Universe
Universe
11th Edition
ISBN: 9781319039448
Author: Robert Geller, Roger Freedman, William J. Kaufmann
Publisher: W. H. Freeman
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Chapter 26, Problem 37Q
To determine

To show: The Jeans length of the early universe was 100 ly and the total mass of the universe was 4×105 M, if the universe was contained in a sphere. The temperature of the universe was 3000 K and the density of the universe was 1018 kg/m3 in the early universe.

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Explanation of Solution

Given data:

Temperature of the early universe is 3000 K and ρm is the density of the universe is 1018 kg/m3.

Formula used:

According to Jeans, an object will only grow if the fluctuation in density crosses the density that is called Jean Length. Jeans length is calculated by the expression.

LJ=πkTmGρm

Here, k is Boltzmann constant, T is the temperature, m is the mass of a single particle in the gas, G is the universal gravitational constant, and ρm is the average density of matter in gas.

The expression for mass is:

m=Vρ

Here, m is the mass, V is the volume, and ρ is the density.

The volume of a sphere is:

V=43πr3

Here, r is the radius of the sphere.

The relation between light year and meter is:

1 ly=9.46×1015 m

The mass of the sun is denoted as:

1 M=1.99×1030 kg

Explanation:

Consider the value of Boltzmann constant, universal gravitational constant, and mass of hydrogen as 1.38×1023 J/K, 6.67×1011 Nm2/kg2, and 1.67×1027 kg, respectively.

Recall the expression for calculating the Jean length.

LJ=πkTmGρm

Substitute 1.38×1023 J/K for k, 6.67×1011 Nm2/kg2 for G, 1.67×1027 kg for m, 1018 kg/m3 for ρm, and 3000 K for T.

LJ=π×(1.38×1023 J/K)×3000 K(1.67×1027 kg)×(6.67×1011 Nm2/kg2)×(1018 kg/m3)=1.0805×1018 m

Since the length of the universe is given as 100 ly, convert the above Jeans length into ly by using the conversion formula.

LJ=1.0805×1018 m×(1 ly9.46×1015 m)=114.21 ly

Consider the sphere of diameter equal to Jean length. Recall the expression for volume.

V=43πr3

Substitute (1.0805×1018 m2) for r.

V=43π(1.0805×1018 m2)3=6.61×1053 m3

Recall the expression of mass.

m=Vρ

Substitute 6.61×1053 m3 for V and 1018 kg/m3 for ρm, and use conversion expression to express it in terms of the mass of the sun.

m=6.61×1053 m3×1018 kg/m3=(0.66×1036 kg)(1 M1.99×1030 kg)=3.3×105 M

Conclusion:

Therefore, by substituting the condition prevalent in the early universe, it can be concluded that the Jeans length for the early universe is nearly 100 ly and the total mass nearly is 4×105 M.

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