<p>The difference between the apparent magnitudes of two stars of the same spectral type (or colour in this case) is equal to the difference in their distance moduli (ignoring any extinction effects).</p>
<p>i.e. If we take two stars of the same spectral type and assume that they have the same absolute magnitude <span class="math-container">$M$</span>:
<span class="math-container">$$ m_1 - M = 5\log d_1 - 5$$</span>
<span class="math-container">$$ m_2 - M = 5\log d_2 -5 $$</span>
subtracting one from the other:
<span class="math-container">$$m_1 - m_2 = 5(\log d_1 - \log d_2)\ . $$</span></p>
<p>Since you know the apparent magnitude and distance to your calibration stars (<span class="math-container">$m_1$</span>, <span class="math-container">$d_1$</span>), then you can work out the distance of stars in the Pleiades that have apparent magnitude <span class="math-container">$m_2$</span> and the same spectral type. If you have many stars to compare or a a range of apparent magnitudes in the Pleiades at the same colour (remember that some of these may be binaries) then you will need to do some averaging.</p>

CC BY-SA 4.0
2023-08-20T12:03:08.613Z
I've found the difference between the apparent magnitudes of the stars respectively: $0, 0.34, 0.61$ and $0.88$. Now if I took a look at the HR-diagram the same $B-V$ index corresponds with respectively apparent magnitudes on the main sequence between: $6-8, 8-9, 9-11$ and $10-12$ with here and there some stars that also have that $B-V$ index but lie way above or beneath the main sequence. But okay I kind of understand what you are saying, but in order to find the distance to the Pleiades with the distance modulus I also need to find the absolute magnitude of the Pleiades, or is that incorrect
lynx_s
CC BY-SA 4.0
2023-08-20T13:24:10.003Z
Ahh I get it, although I don't seem to get the right answer, I found 411.09 ly and the correct distance is 444.2 ly, but could this be because of the way I probably chose the apparent magnitudes of the HR-diagram of the Pleaides?
lynx_s
CC BY-SA 4.0
2023-08-20T11:47:11.610Z
2023-08-20T12:30:36.923Z
2023-08-20T12:30:36.923Z
ProfRob
<p>So I have a table of data for the following stars: Vega, <span class="math-container">$\alpha$</span> Crv, <span class="math-container">$\beta$</span> CVn and <span class="math-container">$\epsilon$</span> Eri. I have been given their parallax, luminosity, <span class="math-container">$m_{B}$</span> magnitude and the <span class="math-container">$m_{V}$</span> magnitude. Now I'm asked to first calculate the distance <span class="math-container">$d$</span> of all the stars aswell as the absolute bolometric magnitude <span class="math-container">$(M_{bol})$</span> of the stars. Now I used the basic parallax formula for the distances: <span class="math-container">$$d=\frac{1 AU}{\tan{\theta}}$$</span> with <span class="math-container">$\theta$</span> the parallax in milli-arcsec. Now as for the absolute bolometric magnitude, I first wanted to use the distance modulus to calculate the absolute magnitude, but since I don't have the bolometric magnitude this isn't possible I think, so I used the formula that uses the luminosity given by: <span class="math-container">$$M_{bol}=M_{bol,\odot}+2.5\log\frac{L_{\odot}}{L}$$</span> I think I found the right values, but now there is a second question where they ask to compare the apparent magnitude in the V-band of the four stars with the observed apparent magnitudes of stars with the same B-V index in the Pleiades cluster, where they've given a figure of the HR-diagram of the Pleiades. Then they ask to calculate the distance to the Pleiades based on the mean apparent visual magnitudes. So as I said the first question I think I have, so the distance and absolute bolometric magnitude, but I don't know how we can find a distance for the Pleiades based on all that.</p>

CC BY-SA 4.0
2023-08-20T11:33:47.753Z
Also the HR-diagram of the Pleiades, the y-axis is $m_{V}$ and the x-axis is the $B-V$ index
lynx_s
<observational-astronomy>
A question about the cosmic distance ladder
lynx_s
# A question about the cosmic distance ladder

(Asked by: username_0 on 2023-08-20)

So I have a table of data for the following stars: Vega, $\alpha$ Crv, $\beta$ CVn and $\epsilon$ Eri. I have been given their parallax, luminosity, $m\_{B}$ magnitude and the $m\_{V}$ magnitude. Now I'm asked to first calculate the distance $d$ of all the stars aswell as the absolute bolometric magnitude $(M\_{bol})$ of the stars. Now I used the basic parallax formula for the distances: $$d=\frac{1 AU}{\tan{\theta}}$$ with $\theta$ the parallax in milli-arcsec. Now as for the absolute bolometric magnitude, I first wanted to use the distance modulus to calculate the absolute magnitude, but since I don't have the bolometric magnitude this isn't possible I think, so I used the formula that uses the luminosity given by: $$M\_{bol}=M\_{bol,\odot}+2.5\log\frac{L\_{\odot}}{L}$$ I think I found the right values, but now there is a second question where they ask to compare the apparent magnitude in the V-band of the four stars with the observed apparent magnitudes of stars with the same B-V index in the Pleiades cluster, where they've given a figure of the HR-diagram of the Pleiades. Then they ask to calculate the distance to the Pleiades based on the mean apparent visual magnitudes. So as I said the first question I think I have, so the distance and absolute bolometric magnitude, but I don't know how we can find a distance for the Pleiades based on all that.

**Tags:** observational-astronomy
> **Comments:**
> **username_0**: Also the HR-diagram of the Pleiades, the y-axis is $m_{V}$ and the x-axis is the $B-V$ index  

---

## 1 answers

### Answer by username_1 on 2023-08-20. Score: 2

The difference between the apparent magnitudes of two stars of the same spectral type (or colour in this case) is equal to the difference in their distance moduli (ignoring any extinction effects).

i.e. If we take two stars of the same spectral type and assume that they have the same absolute magnitude $M$:
$$ m\_1 - M = 5\log d\_1 - 5$$
$$ m\_2 - M = 5\log d\_2 -5 $$
subtracting one from the other:
$$m\_1 - m\_2 = 5(\log d\_1 - \log d\_2)\ . $$

Since you know the apparent magnitude and distance to your calibration stars ($m\_1$, $d\_1$), then you can work out the distance of stars in the Pleiades that have apparent magnitude $m\_2$ and the same spectral type. If you have many stars to compare or a a range of apparent magnitudes in the Pleiades at the same colour (remember that some of these may be binaries) then you will need to do some averaging.

> **Comments:**
> **username_0**: I've found the difference between the apparent magnitudes of the stars respectively: $0, 0.34, 0.61$ and $0.88$. Now if I took a look at the HR-diagram the same $B-V$ index corresponds with respectively apparent magnitudes on the main sequence between: $6-8, 8-9, 9-11$ and $10-12$ with here and there some stars that also have that $B-V$ index but lie way above or beneath the main sequence. But okay I kind of understand what you are saying, but in order to find the distance to the Pleiades with the distance modulus I also need to find the absolute magnitude of the Pleiades, or is that incorrect  
> **username_0**: Ahh I get it, although I don't seem to get the right answer, I found 411.09 ly and the correct distance is 444.2 ly, but could this be because of the way I probably chose the apparent magnitudes of the HR-diagram of the Pleaides?  

---

