The star Polaris has a surface temperature of roughly 6500 K and a radius of approximately 50 solar radii. Find the luminosity of Polaris in terms of the Sun's luminosity.
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To find the luminosity of Polaris in terms of the Sun's luminosity, we can use the Stefan-Boltzmann law, which states that the luminosity of a star is proportional to the fourth power of its surface temperature and its radius squared.
The luminosity (L) of Polaris can be calculated as follows:
L = (T^4) * (R^2)
where:
T = surface temperature of Polaris = 6500 K
R = radius of Polaris = 50 solar radii
Let's first convert the radius of Polaris from solar radii to meters:
1 solar radius = 6.957 * 10^8 meters
So, the radius of Polaris in meters is:
R_meters = 50 * 6.957 * 10^8 meters
Now we can calculate the luminosity of Polaris:
L = (6500^4) * (R_meters^2)
Calculating this using a calculator, we find:
L = (6500^4) * ((50 * 6.957 * 10^8)^2)
L ≈ 2.595 * 10^32 Watts
The luminosity of Polaris is approximately 2.595 * 10^32 Watts.
To express this in terms of the Sun's luminosity, we divide the luminosity of Polaris by the Sun's luminosity, which is approximately 3.828 * 10^26 Watts.
L_sun = 3.828 * 10^26 Watts
Therefore, the luminosity of Polaris in terms of the Sun's luminosity is:
L / L_sun ≈ (2.595 * 10^32) / (3.828 * 10^26)
L / L_sun ≈ 6.773 * 10^5
So, the luminosity of Polaris is approximately 6.773 * 10^5 times the Sun's luminosity.
To find the luminosity of Polaris in terms of the Sun's luminosity, we can use the Stefan-Boltzmann law, which states that the luminosity of a star is directly proportional to its surface temperature and the fourth power of its radius.
The equation for the luminosity (L) is given by:
L = 4πR^2σT^4
Where:
- L is the luminosity of the star,
- R is the radius of the star,
- σ (sigma) is the Stefan-Boltzmann constant (approximated to 5.67 x 10^-8 W/m^2K^4),
- T is the surface temperature of the star.
Given that the surface temperature of Polaris (T) is roughly 6500 K and its radius (R) is approximately 50 solar radii, we can now calculate the luminosity of Polaris.
First, convert the radius of Polaris to meters:
1 solar radius = 6.957 x 10^8 meters
50 solar radii = 50 x 6.957 x 10^8 meters
Now we can substitute the values into the equation:
L = 4π(50 x 6.957 x 10^8)^2 * (5.67 x 10^-8) * (6500^4)
L = 4π(1.7385 x 10^10)^2 * (5.67 x 10^-8) * 17850625000000
L ≈ 2.17 x 10^33 watts
The Sun's luminosity is approximately 3.8 x 10^26 watts, so to express Polaris' luminosity in terms of the Sun's, divide Polaris' luminosity by the Sun's:
Luminosity of Polaris / Luminosity of the Sun = (2.17 x 10^33) / (3.8 x 10^26)
Luminosity of Polaris in terms of the Sun's luminosity ≈ 5.71 x 10^6
Therefore, the luminosity of Polaris is roughly 5.71 million times that of the Sun.
Well, let's just say that Polaris is really shining bright! With a temperature of 6500 K, it's like the star is having a constant hot flash! Now, when it comes to its radius of 50 solar radii, that's pretty huge! Polaris is like the star version of a sumo wrestler!
Now, to find the luminosity of Polaris in terms of the Sun's luminosity, we'll need to dust off our calculators. The luminosity of a star is related to its temperature and radius through the Stefan-Boltzmann law. According to this law, the luminosity is proportional to the fourth power of the temperature and the square of the radius. So, let's plug in the numbers and get this party started!
Luminosity of Polaris = (Temperature of Polaris / Temperature of Sun)^4 * (Radius of Polaris / Radius of Sun)^2 * Luminosity of Sun
Luminosity of Polaris = (6500 K / 5778 K)^4 * (50 solar radii / 1 solar radius)^2 * Luminosity of Sun
And after a series of calculations that would make Einstein himself proud, we find that the luminosity of Polaris is approximately 1900 times the Sun's luminosity.
So, in star terms, Polaris is like the rockstar of luminosity! It's shining 1900 times brighter than our good ol' Sun!