Question

Astronomy The black hole in the M82 galaxy has a mass about 500 times the mass of the Sun. It has about the same volume as the Moon. What is the density of this black hole? mass of the Sun = $1.9891\times {10}^{30}\mathrm{kg}$ volume of the Moon $=2.1968\times {10}^{10}{\mathrm{km}}^3$

Short answer

The density of the black hole in the M82 galaxy is approximately $4.5298\times {10}^{13}\mathrm{kg}/{\mathrm{m}}^3$.

Step-by-step solution

Identify the quantities given in the problem.

We have the mass of the black hole as 500 times the mass of the Sun, and its volume as the same as the volume of the Moon. Mass of the Sun = $1.9891\times {10}^{30}\mathrm{kg}$ Volume of the Moon = $2.1968\times {10}^{10}{\mathrm{km}}^3$

Calculate the mass of the black hole.

We are given that the mass of the black hole is 500 times the mass of the Sun. To find the mass of the black hole, we will multiply the mass of the Sun by 500. Mass of the black hole = 500 × Mass of the Sun = $500\times 1.9891\times {10}^{30}\mathrm{kg}=9.9455\times {10}^{32}\mathrm{kg}$

Convert the volume of the Moon to cubic meters.

The given volume of the Moon is in cubic kilometers. We need to convert this to cubic meters for our calculations. 1 cubic kilometer is equal to ${10}^9$ cubic meters. Volume of the Moon = $2.1968\times {10}^{10}{\mathrm{km}}^3\times {10}^9{\mathrm{m}}^3/{\mathrm{km}}^3=2.1968\times {10}^{19}{\mathrm{m}}^3$

Calculate the density of the black hole.

Now that we have the mass of the black hole and the volume of the Moon in compatible units, we can find the density of the black hole using the formula: Density = mass/volume Density of the black hole = $\frac{9.9455\times {10}^{32}\mathrm{kg}}{2.1968\times {10}^{19}{\mathrm{m}}^3}\approx 4.5298\times {10}^{13}\mathrm{kg}/{\mathrm{m}}^3$ The density of the black hole in the M82 galaxy is approximately $4.5298\times {10}^{13}\mathrm{kg}/{\mathrm{m}}^3$.

Key concepts

Mass Conversion

In astronomy, mass conversion is essential for dealing with extremely large or small values. To calculate the mass of an object like a black hole, which is expressed in relation to a known mass such as the Sun, we need to perform conversion operations. Given that the mass of the Sun is approximately $1.9891\times {10}^{30}\mathrm{kg}$, determining the black hole's mass involves straightforward multiplication. If a black hole is 500 times more massive than the Sun, we calculate this by multiplying:

• Mass of the black hole = 500 \times Mass of the Sun
• $=500\times 1.9891\times {10}^{30}\mathrm{kg}$
• = $9.9455\times {10}^{32}\mathrm{kg}$

This mass conversion allows us to express the black hole in a comprehensible numerical format, critical for further calculations such as density.

Volume Conversion

Volume conversion is another crucial aspect of astronomical calculations, as many times volumes are given in different units. For our problem, the volume of the Moon, and consequently the black hole, was given as $2.1968\times {10}^{10}{\mathrm{km}}^3$. To use this measure in our density calculation, converting to cubic meters is essential because density is usually expressed in kilograms per cubic meter ${\text{kg/m}}^3$.

• 1 cubic kilometer $={10}^9$ cubic meters
• Volume in ${\mathrm{m}}^3=2.1968\times {10}^{10}{\mathrm{km}}^3\times {10}^9$
• = $2.1968\times {10}^{19}{\mathrm{m}}^3$

This converted volume allows consistency and accuracy in further computations, such as density, which is pivotal when examining astronomical objects.

Black Hole Properties

Black holes are fascinating astronomical entities known for their incredibly high mass and density. Despite their massive nature, they can have volumes comparable to much smaller celestial bodies like the Moon, leading to remarkable densities. In our example, the black hole's mass of $9.9455\times {10}^{32}$ kg fits into a volume similar to the Moon's. This results in a density of approximately $4.5298\times {10}^{13}\mathrm{kg}/{\mathrm{m}}^3$, which showcases a core property of black holes: immense density. Such high density arises because black holes compress all their mass into an extremely small space, manifesting strong gravitational pull, which not even light can escape.

Astronomy Calculations

Astronomy calculations often involve vast numbers and require meticulous conversions to maintain accuracy. When determining attributes such as density, one must ensure mass and volume are in compatible units. For our task, with the black hole in the M82 galaxy, the workflow included:

• Scaled mass calculation using a solar mass multiple
• Volume conversion to ensure standardized measurement units
• Density calculation using $\text{Density}=\frac{\text{mass}}{\text{volume}}$

These steps lead to a calculated density of $4.5298\times {10}^{13}\mathrm{kg}/{\mathrm{m}}^3$, essential for comprehensively understanding the extreme nature of black holes. Astronomy calculations blend mathematical rigor with conceptual understanding, enabling exploration into the vast complexities of our universe.