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Unformatted text preview: Ph 444 Solutions for Problem Set 6 1. (Ryden 7.5) The flux, f , received from a standard candle of luminosity L is (Ryden equation 7.21) f = L 4 πd 2 L , (1) where d L is the luminosity distance. The angular diameter, δθ , of a standard yardstick of size ℓ is (Ryden equation 7.33) δθ = ℓ d a , (2) where d a is the angular diameter distance. Thus, the surface brightness, Σ, of an object that is both a standard candle and a standard yardstick is Σ ∝ f ( δθ ) 2 = L 4 πℓ 2 parenleftBigg d a d L parenrightBigg 2 . (3) Now from Ryden equation (7.37), d a = d L / (1 + z ) 2 , so Σ ∝ L ℓ 2 parenleftbigg 1 1 + z parenrightbigg 4 . (4) Note that surface brightness decreases quickly with increasing redshift. Observations of the surface brightness of these objects as a function of redshift cannot determine q because the ratio d a /d L has no dependence on the cosmological model. Of course, measuring either angular diameters or fluxes as a function of redshift does constrain the model. But doing both provides no additional information. 2. This problem explores what is required to measure the equation of state of the dark energy using observations of type Ia supernovae. What we observe is the peak apparent magnitude, m , of each supernova. The distance modulus is then calculated from the corrected peak absolute magnitude of the supernova. The distance modulus to an individual type Ia supernova can be measured with an accuracy of σ sn = . 15 mag. This uncertainty comes mostly from our inability to exactly correct for the intrinsic spread of supernovae peak luminosities rather than the measurement uncertainty in the peak apparent magnitude of a supernova. The basis for using supernovae to probe cosmology is the relation between distance modulus and redshift and this problem adopts the form given by Ryden equation (7.52): m − M = 43 . 17 − 5 log 10 parenleftBigg H 70 kms − 1 Mpc − 1 parenrightBigg + 5 log 10 ( z ) + 1 . 086(1 − q ) z. (5) Here q is the deceleration constant which is given by Ryden equation (7.10):is the deceleration constant which is given by Ryden equation (7....
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- Fall '08
- Dark Matter, Supernova, White dwarf, absolute magnitude