Question

Be sure to show all calculations clearly and state your final answers in complete sentences. White Dwarf Density. A typical white dwarf has a mass of about \(1.0 M_{\text {Sun }}\) and the radius of Earth (about 6400 kilometers). Calculate the average density of a white dwarf, in kilograms per cubic centimeter. How does this compare to the mass of familiar objects?

Short answer

A typical white dwarf has a density of approximately \(1.8 \times 10^3\) kg/cm³, making it incredibly dense compared to most substances.

Step-by-step solution

Understand the Known Values

A typical white dwarf has a mass of about \(1.0 M_{\text{Sun}}\) and a radius equivalent to that of Earth, around 6400 kilometers. We need to convert these into standard units for use in our density formula.

Convert Mass to Kilograms

The mass of the sun is approximately \(1.989 \times 10^{30}\) kilograms. Therefore, the mass of the white dwarf is the same: \(M = 1.989 \times 10^{30}\) kg.

Convert Radius to Centimeters

The radius of Earth is about 6400 kilometers, which we can convert to meters (\(6400 \times 10^3\) meters) and then to centimeters by multiplying by 100, giving us \(6.4 \times 10^8\) cm.

Calculate the Volume of the White Dwarf

Using the formula for the volume of a sphere, \(V = \frac{4}{3}\pi r^3\), where \(r = 6.4 \times 10^8\) cm, we find volume:\[V = \frac{4}{3} \pi (6.4 \times 10^8\, \text{cm})^3\]. Calculating this gives approximately \(1.1 \times 10^{27}\) cubic centimeters.

Calculate the Average Density

Density is mass divided by volume. \(\text{Density} = \frac{M}{V} = \frac{1.989 \times 10^{30} \, \text{kg}}{1.1 \times 10^{27} \, \text{cm}^3}\), which is approximately \(1.8 \times 10^3 \, \text{kg/cm}^3\).

Compare with Familiar Objects

A white dwarf's density \(1.8 \times 10^3 \, \text{kg/cm}^3\) means it is about a million times denser than water (which is \(1 \, \text{kg/cm}^3\)) and can compare to the density of a nucleus, indicating extreme compactness.

Key concepts

Star Mass Calculation

The first key aspect of calculating the density of a white dwarf is understanding its mass. Stars, like white dwarfs, are often measured in terms of the sun's mass, represented as M\(_{\text{Sun}}\). The mass of the sun is a widely accepted scientific standard and is approximately \(1.989 \times 10^{30}\) kilograms. Therefore, when we say a white dwarf has a mass of about \(1.0 M_{\text{Sun}}\), it means its mass is also \(1.989 \times 10^{30}\) kg, similar to the sun's.
Calculating the mass is crucial as it forms a pivotal part of determining the density later in the process. These measurements help astronomers predict behavior and compare other celestial bodies.

Volume of a Sphere

When calculating the density of a white dwarf, knowing the volume is essential. A white dwarf has a radius about the same as Earth's, approximately 6400 kilometers. To calculate volume, convert the radius to centimeters, involving:

• Convert kilometers to meters: \(6400 \times 10^3\) meters.
• Convert meters to centimeters: multiplying by 100 results in \(6.4 \times 10^8\) centimeters.

Next, use the sphere volume formula, \(V = \frac{4}{3}\pi r^3\), where \(r\) stands for radius. For the white dwarf, this gives approximately \(1.1 \times 10^{27}\) cubic centimeters.
This step is significant because knowing the volume allows us to calculate density accurately, defined as mass per unit volume.

Average Density Calculation

Density is a measure of how much mass is contained within a given volume. The average density formula is \(\text{Density} = \frac{M}{V}\), where \(M\) is mass and \(V\) is volume.
For a white dwarf:

• Mass \(M\) is \(1.989 \times 10^{30}\) kg.
• Volume \(V\) is \(1.1 \times 10^{27}\) cm\(^3\).

Thus, the density is \(\frac{1.989 \times 10^{30} \, \text{kg}}{1.1 \times 10^{27} \, \text{cm}^3}\), roughly \(1.8 \times 10^3 \, \text{kg/cm}^3\).
Understanding average density not only gives insight into the compactness of white dwarfs but also aids in understanding the physics governing stellar objects. It indicates how tightly matter is packed. In the case of white dwarfs, it's extremely dense compared to everyday objects on Earth.

Star Comparison

Comparing the density of a white dwarf to more familiar celestial objects or substances can be quite illuminating. Consider that water, a common reference, has a density of about \(1 \, \text{kg/cm}^3\). In comparison, a white dwarf's density is \(1.8 \times 10^3 \, \text{kg/cm}^3\). This reveals that white dwarfs are remarkably dense, much more so than many things we encounter on Earth.
Such high density is on par with atomic nuclei, showing how compact and tightly packed the matter is in such stars.
By understanding these comparisons, we can appreciate how unique and intriguing white dwarfs are in the cosmos, offering insight into the final stages of stellar evolution.