Show that the angular-diameter distance for a flat space (k = 0; Eq. 10.37) out
ID: 3162648 • Letter: S
Question
Show that the angular-diameter distance for a flat space (k = 0; Eq. 10.37) out to redshift z, has a maximum with respect to redshift z, and find that redshift. The angular size on the sky of an object with physical size d is = d/DA. What is the implication of the maximum of DA for the appearance of objects at redshifts beyond the one you found?
3. Show that the angular-diameter distance for a flat space (k 0; Eq. 10.37) out to redshift z, -3/2 has a maximum with respect to redshift z, and find that redshift. The angular size on the sky of an object with physical size d is 0 -d/DA. What is the implication of the maximum of DA for the appearance of objects at redshifts beyond the one you found?Explanation / Answer
Answer The angular diameter distance DA is defined as the ratio of an object’s physical transverse size to its angular size.
wkt,
DA = DM /1 + z
mass density of the Universe and the value of the cosmological constant.
can be made into dimensionless density parameters M and by M 8G0 /3 H20
/3 H2
The redshift z of an object is the fractional doppler shift of its emitted light resulting from radial motion z e /o 1 = o /e 1
1 + z = [ 1 + v/c / 1 v/c]1/2
The comoving distance between two events at the same redshift or distance but separated on the sky by some angle is DM and the transverse comoving distance DM (so-denoted for a reason explained below) is simply related to the line-of-sight comoving distance DC:
DM = DH 1 R sinh hR DC/DH
for R > 0
DC for R = 0
DH 1 /|R| sin |R| DC/DH or R < 0
DM = DH 2[2 M (1 z) (2 M ) 1+M z] / 2M (1 + z) for = 0
THUS, between two objects at redshifts z1 and z2, frequently used in gravitational lensing. It is not found by subtracting the two individual angular diameter distances
correct formula, for R 0, is
DA = 1 / 1 + z2 DM2 (1+R D2 M1/ D2 H)1/2 ( DM1 (1+R D2M2 /D2 H 1/2
Hence applying the given set parameters in the above equation we get the desired result.
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.