Question

A star is estimated to have a mass of \(2 \times 10^{36} \mathrm{~kg}\). Assuming it to be a sphere of average radius \(7.0 \times 10^{5} \mathrm{~km}\), calculate the average density of the star in units of grams per cubic centimeter.

Short answer

The average density of the star is approximately 1.4 grams per cubic centimeter.

Step-by-step solution

Write down the given values

We know the mass of the star is \(2 \times 10^{36} \mathrm{~kg}\) and its average radius is \(7.0 \times 10^{5} \mathrm{~km}\).

Convert the mass to grams

Since 1 kg is equal to 1000 grams, we can convert the mass of the star to grams: $$(2 \times 10^{36} \mathrm{~kg}) \times (1000 \frac{\mathrm{g}}{\mathrm{kg}}) = 2 \times 10^{39} \mathrm{~g}$$

Convert the average radius to centimeters

Since 1 km is equal to \(10^{5}\) centimeters, we can convert the radius to centimeters: $$(7.0 \times 10^{5} \mathrm{~km}) \times (10^{5} \frac{\mathrm{cm}}{\mathrm{km}}) = 7.0 \times 10^{10} \mathrm{~cm}$$

Calculate the volume of the star

The star is assumed to be a sphere, so we will use the formula for the volume V of a sphere: $$V = \frac{4}{3} \pi r^3$$ Plugging in the radius, we get: $$V = \frac{4}{3} \pi (7.0 \times 10^{10} \mathrm{~cm})^3 = \frac{4}{3} \pi (3.43\times 10^{32} \mathrm{~cm^3})$$

Calculate the average density of the star

Using the density formula (Density = Mass / Volume), we can find the average density: $$\rho = \frac{Mass}{Volume} = \frac{2 \times 10^{39} \mathrm{~g}}{\frac{4}{3} \pi (3.43\times 10^{32} \mathrm{~cm^3})}$$ After calculating, we get the average density of the star: $$\rho \approx 1.4 \, \mathrm{g/cm^3}$$ The average density of the star is approximately 1.4 grams per cubic centimeter.