- #1
iloveannaw
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Suppose the luminosity function of galaxies is given by the approximation:
dn = [tex]\varphi[/tex] L dL = [tex]\varphi[/tex]* (L / L*)^-[tex]\alpha[/tex] dL/L* when L < L*
and
dn = [tex]\varphi[/tex] L dL = 0 when L > L*
where [tex]\alpha[/tex] = 1.25, L* = 2.0E10 solar luminosities and [tex]\varphi[/tex]* = (3Mpc)^-3
Integrate the above eqn to find the total number density of galaxies, integrated over all luminosities. Ypu should find that the integral diverges (e.g. the number of galaxies is infinite)
Well the integration part seemed fairly straight forward, taking x = (L/L*) and dx = dL/L* I got:
I = [tex]\int[/tex] [tex]\varphi[/tex]* (L / L*)^-[tex]\alpha[/tex] dL/L* = [tex]\varphi[/tex]*[tex]\int[/tex] x^-[tex]\alpha[/tex] dx
I = [tex]\varphi[/tex]* (x^(1-[tex]\alpha[/tex])) / (1-[tex]\alpha[/tex])
I = ((3 Mpc)^-3 . (2.0E10 SLs)^-0.25) / -0.25 = (very small negative value)(1 SL)
Im pretty sure the number of galaxies is not less than zero just can't figure this question out. As limits I've taken:
upper limit = L*
lower limit = 0
thanks in advance
dn = [tex]\varphi[/tex] L dL = [tex]\varphi[/tex]* (L / L*)^-[tex]\alpha[/tex] dL/L* when L < L*
and
dn = [tex]\varphi[/tex] L dL = 0 when L > L*
where [tex]\alpha[/tex] = 1.25, L* = 2.0E10 solar luminosities and [tex]\varphi[/tex]* = (3Mpc)^-3
Integrate the above eqn to find the total number density of galaxies, integrated over all luminosities. Ypu should find that the integral diverges (e.g. the number of galaxies is infinite)
Well the integration part seemed fairly straight forward, taking x = (L/L*) and dx = dL/L* I got:
I = [tex]\int[/tex] [tex]\varphi[/tex]* (L / L*)^-[tex]\alpha[/tex] dL/L* = [tex]\varphi[/tex]*[tex]\int[/tex] x^-[tex]\alpha[/tex] dx
I = [tex]\varphi[/tex]* (x^(1-[tex]\alpha[/tex])) / (1-[tex]\alpha[/tex])
I = ((3 Mpc)^-3 . (2.0E10 SLs)^-0.25) / -0.25 = (very small negative value)(1 SL)
Im pretty sure the number of galaxies is not less than zero just can't figure this question out. As limits I've taken:
upper limit = L*
lower limit = 0
thanks in advance
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