Question

In 1999 , scientists discovered a new class of black holes with masses 100 to 10,000 times the mass of our sun that occupy less space than our moon. Suppose that one of these black holes has a mass of \(1 \times 10^{3}\) suns and a radius equal to one-half the radius of our moon. What is the density of the black hole in \(\mathrm{g} / \mathrm{cm}^{3}\) ? The radius of our sun is \(7.0 \times 10^{5} \mathrm{~km}\), and it has an average density of \(1.4 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\). The diameter of the moon is \(2.16 \times 10^{3} \mathrm{mi}\).

Short answer

The density of the black hole is approximately \(1.85 \times 10^{12}\ \mathrm{g}/\mathrm{cm}^3\).

Step-by-step solution

- Convert the Radius of the Moon to Centimeters

The radius of the moon is given in miles. First, convert the diameter to the radius in miles by dividing it by 2. Then convert the radius from miles to kilometers, remembering there are 1.60934 kilometers in a mile. Finally, convert the kilometers to centimeters by noting that there are 100,000 centimeters in a kilometer.

- Calculate the Volume of the Black Hole

Using the radius obtained from Step 1, calculate the volume of the black hole using the formula for the volume of a sphere, which is \(\frac{4}{3}\pi r^3\).

- Calculate the Mass of the Black Hole in Grams

The mass of the black hole is given in terms of solar masses. Multiply the number of solar masses by the mass of the sun in kilograms (mass of sun = average density of the sun \(\times\) volume of the sun), then convert the mass from kilograms to grams by multiplying by 1,000.

- Calculate the Density of the Black Hole

Density is mass divided by volume. Using the mass from Step 3 in grams and the volume from Step 2 in cubic centimeters, calculate the density of the black hole in \(\mathrm{g}/\mathrm{cm}^3\).

Key concepts

Astronomical Mass Conversion

When dealing with celestial bodies like suns, moons, and black holes, we often encounter measurements in 'solar masses' or the mass of our sun.
In the context of black holes, astronomers quantify mass relative to our sun because the numbers are so astronomically large that using regular units like kilograms would be impractical. One solar mass is equivalent to approximately 1.989 x 10^30 kilograms. For our exercise, it's crucial to convert this mass into a more standard unit we can work with, like grams, to assess the density of the black hole.
To convert the mass of the black hole from solar masses to grams, as shown in Step 3 of the provided solution, you would first determine the mass of the sun in kilograms based on its density and volume and then multiply by the number of solar masses. After obtaining the mass in kilograms, multiply that figure by 1,000 to convert it to grams.

Volume of a Sphere

Calculating the volume of a sphere is essential when trying to understand the scale and size of round celestial objects like planets, moons, and black holes. In our exercise, we measure how much space a black hole occupies.
To find this volume, we use the formula for the volume of a sphere, which is \br \[\frac{4}{3} \times \pi \times r^3\], where \(r\) is the radius.
It's important to ensure that the radius is in consistent units across calculations. Our exercise requires converting the moon's radius from miles to centimeters before applying it to the volume formula. Since a sphere's volume increases with the cube of its radius, a seemingly small error in the radius measurement can lead to a large discrepancy in the calculated volume, and therefore in the calculated density.

Density Formula

Density is a measure of how much mass is contained in a given volume. Commonly, density is expressed as grams per cubic centimeter (\(g/cm^3\)) in science. The formula to calculate density is simple but powerful: \[\text{Density} = \frac{\text{Mass}}{\text{Volume}}\]. In our exercise, we have the mass of the black hole in grams and the volume in cubic centimeters. By dividing the mass by the volume, we get the density of the black hole, which allows us to understand how compact it is.
For astronomical objects like black holes, which are extremely dense, this calculation provides insight into the extreme conditions within these fascinating entities. A high density implies that a staggering amount of mass is compressed into a small space, which is characteristic of black holes and is essential for their strong gravitational pull. The calculation process in Steps 1 through 4 leads us to understand just how dense such an object can be.