University:
Massachusetts Institute of Technology
Course:
8.282J | Astrophysics
Academic year:

2023
Views:

121

Pages:

3
Author:

gerahsnottiness

Earth, Atmospheric, and Planetary Sciences Department Problem Set 13 Not To Be Turned In Reading: Zeilik & Gregory: Chapters 24, 25, and 26. Problem 1 “Relativistic Doppler Eﬀect” Zeilik & Gregory, Chapter 24, Problem #1, page 486. Refer to equation 814a in the text. Problem 2 “Cygnus A Radio Galaxy” Zeilik & Gregory, Chapter 24, Problem #3, page 486. The units for ﬂux density are W m−2 Hz−1 . In this problem it is adequate to use the nonrelativistic form Δλ/λ ≡ z � v/c. Problem 3 “Feeding a Black Hole” Zeilik & Gregory, Chapter 24, Problem #8, page 486. a. If a star of mass ms and radius R passes a black hole of mass MBH(� ms) at a distance D, the magnitude of its acceleration is GMBH/D2 . The force per unit mass felt by an observer at rest on the surface of the star at a point nearest the black hole is: GMBH Gms GMBH − 2 − , 2 (D − R) R D2 where the ﬁrst and second terms are the opposing forces of gravity from the black hole and the star, respectively, while the third term is the ﬁctitious force associated with being in the frame of reference that is accelerating with the center of mass of the star. The star will be pulled apart near the point where the sum of these forces goes to zero. For the case where 1 D � R this occurs when D � (2MBH /ms )1/3 R. You can derive this condition by using a Taylor’s series expansion for 1/(D − R)2 . For a related discussion see Z&G, chapter 34D, and equation 310. b. Use an energy conversion factor of ∼ 1020 ergs/gram for matter falling into a black hole since at least 7% of its rest energy is released as radiation. Problem 4 “Distance–Redshift Relation” Zeilik & Gregory, Chapter 24, Problem #13, page 487. (Note that 1 nm = 10Å= 10−7 cm.) Compare the answer you get using d = v/H0 (where the special relativistic relation between λ/λ0 and v at high speeds is deﬁned by equation 814a) with the answer obtained via equation (223d), where d = cz(1 + z/2)/[H0 (1 + z)2 ]. Problem 5 “H0 , τ0 , and ρcrit ” Zeilik & Gregory, Chapter 25, Problem #3, page 501. Problem 6 “Expanding Balloon Analogy” Zeilik & Gregory, Chapter 25, Problem #4, page 501. In part (b) it’s easier if you leave s/R as θ. Problem 7 “Scale Factor and Age vs. Redshift” Use the formalism that was developed in lecture for the evolution of the scale factor, a, as a function of time, t (after the formation of the Universe), to show that: � �2/3 3 H0 t , a= 2 for the special case where Ω0 = 1 (Ωm = 1; ΩΛ = 0). Use the “Lemaitre equation” that connects z and the scale factor: 1 1+z = , a to ﬁnd the relation between the z and the “age”, t: � �3/2 2 −1 1 t = H0 . 3 1+z Find the age, t, of (i) our Galaxy, with z = 0, (ii) a quasar with z = 6.4, and (iii) the cosmic microwave background, with z � 1000. 2 Problem 8 “Dust in the Universe” Zeilik & Gregory, Chapter 26, Problem #1, page 515. 2 Take σ = πrgrain as the scattering cross section. Assume a mass density of 1 g cm−3 (or 103 −3 kg m ) for the material that makes up a dust grain. 3