Difference in power and absolute magnitude of a star

[Rohan1997]

My physics book mentions that a stars luminosity is its total power output at all wavelengths and that
absolute magnitude is defined as inherent brightness and NOT luminosity.

It then mentions that two stars of the same power output have the same absolute magnitude.

Since luminosity is the power output of a star how is this possible? the book contradicts itself by saying that absolute magnitude cannot be defined as luminosity and then mentions that two stars with same absolute magnitude have the same power/luminosity?

To be honest the whole thing with intensity, power output, luminosity and brightness of stars is very confusion for me and I have spent the past hour just trying to grasp these simple concepts, it would be much appreciated if anyone could help me in understanding this.

Many thanks

 

[|Glitch|]

My physics book mentions that a stars luminosity is its total power output at all wavelengths and that
absolute magnitude is defined as inherent brightness and NOT luminosity.

It then mentions that two stars of the same power output have the same absolute magnitude.

Since luminosity is the power output of a star how is this possible? the book contradicts itself by saying that absolute magnitude cannot be defined as luminosity and then mentions that two stars with same absolute magnitude have the same power/luminosity?

To be honest the whole thing with intensity, power output, luminosity and brightness of stars is very confusion for me and I have spent the past hour just trying to grasp these simple concepts, it would be much appreciated if anyone could help me in understanding this.

Many thanks

Absolute magnitude is typically broken down into different wavelengths, whereas luminosity (as you mentioned) includes all wavelengths. However, absolute bolometric magnitude does include all wavelengths (whether observed or not), with a correction depending on the star, and therefore does correspond to luminosity as follows:

L_star = L_☉ × 10^{((M_bolsun - M_bolstar) / 2.5)}​

Where:

M_bolsun = Absolute bolometric magnitude of the sun;
M_bolstar = Absolute bolometric magnitude of the star.​

A bolometric correction must be made to the absolute magnitude in order to convert an objects absolute visible magnitude to its absolute bolometric magnitude. In 1999 the IAU defined absolute bolometric magnitude zero to correspond to a bolometric luminosity of 3.055 × 10²⁸ Watts. This particular luminosity was selected as the zero-point for the absolute bolometric magnitude scale so that the sun's luminosity would correspond to absolute bolometric magnitude of 4.75. Since the sun has an absolute visual magnitude of 4.82, the bolometric correction for the sun is -0.07 magnitude.

 

[Rohan1997]

Absolute magnitude is typically broken down into different wavelengths, whereas luminosity (as you mentioned) includes all wavelengths. However, absolute bolometric magnitude does include all wavelengths (whether observed or not), with a correction depending on the star, and therefore does correspond to luminosity as follows:

Thanks for the quick reply, I'm only doing high school physics, but your answer did make sense to me at the beginning.

So what you're saying is that bolometric absolute magnitude is a logarithmic scale of the luminosity of a star (and therefore the power output of a star)?
and therefore you can say two stars of the same bolometric absolute magnitude have the same power output?

For my course it only mentions absolute magnitude (defined as the apparent magnitude of a star 10 pc away in my textbook) and not bolometric, do you think this mean that it's only taking about the absolute magnitude of visible light wavelengths then?

Does this also apply for apparent magnitude and intensity?
So the bolometric apparent magnitude is a logarithmic scale of the intensity and normal apparent magnitude is the brightness of a star?
and therefore could you say two stars of same intensity have the same bolometric apparent magnitude?

 

[|Glitch|]

Thanks for the quick reply, I'm only doing high school physics, but your answer did make sense to me at the beginning.

So what you're saying is that bolometric absolute magnitude is a logarithmic scale of the luminosity of a star (and therefore the power output of a star)?
and therefore you can say two stars of the same bolometric absolute magnitude have the same power output?

Precisely. When referring to the total energy output of a star, the proper term is bolometric absolute magnitude.

For my course it only mentions absolute magnitude (defined as the apparent magnitude of a star 10 pc away in my textbook) and not bolometric, do you think this mean that it's only taking about the absolute magnitude of visible light wavelengths then?

To be clear what is meant they should state more than just absolute magnitude. There are different absolute magnitudes (fluxes) for different electromagnetic wavelengths. The Johnson-Cousins-Glass photometric system is the most common, but there are other photometric systems used as well. It is therefore necessary to specify to which type of absolute magnitude is being referred: M_bol, M_u-b, M_v, etc.

Using the bolometric luminosity of Vega (40.12 ± 0.45) the bolometric absolute magnitude can be calculated as:

M_bolstar = M_bolsun + (-2.5 × log₁₀(L_star / L_☉)) = 4.75 + (-2.5 × 1.603 ± 0.0048) = 0.7415 ± 0.012​

The visual absolute magnitude of Vega (0.58), subtracting the bolometric absolute magnitude (0.74), should give you the bolometric correction used for that star (-0.16)

Does this also apply for apparent magnitude and intensity?

By "intensity" I assume you are referring to the total electromagnetic energy output of the star. In which case the bolometric apparent magnitude and "intensity" of the star would diminish based upon distance using the inverse-square law.

So the bolometric apparent magnitude is a logarithmic scale of the intensity and normal apparent magnitude is the brightness of a star?

Apparent magnitude, whether bolometric or not, is dependent upon distance, absolute magnitude is not.

and therefore could you say two stars of same intensity have the same bolometric apparent magnitude?

You can say two stars with the same total electromagnetic energy output have the same bolometric absolute magnitude. The bolometric apparent magnitude will vary for each star depending on the distance each star is away from the observer.