Cepheid Variables and the Small Magellanic Cud

[Berdi]

Homework Statement

The Galactic Cepheid variable δ Cephei has a parallax of 0.0033 arcseconds. An astronomer locates another Cepheid, with the same period as δ Cephei, in the Small Magellanic Cloud (SMC). If this Cepheid appears 11.5 magnitudes fainter than δ Cephei, calculate the distance to the SMC.

Homework Equations

$$m-M=5\log(d)-5$$

$$L=4\pi R^2F$$

$$m_{1}-m_{2}=2.5\log( L_{2}/L_{1})$$

The Attempt at a Solution

Well, I know the distance to the first Cepheid, as it is 1/Parallax = 303pc. I understand that the same period means the same luminosity, and that 11.5 fainter will mean that the m will be actually 11.5 larger, But I'm unsure how this will help me. Unless you say that $$m_{1}-m_{2}$$ is 11.5 and change the L's for 1/d^2 ?

 

[mgb_phys]

The absolute magnitudes of the two variables are the same (same period)
If one appears 11.5 mag fainter then it's obvisouly further away.

The first equation tells you have far away an object would have to be to be less bright by (m-M) magnitudes compared to an object 1pc away.

Another way to think of it. Work out the relative apparent luminosity of the two stars, then if the second one was 1/4 as bright it woudl have to be twice as far away.

 

[kerimek]

I would approach this problem by first understanding the concept of Cepheid variables and their use as standard candles for distance measurements. Cepheid variables are pulsating stars that have a relationship between their period of pulsation and their luminosity, making them useful for calculating distances in the universe.

In this case, we have two Cepheid variables, δ Cephei and the one in the SMC, with the same period. This means they have the same luminosity. We also know that the SMC Cepheid appears 11.5 magnitudes fainter than δ Cephei. This means that the difference in their apparent magnitudes, m_{1}-m_{2}, is 11.5.

Using the equation m_{1}-m_{2}=2.5\log( L_{2}/L_{1}), we can rewrite this as:

11.5 = 2.5\log( L_{2}/L_{1})

We also know that the luminosity of a star is related to its distance by the equation L=4\pi R^2F, where R is the radius of the star and F is its flux. Since both Cepheids have the same period, their radii are likely to be similar. This means that the difference in their luminosities is mainly due to the difference in their distances.

Therefore, we can rewrite the equation as:

11.5 = 2.5\log( d_{2}^2/d_{1}^2)

Solving for d_{2}, the distance to the SMC, we get:

d_{2} = d_{1}\times 10^{11.5/2.5}

Substituting d_{1} = 303pc, we get:

d_{2} = 303\times 10^{11.5/2.5} = 3.99\times 10^5 pc

Therefore, the distance to the SMC is approximately 399,000 parsecs.