Arcturus is about half as hot as the Sun but is about 100 times more luminous. What is its radius campared to the sun.
If Arcturus is half as hot, it radiates (1/2)^4 = 1/16 as much as the sun PER UNIT AREA. But since it radiates 100 times more totally, it must have 1600 times the area of the sun. That means it must have sqrt 1600 = 40 times the radius, since radius is proportional to the square root of area
To find the radius of Arcturus compared to the Sun, we can use the information given: Arcturus is about half as hot as the Sun but is about 100 times more luminous.
Since Arcturus is half as hot as the Sun, it radiates (1/2)^4 = 1/16 as much as the Sun per unit area. However, since it is 100 times more luminous, it must radiate 100 times more energy in total.
To determine the relationship between the areas of Arcturus and the Sun, we can equate the total energy radiated by both stars. Given that Arcturus radiates 100 times more energy than the Sun, it must have an area 100 times larger than the Sun.
Therefore, the area of Arcturus is 100 times that of the Sun, which means it must have a radius that is sqrt(100) = 10 times larger than the Sun, as the radius is proportional to the square root of the area.
So, the radius of Arcturus is 10 times the radius of the Sun.