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Chapter 1: Problem 67

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

Expert verified
The average density of the star is approximately \( 1.39 \times 10^{6} \mathrm{~g/cm^3} \).

Step by step solution

01

(Convert kg to g)

We are given the mass of the star in kilograms, and we need to convert it to grams. 1 kg = 1000 g So our conversion will be: \( (2 \times 10^{36} \mathrm{~kg}) * (1000 \mathrm{~g/kg}) = 2 \times 10^{39} \mathrm{~g} \) #Step 2: Convert volume of the star to cubic centimeters#
02

(Convert km to cm)

We need to convert the radius of the star from kilometers to centimeters in order to calculate the volume in cubic centimeters. 1 km = 100,000 cm So our conversion will be: \( (7 \times 10^{5} \mathrm{~km}) * (10^{5} \mathrm{~cm/km}) = 7 \times 10^{10} \mathrm{~cm} \) #Step 3: Calculate the volume of the star#
03

(Calculate volume of sphere)

Now that we have the radius of the star in centimeters, we can calculate its volume using the formula for the volume of a sphere: \( V = \frac{4}{3} \pi r^{3} \) where \( V \) is the volume and \( r \) is the radius. Plug in the values we just found: \( V = \frac{4}{3} \pi (7 \times 10^{10} \mathrm{~cm})^{3} \) Calculate the volume: \( V \approx 1.44 \times 10^{33} \mathrm{~cm^3} \) #Step 4: Calculate the average density of the star#
04

(Calculate density)

Now we have the mass of the star in grams, and its volume in cubic centimeters. We can calculate the average density using the formula: \( \rho = \frac{m}{V} \) where \( \rho \) is the density, \( m \) is the mass, and \( V \) is the volume. Plug in the values we found: \( \rho = \frac{2 \times 10^{39} \mathrm{~g}}{1.44 \times 10^{33} \mathrm{~cm^3}} \) Calculate the average density: \( \rho \approx 1.39 \times 10^{6} \mathrm{~g/cm^3} \) The average density of the star is approximately \( 1.39 \times 10^{6} \) grams per cubic centimeter.

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