Arcturus is about half as hot as the Sun but is about 100 times more luminous. What is its radius campared to the sun.
To compare the radius of Arcturus to the Sun, we can make use of the Stefan-Boltzmann law, which states that the luminosity (L) of a star is related to its surface area (A) and temperature (T) by the equation L = σAT^4. Here, σ is the Stefan-Boltzmann constant.
First, let's compare the luminosity of Arcturus to that of the Sun. It is mentioned that Arcturus is about 100 times more luminous than the Sun, so we can express this as:
L_arcturus = 100 * L_sun (Equation 1)
Next, let's compare the temperatures of Arcturus and the Sun. It is stated that Arcturus is about half as hot as the Sun. We can express this as:
T_arcturus = 0.5 * T_sun (Equation 2)
Since luminosity is proportional to the surface area and temperature raised to the fourth power, we can combine Equations 1 and 2:
σAT_arcturus^4 = 100 * σAT_sun^4
Cancelling out the Stefan-Boltzmann constant (σ) and re-arranging the equation, we obtain:
(T_arcturus / T_sun)^4 = 100
Taking the fourth root of both sides of the equation:
T_arcturus / T_sun = ∛(100)
T_arcturus / T_sun = 4.6416
Now, since we know the relationship between temperature and radius for stars (higher temperature correlates with a larger radius), we can infer that the ratio of the radii of Arcturus and the Sun will be the same as the ratio of their temperatures:
R_arcturus / R_sun = T_arcturus / T_sun
Substituting the temperature ratio we found earlier:
R_arcturus / R_sun = 4.6416
Therefore, the radius of Arcturus is approximately 4.6416 times that of the Sun.