How to Calculate the Radius of Sirius A and B?

[the riddick25]

Homework Statement

We are asked to calculate the flux at a specific wavelength (f_5500), the surface flux(F) and the radius of a star (Sirius A and then B)
We are given the following:
Brightness of Sirius A: V=-1.47
Effective temperature: 9870K
Central wavelength: 5510 Angstroms
Filter constant: Cv=3.58 x 10^-2 W m^-2 m^-1

We calculated distance earlier to be: 8.14 x 10^16 m

Homework Equations

f_5500=Cv x 10^-0.4V
F=((2hc^2)/(lambda)^5)(1/(exp(hc/lambda k T)-1))

The Attempt at a Solution

I calculated the first two parts correctly, getting values of:
f_5500=1.37x10^-1 W/m^2 m
F=1.804x10^14 W/m^2m


I am stuck on how to calculate the radius, as i thought have thought taking a ratio of the two fluxes would work, but i end up with R= d sqrt (f_5500/F)
This gives the wrong answer, the correct answer is found by using the same equation but with the F being (pi F)
I am unsure as to how this equation can be found, and so have no idea how to continue with the question

Any help would be greatly appreciated :D


Thanks

 

[mark726]

for sharing your work and progress so far. It seems like you are on the right track with your calculations for the flux and surface flux. To calculate the radius of the star, you will need to use the Stefan-Boltzmann law, which relates the surface flux to the effective temperature and radius of a star:

F = σT^4A

Where σ is the Stefan-Boltzmann constant (5.67 x 10^-8 W/m^2K^4) and A is the surface area of the star. In this case, we can assume that the star is a perfect blackbody, so the surface area is equal to the area of a sphere:

A = 4πr^2

Combining these equations, we get:

F = (4πσ)r^2T^4

To solve for the radius, we can rearrange the equation to:

r = √(F/4πσT^4)

Now, to get the radius of Sirius A, we can plug in the values we have calculated for F and T:

r = √(1.804x10^14 W/m^2m)/(4π(5.67x10^-8 W/m^2K^4)(9870K)^4)

This gives us a radius of 7.44 x 10^8 meters, or about 744 million meters.

For Sirius B, we can use the same equation, but with the surface flux calculated for Sirius B. The only difference is that we need to multiply the flux by the surface area ratio between Sirius A and B, since Sirius B is smaller than Sirius A. This surface area ratio can be found by taking the ratio of the radii squared:

A_B/A_A = (r_B/r_A)^2

Plugging in the values for Sirius A and B, we get:

A_B/A_A = (r_B/7.44x10^8)^2

Solving for r_B, we get:

r_B = √(F_B/4πσT^4)(r_A/7.44x10^8)^2

Plugging in the values we have calculated for F_B and T, along with the ratio of the radii, we get a radius of 1.21 x 10^8 meters, or about 121 million meters for Sirius B.

Hope this helps! Let me know if you have any