# Distance to a Star cluster

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.

## Answers and Replies

Homework Helper
Gold Member
2022 Award
$$0-M_v=5log(7.76/10)=0.55$$
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 $$M_v=0.55$$

Homework Helper
Gold Member
2022 Award
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
mv
Mv
Av

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