole in solar mass units


#          v = H*`0`*x:  #  Hubble law, v is the velocity in km/s, H0 is the Hubble constant, and x is the distance in Mpc



(1) To find the distance (z) to the quasar, use the formula


where "  lambda[0] = "rest-frame wavelength, and "lambda[obs] = "observed wavelength. For this quasar, the magnesium line at 1215.67 is observed at a red-shifted wavelength of 9550 . Therefore, its distance is


  solve(1215.67*(1+z) = 9550, z)



or approximately z = 6.86.

(2) For objects at such great distances, a relativistic formula must be used to calculate the recession velocity:


NULLz = sqrt((c+v)/(c-v))-1

z = ((c+v)/(c-v))^(1/2)-1


where c is the speed of light.


solve(6.86 = sqrt((c+v)/(c-v))-1, v)



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, H0 is the Hubble constant, and x is the distance in Mpc.

solve(.968*300000 = 70*x, x)



and in light years:




or about 13.5 billion light years, equivalent in look-back time to near the beginning of the Universe.

(4) To find the mass of the super-massive black hole in this quasar, a formula from De Rosa et al (2014) is used.




where `λL`[lambda]for this quasar is measured at 0.94*1046 erg s-1, and FWHM (full width half maximum) broadening for the magnesium II line is found to be 5446 km s-1.


M = 10^6.86*(5446/10^3)^2*(.94*10^46/10^44)^.5

M = 2083143562.


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 * 106, using the distance modulus formula.

solve(24-M = -2.5*log[10]((10/(4149*10^6))^2)+6, M)



At a distance of 10 pc, this quasar would be only a little less bright than the daytime Sun at its present distance.



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 emission-line 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., González-Solares, E., Kuijken, K., and Lewis, J. (2013). Discovery of three z > 6.5 quasars in the Vista Kilo-degree Infrared Galaxy (Viking) Survey. Astrophysical Journal, 779, 24.