By Edward L. Wright, Phd, UCLA

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**Additional info for Astronomy 275 Lecture Notes, Spring 2009**

**Example text**

This is normally defined for the case where hν << kTe << me c2 . Since the x variable is defined 60 µ Residual [kJy/sr] 100 y 0 -100 0 5 10 ν [/cm] 15 20 Fig. 5 × 10−5 and µ = 9 × 10−5 . as hν/kTe , these assumptions make dx very small, and thus the only term that contributes significantly to ∂n/∂y is the ∂n/∂x term: the n and n2 terms are unimportant in this limit. For a blackbody n = 1/(ex − 1) we get ∂n ∂ ∂n = x−2 x4 ∂y ∂x ∂x ∂ x4 ex = −x−2 ∂x (ex − 1)2 2x4 e2x − (ex − 1)(4x3 ex + x4 ex ) = x−2 (ex − 1)3 x2 e2x − 4xe2x + 4xex + x2 ex = (ex − 1)3 2 x x x x (e + 1) − 4x(e − 1) = e (ex − 1)3 61 = xex (ex − 1)2 x ex + 1 −4 ex − 1 (193) Thus the change in intensity due to y is (to first order) 2hν 3 ∂Iν xex = 2 ∂y c (ex − 1)2 x ex + 1 −4 ex − 1 with x = hν/kT◦ (194) The derivative of the Planck function with respect to T is given by T so ∂Iν ∂y 2hν 3 xex ∂Bν (T ) = 2 ∂T c (ex − 1)2 (195) can be described as a changed Planck brightness temperature: Tν = T◦ 1 + y x ex + 1 − 4 + ...

And a(to + ∆to ) = a(to )(1 + H(to )∆to + . ). But we also know that ∆te = ∆to /(1 + z). Combining gives a(to + ∆to ) a(te + ∆te ) a(to )(1 + H(to )∆to + . . = a(te )(1 + H(te )∆to /(1 + z) + . . a(to ) (1 + [H◦ − H(z)/(1 + z)]∆to + . ) = a(te ) = (1 + z)(to ) + [(1 + z)H◦ − H(z)]∆to (1 + z)(to + ∆to ) = (80) Thus the rate of change of the redshift of a comoving object is d(1 + z) = (1 + z)H◦ − H(te ) dto = H◦ (1 + z) 1 − (81) [1 − Ωtot,◦ ] + Ωv◦ /(1 + z)2 + Ωm◦ (1 + z) + Ωr◦ (1 + z)2 For example, consider a source with z = 3 in a Universe with Ωm◦ = 1.

1 Redshift z 1 Fig. — Distance modulus vs. redshift for high redshift Type Ia supernovae. Data are taken from the Union catalog published by Kowalski et al. (2008, ApJ, 686, 749). The distance modulus is DM = 5 log(DL (z)/D◦ ), normalized to D◦ = 10 pc for this plot. 27; green for Ωv◦ = 0, Ωm◦ = 0; black for Ωv◦ = 0, Ωm◦ = 1; and red for Ωv◦ = 0, Ωm◦ = 2. The large number of objects, and the large errorbars for individual objects, make it difficult to see the goodness of fits for these models. 2 Ned Wright - 13 May 2008 Fig.

### Astronomy 275 Lecture Notes, Spring 2009 by Edward L. Wright, Phd, UCLA

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