Question

Toward the end of their lives many stars become much bigger. Assume that they remain spherical in shape and that their masses do not change in this process. If the radius of a star increases by a factor of 11.4 , by what factors do the following change: a) its surface area, b) its circumference, c) its volume?

Short answer

Answer: The surface area changes by a factor of \(11.4^2\), the circumference changes by a factor of 11.4, and the volume changes by a factor of \(11.4^3\).

Step-by-step solution

Calculate the change in surface area

Given that the radius increases by a factor of 11.4, let's denote the initial radius as r and the final radius as R, where R = 11.4r. Now, we will calculate the new surface area (A') using the formula \(A' = 4\pi R^2\). Since R = 11.4r, we can substitute this value into the formula to evaluate A': \(A' = 4\pi (11.4r)^2\) \(A' = 4\pi (11.4^2 r^2)\) Now, let's divide the new surface area (A') by the initial surface area (A): \(\frac{A'}{A} = \frac{4\pi (11.4^2 r^2)}{4\pi r^2}\) This simplifies to: \(\frac{A'}{A} = 11.4^2\) So, the surface area changes by a factor of \(11.4^2\).

Calculate the change in circumference

We will now calculate the new circumference (C') using the formula \(C' = 2\pi R\). Since R = 11.4r, we can substitute this value into the formula to evaluate C': \(C' = 2\pi (11.4r)\) Now, let's divide the new circumference (C') by the initial circumference (C): \(\frac{C'}{C} = \frac{2\pi (11.4r)}{2\pi r}\) This simplifies to: \(\frac{C'}{C} = 11.4\) So, the circumference changes by a factor of 11.4.

Calculate the change in volume

Lastly, we will calculate the new volume (V') using the formula \(V' = \frac{4}{3}\pi R^3\). Since R = 11.4r, we can substitute this value into the formula to evaluate V': \(V' = \frac{4}{3}\pi (11.4r)^3\) \(V' = \frac{4}{3}\pi (11.4^3 r^3)\) Now, let's divide the new volume (V') by the initial volume (V): \(\frac{V'}{V} = \frac{\frac{4}{3}\pi (11.4^3 r^3)}{\frac{4}{3}\pi r^3}\) This simplifies to: \(\frac{V'}{V} = 11.4^3\) So, the volume changes by a factor of \(11.4^3\). In summary, the surface area changes by a factor of \(11.4^2\), the circumference changes by a factor of 11.4, and the volume changes by a factor of \(11.4^3\).

Key concepts

Surface Area Calculation in Stellar Evolution

Understanding how a star's surface area changes as it evolves is crucial when studying astronomy and physics. The surface area of a spherical object like a star is determined by the formula \( A = 4\pi r^2 \), where \( r \) is the radius of the sphere. As a star ends its life cycle, it often expands to become a giant or a supergiant, increasing its radius significantly.

From the exercise, we know the radius increases by a factor of 11.4, leading to a larger surface area. Since the surface area formula includes the radius squared, the new surface area is proportional to the square of the radius increase. Therefore, if the radius grows by a factor of 11.4, the surface area expands by \( 11.4^2 \), which is approximately 129.96 times the original size. This quadratic relationship between radius and surface area has immense implications for a star's thermal radiation and its interactions with surrounding space.

Circumference Calculation and Its Role in Stellar Measurements

The circumference of a circle—or in our cosmic example, a sphere such as a star—is the distance around it, which can be found using the formula \( C = 2\pi r \), with \( r \) representing the radius. When a star expands, its circumference, which roughly corresponds to its equator, also increases.

Following the problem provided, the stellar radius inflates by a factor of 11.4. Since the circumference is directly proportional to the radius, when the radius multiplies by 11.4, so does the circumference. Hence, the circumference of a star after it expands will be 11.4 times larger than before. Knowing the circumference is essential for astronomers as they measure and compare different stages of stellar evolution, contributing to our understanding of star sizes and rotational dynamics.

Volume Calculation in the Context of Expanding Stars

In the vast domains of space, when a star expands, its volume also changes dramatically. Volume is a measurement of how much three-dimensional space an object occupies. For spheres, the volume \( V \) is calculated using \( V = \frac{4}{3}\pi r^3 \), with \( r \) being the radius. As stars balloon in size towards the end of their life cycles, not only does the surface area and circumference expand, but the volume increases exponentially.

The provided exercise shows that when a star's radius is magnified by a factor of 11.4, its volume grows by \( 11.4^3 \) because of the cubic relationship in the volume formula. This results in the volume becoming roughly 1481.54 times its original size. The dramatic change in volume affects a star's density, luminosity, and the process of shedding its outer layers, which are critical factors in the later stages of stellar evolution.