Universe
Universe
11th Edition
ISBN: 9781319039448
Author: Robert Geller, Roger Freedman, William J. Kaufmann
Publisher: W. H. Freeman
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Chapter 25, Problem 41Q

(a)

To determine

The critical density of the matter in the universe (ρc), if the value of Hubble constant is 50km/s/Mpc.

(a)

Expert Solution
Check Mark

Answer to Problem 41Q

Solution:

4.7×1027 kg/m3

Explanation of Solution

Given data:

The Hubble constant is 50km/s/Mpc.

Formula used:

The relation between the Hubble constant and the critical density is:

ρc=3(H0)28πG

Here, H0 is the Hubble constant and G is universal gravitational constant having a value of 6.67×1011 Nm2/kg2

Explanation:

The conversion process of a unit of Hubble constant from km/s/Mpc to s-1 is:

H0=50 km/s/Mpc=5×104msMpc(1 Mpc3.09×1022 m)=1.62×1018 s1

Recall the relation between the Hubble constant and the critical density.

ρc=3(H0)28πG

Substitute 1.62×1018 s1 for H0 and 6.67×1011 Nm2/kg2 for G.

ρc=3(1.62×1018 s1)28π(6.67×1011 Nm2/kg2)=7.87×10361.67×109=4.7×1027 kg/m3

Conclusion:

Therefore, the critical density of matter is 4.7×1027 kg/m3, if the value of Hubble constant is 50km/s/Mpc.

(b)

To determine

The critical density of the matter in the universe (ρc) if the value of Hubble constant is 100km/s/Mpc.

(b)

Expert Solution
Check Mark

Answer to Problem 41Q

Solution:

1.9×1026 kg/m3

Explanation of Solution

Given data:

Hubble constant is 100km/s/Mpc.

Formula used:

The relation between Hubble constant and the critical density is:

ρc=3(H0)28πG

Here, H0 is the Hubble constant and G is universal gravitational constant having a value of 6.67×1011 Nm2/kg2.

Explanation:

The conversion process of unit of Hubble constant from km/s/Mpc to s-1 is:

H0=100 km/s/Mpc=105msMpc(1 Mpc3.09×1022 m)=3.24×1018 s1

Recall the relation between the Hubble constant and the critical density.

ρc=3(H0)28πG

Here, H0 is the Hubble constant (100km/s/Mpc) and ρc is the critical density for H0.

Substitute 3.24×1018 s1 for H0 and 6.67×1011 Nm2/kg2 for G.

ρc=3(3.24×1018 s1)28π(6.67×1011 Nm2/kg2)=3.14×10351.67×109=1.9×1026 kg/m3

Conclusion:

Therefore, the critical density of matter is 1.9×1026 kg/m3, if the value of Hubble constant is 100km/s/Mpc.

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