Apparent Magnitude, Absolute Magnitude and Distance
Apparent magnitude and Absolute magnitude are two ways of comparing an object's brightness. In this example we look at the relationship between absolute magnitude, apparent magnitude and luminosity.
There are two main types of magnitude commonly used in astronomy. The first of these, apparent magnitude, is the brightness of the object as seen by an observer on the Earth. The apparent magnitude of a star is dependent on two factors:
- The luminosity of the star (total energy per second radiated)
- The distance of the star from Earth
The second, absolute magnitude, is dependent solely on the stars luminosity and can be regarded as an intrinsic property of the star. Absolute magnitude is defined as the apparent magnitude of an object if it were a standard distance from the Earth. The standard distance is 10 parsecs. Since distance is always equal when comparing absolute magnitudes, it can be removed as a factor in the stars brightness which is why it can be regarded as an intrinsic property.
Absolute magnitude and Luminosity
A stars luminosity, L, is the total amount of energy radiated per unit time. The absolute magnitude of a star is related to its luminosity in the same way as apparent magnitude is related to flux. If we compare the ratio of the brightness of two stars, expressed in terms of their luminosities, then we obtain a relation for the difference in their absolute magnitudes.
Equation 23 - Absolute Magnitude Relation
Capital letters are used to indicate absolute magnitudes and lower case letters are used to identify apparent magnitudes.
As we have previously stated, absolute magnitude is the apparent magnitude of an object if it was a distance of 10 parsecs from the Earth.
It is clear from this definition that a star located at 10 parsecs from the Earth will have the same apparent and absolute magnitude. A star that is further away than 10 parsecs will have a fainter apparent magnitude than absolute magnitude and a star that is closer than 10 parsecs will have a brighter apparent magnitude than absolute magnitude.
How do we know a stars absolute magnitude? We could travel to every star and measure the apparent brightness from a distance of 10 parsecs, but at the moment that really isn't a practical solution. Luckily for us however, the apparent and absolute magnitudes are related by a very important formula.
Distance Modulus is the difference between the apparent and absolute magnitudes. This can be obtained by combining the definition of absolute magnitude with an expression for the inverse square law and Pogson's relation. Using the distance modulus it is possible to establish a relationship between the absolute magnitude, M, of a star, its apparent magnitude, m, and its distance, d.
The inverse square law tells us that for a star at distance d (parsecs), with observed flux Fm, then its flux FM at 10 parsecs would be given by:
Equation 24 - Inverse Square Law for Flux
We can combine this with equation 23 above to give the distance modulus equation.
Equation 25 - Distance Modulus
If we measure a stars apparent magnitude, and its distance in parsecs is known, then we can determine the absolute magnitude and hence the luminosity of the star. If we know the stars absolute and apparent magnitudes we can use distance modulus to calculate the distance to the star. This equation is very powerful and will be used a great many times in upcoming tutorials.
The formula for calculating Absolute Magnitude within our galaxy is:
Equation 31 - Absolute Magnitude
Where D is the distance to the star in parsecs.
Barnard's Star lays 1.82 parsecs away and has an observed (apparent) magnitude of 9.54.
m - M = 5((log10 D)-1) M = 9.54 * 5((log10 1.82)-1) M = 9.54 - (-3.7) M = 13.24
If Barnard's Star were to be moved to a distance of 10 parsecs from the Earth it would then have a magnitude of 13.24.
If we already know both Apparent and Absolute magnitudes, it is possible to calculate the distance to the star:
d = 100.2(m - M + 5)
Using Barnard's Star again,
d = 100.2(9.54-13.24+5) d = 100.26 d = 1.82 parsecs
Another type of magnitude of interest to astronomers is the bolometric magnitude. So far the absolute and apparent magnitudes are based on the total visible energy radiated from the star. We know that not all of that energy is received on Earth since it is filtered out by our atmosphere.
Bolometric magnitude is a based on the flux throughout the entire electromagnetic spectrum. The term absolute bolometric magnitude is based specifically on the luminosity (or total rate of energy output) of the star. We will look at bolometric magnitudes in a later tutorial.