To: (a) show that to prevent unlimited expansion of the universe, its average density must at least be equal to 3 H 2 8 π G (b) evaluate the ‘critical density’ numerically in terms of H-atoms per cubic meters. .

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Answer 1

Question

Chapter 44, Problem 39P

To determine

To:

(a) show that to prevent unlimited expansion of the universe, its average density must at least be equal to 3H28πG

(b) evaluate the ‘critical density’ numerically in terms of H-atoms per cubic meters.

.

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*39 Will the universe continue to expand forever? To attack this question, assume that the theory of dark energy is in error and that the recessional speed v of a galaxy a distance r from us is determined only by the gravitational interaction of the matter that lies inside a sphere of radius r centered on us. If the total mass inside this sphere is M, the escape speed v, from the sphere is v. = V2GMIT (Eq. 13-28). (a) Show that to prevent unlimited expansion, the aver- age density p inside the sphere must be at least equal to ЗН 87G (b) Evaluate this "critical density" numerically; express your an- swer in terms of hydrogen atoms per cubic meter. Measurements of the actual density are difficult and are complicated by the pres- ence of dark matter.

Another commonly calculated velocity in galactic dynamics is the escape velocity vesc, that is the minimum velocity a star must have in order to escape the gravitational field of the galaxy. (a) Starting from the work required to move a body over a distance dr against f show that the escape velocity from a point mass galaxy is vsc = 2GM/r where r is your initial distance. (b) Since we know galaxies aren't actually point-masses, also show that vesc from r for a galaxy with a p(r) xr¯² density profile is vese that R is a cutoff radius at which the mass density is zero. = 2v(1+ ln(R/r)). Here you must assume (c) The largest velocity measured for any star in the solar neighbourhood, at r=8 kpc, is 440 km/s. Assuming that this star is still bound to the galaxy, find the lower limit (in kiloparsecs), to the cutoff radius R and a lower limit (in solar units) to the mass of the galaxy. Note the solar rotation velocity is 220 km/s.

Another commonly calculated velocity in galactic dynamics is the escape velocity vesc, that is the minimum velocity a star must have in order to escape the gravitational field of the galaxy. (a) Starting from the work required to move a body over a distance dr against f show that the escape velocity from a point mass galaxy is vse = 2GM/r where r is your initial distance. (b) Since we know galaxies aren't actually point-masses, also show that vesc from r for a galaxy with a p(r) x r-² density profile is vse = 2v²(1+ ln(R/r)). Here you must assume that R is a cutoff radius at which the mass density is zero. (c) The largest velocity measured for any star in the solar neighbourhood, at r=8 kpc, is 440 km/s. Assuming that this star is still bound to the galaxy, find the lower limit (in kiloparsecs), to the cutoff radius R and a lower limit (in solar units) to the mass of the galaxy. Note the solar rotation velocity is 220 km/s.

Answer 2

Question

Chapter 44, Problem 39P

To determine

To:

(a) show that to prevent unlimited expansion of the universe, its average density must at least be equal to 3H28πG

(b) evaluate the ‘critical density’ numerically in terms of H-atoms per cubic meters.

.

Expert Solution & Answer

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Check out a sample textbook solution

See solution

Answer 3

Question

Chapter 44, Problem 39P

To determine

To:

(a) show that to prevent unlimited expansion of the universe, its average density must at least be equal to 3H28πG

(b) evaluate the ‘critical density’ numerically in terms of H-atoms per cubic meters.

.

Expert Solution & Answer

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Check out a sample textbook solution

See solution

Answer 4

Question

Chapter 44, Problem 39P

To determine

To:

(a) show that to prevent unlimited expansion of the universe, its average density must at least be equal to 3H28πG

(b) evaluate the ‘critical density’ numerically in terms of H-atoms per cubic meters.

.

Expert Solution & Answer

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Check out a sample textbook solution

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Answer 5

Chapter 44, Problem 39P

To determine

To:

(a) show that to prevent unlimited expansion of the universe, its average density must at least be equal to 3H28πG

(b) evaluate the ‘critical density’ numerically in terms of H-atoms per cubic meters.

.

Answer 6

Chapter 44, Problem 39P

To determine

To:

(a) show that to prevent unlimited expansion of the universe, its average density must at least be equal to 3H28πG

(b) evaluate the ‘critical density’ numerically in terms of H-atoms per cubic meters.

.

Answer 7

Chapter 44, Problem 39P

To determine

To:

(a) show that to prevent unlimited expansion of the universe, its average density must at least be equal to 3H28πG

(b) evaluate the ‘critical density’ numerically in terms of H-atoms per cubic meters.

.

Answer 8

Expert Solution & Answer

Answer 9

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Answer 10

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Answer 11

Ch. 44 - Prob. 1Q Ch. 44 - Prob. 2Q Ch. 44 - Prob. 3Q Ch. 44 - Prob. 4Q Ch. 44 - Prob. 5Q Ch. 44 - Prob. 6Q Ch. 44 - Prob. 7Q Ch. 44 - Prob. 8Q Ch. 44 - Prob. 9Q Ch. 44 - Prob. 10Q

To: (a) show that to prevent unlimited expansion of the universe, its average density must at least be equal to 3 H 2 8 π G (b) evaluate the ‘critical density’ numerically in terms of H-atoms per cubic meters. .

Concept explainers

To:

(a) show that to prevent unlimited expansion of the universe, its average density must at least be equal to 3H28πG

(b) evaluate the ‘critical density’ numerically in terms of H-atoms per cubic meters.

.

Want to see the full answer?

Chapter 44 Solutions

To: (a) show that to prevent unlimited expansion of the universe, its average density must at least be equal to 3 H 2 8 π G (b) evaluate the ‘critical density’ numerically in terms of H-atoms per cubic meters. .

Concept explainers

To: (a) show that to prevent unlimited expansion of the universe, its average density must at least be equal to 3H28πG (b) evaluate the ‘critical density’ numerically in terms of H-atoms per cubic meters. .

Want to see the full answer?

Chapter 44 Solutions

To:

(a) show that to prevent unlimited expansion of the universe, its average density must at least be equal to 3H28πG

(b) evaluate the ‘critical density’ numerically in terms of H-atoms per cubic meters.

.