Universe: Stars And Galaxies
Universe: Stars And Galaxies
6th Edition
ISBN: 9781319115098
Author: Roger Freedman, Robert Geller, William J. Kaufmann
Publisher: W. H. Freeman
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Chapter 26, Problem 37Q
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The proof that the Jeans length for the universe at the era of recombination was about 100ly and the total mass contained in a sphere with this diameter was about 4×105M.

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Answer to Problem 37Q

The proof that the that the Jeans length for the universe at the era of recombination was about 100ly and the total mass contained in a sphere with this diameter was about 4×105M is as stated below.

Explanation of Solution

Given:

The temperature of the early universe is, T=3000K.

The density of the universe is, ρm=1018kg/m3.

Formula Used:

The expression for Jeans length is given by,

LJ=πkTmGρm

Here, k is the Boltzmann constant and m is the mass of a single particle in the gas.

The expression for volume is given by,

V=43πr3

The expression for mass density is given by,

ρ=mV

Rewrite the above expression for mass.

m=ρV=ρ(43πr3)

Calculation:

The Jeans length is calculated as,

LJ= πkT mG ρ m = π( 1.38× 10 23 J/K )( 3000K ) ( 1.67× 10 27 kg )( 6.67× 10 11 N m 2 / kg 3 )( 10 18 kg/ m 3 )= 1.30× 10 19 1.11× 10 55 m=1.08×1018m

Solve further,

LJ=(1.08× 10 18m× 1ly 9.46× 10 15 m)=114.16ly100ly

The mass contained in the sphere is calculated as

m=ρ(43πr3)=( 10 18kg/ m 3)(43π ( 1.08× 10 18 m 2 )3)=( 10 18kg/ m 3)(6.60× 10 53m3)=6.60×1035kg

Solve further,

m=(6.60× 10 35kg× 1 M 1.99× 10 30 kg)=3.32×105M

This value is nearly equal to 4×105M.

Conclusion:

Therefore, it is proved that the Jeans length for the universe at the era of recombination was about 100ly and the total mass contained in a sphere with this diameter was about 4×105M.

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