How Far Is the Star Cluster with Apparent Magnitude 15 mag and B-V of 1.39 mag?

[Taylor_1989]

Homework Statement

Question: Vega is a star of spectral type A0V at a distance of 7.76pc and has by definition
$m V =m B =0.0\space mag.$

Observations of a star cluster show, that all stars in the cluster with a spectral type A0V have an apparent magnitude of $m V =15 \space mag$ and a colour index of $B-V=1.39 \space mag$. Assuming an interstellar reddening law of $A V /E(B-V)=3.2$, calculate the distance of this star cluster.

Homework Equations

$$m-M=5log(d/10) \space [1]$$
$$E(B-V)=(B-V)-(B-V)_0 \space [2]$$
$$m_v=M_v+5log(d)-5+A_v \space [3]$$

3. The Attempt at a Solution
Step 1: To calculate the intrinsic brightness which is $(B-V)_0$ so I used vega to do so.
I did this because according to wiki intrinsic brightness is a measure of absolute magnitude. So my reasoning is that if as the cluster and vega are in the same spectral band then they would sahre the same absolute magnitudes.

$$0-M_v=5log(7.76/10)=0.55$$

Step 2: Substituting this into my equation 2 as follows i.e colour excess equation.

$$E(B-V)=1.39-0.55=0.84$$

Step 3: Rearange the equation $A V /E(B-V)=3.2$ to calculate the extinction value $A_v$

$$A_v=(3.2)(0.84)=2.69$$

Step 4: Using equation $[3]$ the modified distance modulus and rearanging to make $d$ the subject, I make $d$ the following:

$$10^{\frac{m_v -M_v+5-A_v}{5}}=d$$$$10^{\frac{15 -0.55+5-2.69}{5}}=2249pc$$

This to me seem quite far out, and in practical term I don't think it is possible. Have I made a wrong calculation or is my assumption in the first step completely wrong, or have I complete miss understood the question.

Any advice would be much appreciated thanks in advance.

 

[haruspex]

$$0-M_v=5log(7.76/10)=0.55$$

Sign?

 

[Taylor_1989]

@haruspex what do you mean by Sign?

 

[Alloymouse]

@haruspex what do you mean by Sign?

I think you forgot a negative sign here:
log(0.776) is negative.

 

[Taylor_1989]

I think you forgot a negative sign here:
log(0.776) is negative.

Sorry I automatically carred it though so, that my fault for missing the step, so it should read $$M_v=0.55$$

 

[haruspex]

Sorry I automatically carred it though so, that my fault for missing the step, so it should read $$M_v=0.55$$

Ok.
But I am not a cosmologist, so I am snowed by the plethora of variables, none of which are defined.
Would you mind adding some clarification, or maybe a link?
mV
mB
B
V
E, or is it a function E()?
A, or is it a variable AV?
m
M
(B-V)₀
m_v
M_v
A_v

 

[Alloymouse]

Homework Statement

Question: Vega is a star of spectral type A0V at a distance of 7.76pc and has by definition
$m V =m B =0.0\space mag.$

Observations of a star cluster show, that all stars in the cluster with a spectral type A0V have an apparent magnitude of $m V =15 \space mag$ and a colour index of $B-V=1.39 \space mag$. Assuming an interstellar reddening law of $A V /E(B-V)=3.2$, calculate the distance of this star cluster.

Homework Equations

$$m-M=5log(d/10) \space [1]$$
$$E(B-V)=(B-V)-(B-V)_0 \space [2]$$
$$m_v=M_v+5log(d)-5+A_v \space [3]$$

3. The Attempt at a Solution
Step 1: To calculate the intrinsic brightness which is $(B-V)_0$ so I used vega to do so.
I did this because according to wiki intrinsic brightness is a measure of absolute magnitude. So my reasoning is that if as the cluster and vega are in the same spectral band then they would sahre the same absolute magnitudes.

$$0-M_v=5log(7.76/10)=0.55$$

Step 2: Substituting this into my equation 2 as follows i.e colour excess equation.

$$E(B-V)=1.39-0.55=0.84$$

Step 3: Rearange the equation $A V /E(B-V)=3.2$ to calculate the extinction value $A_v$

$$A_v=(3.2)(0.84)=2.69$$

Step 4: Using equation $[3]$ the modified distance modulus and rearanging to make $d$ the subject, I make $d$ the following:

$$10^{\frac{m_v -M_v+5-A_v}{5}}=d$$$$10^{\frac{15 -0.55+5-2.69}{5}}=2249pc$$

This to me seem quite far out, and in practical term I don't think it is possible. Have I made a wrong calculation or is my assumption in the first step completely wrong, or have I complete miss understood the question.

Any advice would be much appreciated thanks in advance.

I think for the step on finding E(B-V), you just need to sub (B-V) as 0 as the Stars in the cluster have the same spectral type as Vega. Hence you should get E(B-V) to be -1.39.

Take my point with a pinch of salt though, it's been a while since I last did stuff like this.

 

[Taylor_1989]

Okay first @haruspex sorry for not labelling the equations correctly, but after redoing the question and a bit of research through some books @Alloymouse is correct I should have set $(B-V)_0=0$. I am new to the como stuff as it something I am not very intresed in but unfortunately a compulsory module.

But thank you all for the help