91Ó°ÊÓ

The surface temperature of a star is crucial in determining the amount of energy the star emits. This concept is elegantly described by the Stefan-Boltzmann Law, which states that the power radiated per unit area by a star is proportional to the fourth power of its surface temperature. Mathematically, this can be expressed as:
\[ P = \, \epsilon\sigma AT^4 \]
where:

• \( P \) is the power radiated by the star.
• \( \epsilon \) is the emissivity of the star, which indicates how efficiently it can emit radiation compared to a perfect black body.
• \( \sigma \) is the Stefan-Boltzmann constant.
• \( A \) is the surface area of the star.
• \( T \) is the surface temperature of the star.

In this equation, even a small increase in temperature greatly increases the power emitted due to the exponent of four. For Sirius B, having a higher surface temperature means it has the potential to emit more power per unit area than the Sun, assuming other factors remain constant.

Emissivity is a measure of how effectively a star radiates energy compared to a perfect emitter, known as a black body. This factor, denoted as \( \epsilon \), can range between 0 and 1, where 1 signifies perfect emission. In the case of both Sirius B and our Sun, it's assumed they have equal emissivity.
This assumption simplifies our calculations by allowing us to focus solely on the effects of surface temperature and surface area without the need to consider different efficiencies in energy release. With equal emissivity values:

• Both stars release their radiation similarly.
• The key differences in radiation power arise from differences in temperature and size, not from efficiency discrepancies.

Thus, the higher temperature of Sirius B should theoretically result in higher radiation power if all other factors, like surface area, didn't differ. But we know that's not the case due to its smaller surface area.

Surface area plays a vital role in the amount of total power a star can radiate. For a spherical star, this is calculated as \( A = 4\pi r^2 \). While a star may have a high temperature, like Sirius B, its total radiation power can still be less than a cooler star's if its surface area is considerably smaller.
In our exercise, it's stated that Sirius B radiates less power than the Sun despite a higher surface temperature. This surprising fact means Sirius B must have a much smaller surface area.

• From the power relation \( P = \epsilon \sigma A T^4 \), lower overall power (\( P_B < P_\text{Sun} \)) given a higher temperature (\( T_B > T_\text{Sun} \)) can only result from a smaller surface area (\( A_B < A_\text{Sun} \)).
• The ominous fact that \( A_B < A_\text{Sun} \) implies that the radius of Sirius B is also less than that of our Sun, since surface area depends on \( r^2 \).

Thus, in this context, a star's surface area is pivotal in defining its total radiated power, even when the temperature might suggest otherwise.