Assume that the average galaxy contains 10″ Msun and that the average distance between galaxies is 10 million light-years. Calculate the average density of matter (mass per unit volume) in galaxies. What fraction is this of the critical density we calculated in the chapter?
The Correct Answer and Explanation is :
To calculate the average density of matter in galaxies and compare it to the critical density, we follow these steps:
1. Average Mass of a Galaxy
The average galaxy contains ( 10^{11} M_\odot ), where ( M_\odot = 2 \times 10^{30} \, \text{kg} ).
Thus, the average galaxy mass is:
[
M = 10^{11} \times 2 \times 10^{30} \, \text{kg} = 2 \times 10^{41} \, \text{kg}.
]
2. Volume of the Universe Per Galaxy
The average distance between galaxies is ( 10 \, \text{million light-years} = 10^7 \, \text{ly} ).
Converting light-years to meters:
[
1 \, \text{ly} = 9.46 \times 10^{15} \, \text{m}.
]
So,
[
10^7 \, \text{ly} = 10^7 \times 9.46 \times 10^{15} = 9.46 \times 10^{22} \, \text{m}.
]
The average volume per galaxy is roughly the cube of this distance:
[
V = (9.46 \times 10^{22})^3 = 8.46 \times 10^{68} \, \text{m}^3.
]
3. Average Density of Matter in Galaxies
Density ( \rho ) is mass per unit volume:
[
\rho = \frac{M}{V} = \frac{2 \times 10^{41}}{8.46 \times 10^{68}} \, \text{kg/m}^3.
]
[
\rho = 2.36 \times 10^{-28} \, \text{kg/m}^3.
]
4. Critical Density
The critical density ( \rho_c ) (from cosmology) is:
[
\rho_c = \frac{3H_0^2}{8\pi G},
]
where ( H_0 ) is the Hubble constant (( 70 \, \text{km/s/Mpc} = 2.27 \times 10^{-18} \, \text{s}^{-1} )) and ( G = 6.67 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} ). Substituting,
[
\rho_c = \frac{3 (2.27 \times 10^{-18})^2}{8 \pi (6.67 \times 10^{-11})} = 8.55 \times 10^{-27} \, \text{kg/m}^3.
]
5. Fraction of Critical Density
The fraction is:
[
\frac{\rho}{\rho_c} = \frac{2.36 \times 10^{-28}}{8.55 \times 10^{-27}} \approx 0.0276 \, (\sim 2.8\%).
]
Explanation (300 Words)
The calculated density of matter in galaxies (( 2.36 \times 10^{-28} \, \text{kg/m}^3 )) represents the mass per unit volume distributed across the average separation between galaxies. This value is derived by dividing the average galaxy mass (( 10^{11} M_\odot )) by the volume defined by the cube of the average intergalactic distance. In cosmology, such densities are typically compared to the critical density, which determines the geometry of the universe.
The critical density (( \sim 8.55 \times 10^{-27} \, \text{kg/m}^3 )) is the theoretical density required for the universe to have a flat geometry. Our result shows that the density of matter in galaxies is only about ( 2.8\% ) of the critical density. This suggests that most of the critical density is not accounted for by visible matter in galaxies. This discrepancy is a cornerstone of modern astrophysics, as it points to the existence of dark matter and dark energy, which dominate the mass-energy content of the universe.
The small fraction reflects that while galaxies are massive, the vast distances between them dilute their average density significantly. This also underscores the universe’s vast emptiness and supports observations that suggest ordinary matter makes up only a small fraction of the total cosmological density.