Distance to Stars: Calculating the Distance from Sun

[hhhmortal]

Homework Statement

Two white dwarfs are found to lie within 16 arcseconds of each other on the sky. The probability that such an alignment occurs by chance is very small. Therefore, they are almost certainly physically associated, lying at the same distance fr0m the Sun.

One of the stars is relatively cool, with an effective temperature of 16,000 K and a measured radius of 0.01 times that of the Sun.

a) Calculate the distance to the star, given that the apparent visual magnitude of the star is 14.11 and the bolometric correction for 16,000 K is –1.3.

Homework Equations

I use m_bol(2) - m_bol(1) = 5log(R_1/R_2) + 10log(T_1/T_2)

To find the apparent bolometric magnitude of the sun but how do I find the distance after this?

Thanks.

The Attempt at a Solution

 

[Hao]

To find distance, you need the Apparent Visual Magnitude, and the Absolute Visual Magnitude.

As you can calculate the Bolometric Magnitude of the white dwarf (black body), and you know its Bolometric Correction, you will be able to obtain the Absolute Magnitude from both factors.

 

[hhhmortal]

To find distance, you need the Apparent Visual Magnitude, and the Absolute Visual Magnitude.

As you can calculate the Bolometric Magnitude of the white dwarf (black body), and you know its Bolometric Correction, you will be able to obtain the Absolute Magnitude from both factors.

Ok. How does that work though? I know the apparent visible magnitude. I can work out the apparent bolometric magnitude using bolometric correction. But How do i work out the absolute bolometric magnitude? do i use the equation I displayed above, then use the bolometric correction to find the absolute visivle magnitude?


Thanks!

 

[Hao]

I don't think the equation displayed above is necessary.

The Bolometric Magnituide is a measure of the total (electromagnetic; both visible and invisible) power emmited by a star.

If we assume that the white dwarf is a perfect black body radiator with a known effective temperature, and a known surface area, we can calculate its power, and hence find the Absolute Bolometric Magnituide (conversion of units).

This can then be directly converted to the Absolute Magnitude.

 

[hhhmortal]

I don't think the equation displayed above is necessary.

The Bolometric Magnituide is a measure of the total (electromagnetic; both visible and invisible) power emmited by a star.

If we assume that the white dwarf is a perfect black body radiator with a known effective temperature, and a known surface area, we can calculate its power, and hence find the Absolute Bolometric Magnituide (conversion of units).

This can then be directly converted to the Absolute Magnitude.

Oh ok. So I calculate the luminosity of the star. This will give me the bolometric luminosity since I'm taking into account all wavelengths, hence I then use b = L/A to get the brightness which is the absolute bolometric magnitude..and then convert this to absolute magnitude using bolometric correction?

Is bolometric correction of Mb -Mv valid for both when working apparent and absolute magnitudes?

Thanks.

 

[Hao]

That is what I would do.

Actually, while the equation you gave isn't necessary (as it is just a restatement of results that we derive by assuming black body radiation), you can still apply it to find the Absolute Bolometric Magnitude as long as the reference m_bol(1) is the Abolsute Bolometric Magnitude of a known star, ie. the sun.

The Bolometric correction is a multiplicative factor that acts on the luminosity.

As magnitudes take the logarithm of luminosity, one will find that the Bolometric correction appears as a additive constant that works regardless of distance.

So, yes the correction is valid for both apparent and absolute magnitudes.

 

[hhhmortal]

That is what I would do.

Actually, while the equation you gave isn't necessary (as it is just a restatement of results that we derive by assuming black body radiation), you can still apply it to find the Absolute Bolometric Magnitude as long as the reference m_bol(1) is the Abolsute Bolometric Magnitude of a known star, ie. the sun.

The Bolometric correction is a multiplicative factor that acts on the luminosity.

As magnitudes take the logarithm of luminosity, one will find that the Bolometric correction appears as a additive constant that works regardless of distance.

So, yes the correction is valid for both apparent and absolute magnitudes.

Thanks! that was very useful. So it will just be a simple case of using the distance modulus after finding the magnitudes. Couldn't I have used the apparent and absolute bolometric magnitudes in the distance modulus equation instead of converting them into visible magnitudes?

 

[Hao]

You could do that, but the given information is that of the Apparent Magnitude of the white dwarf, which means that you either:

1) Convert the Apparent Magnitude of the white dwarf to Apparent Bolometric Magnitude

Or

2) Convert the Absolute Bolometric Magnitude to the Absolute Magnitude.

