## How bright is a star? Introducing apparent and absolute magnitude

In ancient Greece, for approx 2000 years ago, the Greek astronomer Hipparchus was the first to make a catalog of stars according to their brightness. The brightest stars were ‘magnitude 1’, the next brightest were ‘magnitude 2’, etc., down to ‘magnitude 6′, which were the faintest stars Hipparchus and ancient astronomers could see.

The Hipparchus scale is based on the fact that your eyes perceive equal ratios of intensity as equal intervals of brightness. On the quantified magnitude scale, a magnitude interval of 1 unit of magnitude corresponds to a factor of 100^1/5 or approximately 2.512 times the amount in actual intensity. For example, first magnitude stars are about 2.512 times brighter than 2nd magnitude stars, first magnitude stars are 2.512^2 times brighter than 3rd magnitude stars, etc.

Nowadays, CCDs inside digital cameras measure the amount of light coming from stars, and can provide a more precise definition of brightness. Using this scale, modern astronomers now define five magnitudes’ difference as having a brightness ratio of 100. Vega was used as the reference star for the scale. Initially Vega had a magnitude of 0, but more precise instrumentation changed that to 0.3.

The brighter an object appears, the lower the value of its apparent magnitude, with the brightest objects reaching negative values. The thing to remember is that brighter objects have smaller magnitudes than fainter objects and it can have negative values for very bright objects. The magnitude system is based on tradition and therefore seems a little bit odd.

When taking Earth as a reference point, however, the scale of magnitude fails to account for the true differences in brightness between stars. The apparent magnitude, depends on the location of the observer. Different observers will come up with a different measurement, depending on their locations and distance from the star. Stars that are closer to Earth, but fainter, could appear brighter than far more luminous ones that are far away. Therefore, it is useful to establish a convention whereby we can compare two stars on the same footing, without variations in brightness due to differing distances complicating the issue.

The solution was to implement an absolute magnitude scale to provide a reference between stars. To do so, astronomers calculate the brightness of stars as they would appear if it were 32.6 light-years, or 10 parsecs from Earth.

The Sun has for example an apparent magnitude of -26.8 and an absolute magnitude of 4.77. Here is how one can calculate the absolute magnitude of the Sun:

The basic formula relating the apparent (m) and absolute (M} magnitudes is

$M = m + 5 - 5 \log{D}$

where D is the distance to the object in pc.

Sun has m = -26.8 and it is located at 1 A.U. (astronomical unit) from us. 1 A.U. = 1.5 x 10^13 cm = 4.85 x 10-6 pc. Thus, the absolute magnitude of the sun is 4.77.

However, as we have seen in previous posts, since stars can emit light at different wavelengths ranging from high-energy X-rays to low-energy infrared radiation, depending on the type of star, they could be bright in some of these wavelengths and dimmer in others. To address this problem, astronomers must specify which wavelength they are using to make the absolute magnitude measurements. Another limitation is the sensitivity of the instrument (CCDs) used to make the measurement.

To conclude, and make a connection with the previous post on calculating distances to stars, let’s calculate the distance to Tau Ceti (ref) using parallax spectroscopy that we introduced in the previous post on luminosity. Tau Ceti is an interesting nearby star because since December 2012, there has been evidence of possibly five planets orbiting Tau Ceti, with two of these being potentially in the “habitable zone”, which is the zone were you could find among other things liquid water.

Tau Ceti has an apparent magnitude, m = 3.49, and its spectral type is a “G type main sequence star” (ref). From a Hertzsprung-Russell diagram which links spectral type to temperature/luminosity, this indicates an absolute magnitude, M, in the range: +5.0 to +6.5. Using the equation above:

$D = 10 \times \frac{(m-M+5)}{5}$

With M = +5.0, D = 5.00 pc and With M = +6.5, D = 2.50 pc. So the Tau Ceti’s distance to earth is between 2.5 and 5 pc. The Hipparcos/parallax measurements give d = 3.64 pc.

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