You will need to use the conversion factor inevitably.

Alternatively, if they provided the Apparent Bolometric Magnitude instead, no conversion would be necessary.

 

[hhhmortal]

You could do that, but the given information is that of the Apparent Magnitude of the white dwarf, which means that you either:

1) Convert the Apparent Magnitude of the white dwarf to Apparent Bolometric Magnitude

Or

2) Convert the Absolute Bolometric Magnitude to the Absolute Magnitude.

You will need to use the conversion factor inevitably.

Alternatively, if they provided the Apparent Bolometric Magnitude instead, no conversion would be necessary.

Oh that makes perfect sense.

I have one more question to ask, if that's ok

How would I be able to get the absolute visible magnitude for type I and II cepheids? E.g. from the following question:

(2) A Cepheid variable has been observed in a nearby galaxy. It has a pulsation period of 10 days and an apparent visual magnitude of 18. We do not know whether this is a Population I or II star.

i) Neglecting the effects of interstellar absorption, estimate the two possible distances to the galaxy.



Is there a separate relationship of period luminosity for type I and II Cepheids?

 

[Hao]

There are two possible distances, each one corresponding to whether the star is Type I or Type II.

When they say type, they refer to the Metallicity of a star
http://en.wikipedia.org/wiki/Metallicity

The use of Cepheid variable stars is not without its problems however. The largest source of error with Cepheids as standard candles is the possibility that the period-luminosity relation is affected by metallicity. For Galactic use only, the following relation is also valid in addition to those highlighted above:

Empirical formulas relating Absolute Magnitude to period for Type I may be found at:
http://en.wikipedia.org/wiki/Cepheid_variable

Unfortunately, I do not have the formula for Type II. It may be in your notes.

 

[hhhmortal]

There are two possible distances, each one corresponding to whether the star is Type I or Type II.

When they say type, they refer to the Metallicity of a star
http://en.wikipedia.org/wiki/Metallicity




Empirical formulas relating Absolute Magnitude to period for Type I may be found at:
http://en.wikipedia.org/wiki/Cepheid_variable

Unfortunately, I do not have the formula for Type II. It may be in your notes.

Yes, I was able to find the relationship for Type I Cepheids but not for Type II , I have tried looking at a few books and browsing on the internet, but no luck.

 

[Hao]

This graph might help:
http://books.google.co.uk/books?id=...4tyLDQ&sa=X&oi=book_result&ct=result&resnum=2

This is larger, but has a Log scale:
http://outreach.atnf.csiro.au/education/senior/astrophysics/variable_cepheids.html

There is a formula for Type II in this research paper:
http://arjournals.annualreviews.org/doi/pdf/10.1146/annurev.astro.43.072103.150612

 

[hhhmortal]

This graph might help:
http://books.google.co.uk/books?id=...4tyLDQ&sa=X&oi=book_result&ct=result&resnum=2

This is larger, but has a Log scale:
http://outreach.atnf.csiro.au/education/senior/astrophysics/variable_cepheids.html

There is a formula for Type II in this research paper:
http://arjournals.annualreviews.org/doi/pdf/10.1146/annurev.astro.43.072103.150612

I'm hoping they will give me either a graph or a formula tomorrow in the exam lol. anyhow the gradient for both seems the same so I would just need to alter the one for Type I slightly if worse comes to worse. So could I assume for type II it's:

Mv = -2.81logP - 0.8 ?

 

[Hao]

I'm afraid that would be your best bet.

It would be strange for an exam to not provide you with the relevant numerical factors.

 

[hhhmortal]

I'm afraid that would be your best bet.

It would be strange for an exam to not provide you with the relevant numerical factors.

Yea I know, this mock paper I am going over is a bit strange to be honest. The following question has been very confused:

b) A typical telescope limiting magnitude is 22. What is the maximum distance over which we can use the Cepheid variables to provide our distance scale.

(The absolute magnitude of the Sun = +4.83.)

Hint. Use the relationship between magnitude and luminosity to estimate the absolute magnitude of the Cepheid.


Well assumming I know the luminosity of the star I still need to be given luminosity of star to find its absolute magnitude...

 

[Hao]

The question gives you the maximum apparent magnitude, and applying this to the distance equation gives you a relationship between the distance and absolute magnitude.

We want to find the distance - the problem to to estimate the absolute magnitude.

We know that a Cepheid is usually a yellow star, and that gives us an estimate of the absolute magnitude.

(They provided the absolute magnitude of the sun for a reason).