Star Brightness and the Magnitude System

The stars we see at night appear to have a wide range of brightnesses. Stellar brightnesses were first put on a quantitative scale by the Greek astronomer Hipparchus around 130 BC. He arranged the visible stars in order of apparent brightness on a scale which ran from 1 to 6, with stars ranked ``1'' being the brightest. The ranks were called magnitudes; a star was said to be of the first magnitude, third magnitude, and so forth.

When the telescope was invented, around 1610 AD, astronomers realized that stars existed which were many times fainter than those that could be seen with the naked eye. Thus, Hipparchus' ancient scale, which ended with 6th magnitude (the faintest stars visible to the eye under very dark sky conditions), had to be extended to higher (fainter) values. The faintest objects now detected by large telescopes are around magnitude 29. Although Hipparchus' scale stopped at 1 on the bright end, modern recalibrations in terms of flux (see below) have forced changes. Bright objects now have zero or negative magnitudes. The brightest star, Sirius, has a magnitude of -1.4. The Sun and planets were not included in Hipparchus' scale. Now we say that the Sun is -26 magnitude, while Venus, at its brightest, is -4.4.

Astronomers still use this magnitude system even though it has two awkward features. The first is that it runs ``backwards'': fainter stars have larger magnitudes. The second is that because it was originally based on the appearance of stars to the human eye, it is a logarithmic system. That is, the response of the eye is proportional not to the amount of light received but to the logarithm of the of the amount of light received. Most sensory responses in living organisms have this ``non-linear'' characteristic.

You should become familiar with the magnitude scale by comparing the star magnitudes given on star charts with the appearance of the same stars in the sky with the naked eye. It is useful to memorize the magnitudes of a few conspicuous stars to use as a guide. One handy set of calibrators visible most of the year is the ``pan'' of the Little Dipper (Ursa Minor), whose four stars are of magnitudes 2, 3, 4, 5.

The modern magnitude scale is based on quantitative measurement of star brightnesses with special light-sensitive detectors, such as charge-coupled devices (CCD's). Any light detector, including the human eye, responds to the total amount of light incident on it in a given amount of time. This, in turn, depends on the collecting area of the instrument. The collecting area of your eye is the open area of the iris called the pupil; your pupil becomes larger in darker conditions, but it cannot open beyond a certain maximum size. The maximum diameter of your pupil under dark conditions is about 7 mm (0.7 cm), so the collecting area of your eye is about $\pi$r² = $\pi$(0.7/2)² cm² = 0.385 cm². The collecting area of a telescope can be much larger. An 8-in (20 cm) diameter telescope, for instance, has a collecting area of $\pi$(20/2)² cm² = 314 cm² which is about 800 times larger than your pupil. Thus with an 8-in telescope you can see stars which produce 800 times less flux than the faintest stars visible to your unaided eye. Modern instruments can have a diameter of up to 10 m, giving them well over a million times the collecting area of your eye!

Today, the magnitude of a star is defined in terms of the flux of light from the star reaching the Earth. Flux is the measure of light energy that is incident on some detector per unit area per unit time. The standard units for this quantity used by astronomers are ``ergs per square centimeter per second'' [erg cm^-2 s^-1]. An erg is a small unit of energy, about equal to that of a jumping flea. As a result of the large distance between us and the stars, we receive little flux from them. The flux from a first magnitude star is only about 2 millionths of an erg per square centimeter per second. It would take 170 million years to warm up a cup of coffee using this starlight!

The magnitude scale is fixed so that two stars which are 5 magnitudes different have a flux ratio of 100. For example, a 1^st magnitude star has a flux 100 times larger than a 6^th magnitude star, which itself has a flux 100 times larger than an 11^th magnitude star, and so on. Thus a 1^st magnitude star has a flux 100 x 100 = 10,000 times larger than an 11^th magnitude star. In general, a star n magnitudes brighter than another has a flux 100^n/5 times larger. In other words, if star 1 has flux f₁ and magnitude m₁, and star 2 has flux f₂ and magnitude m₂, then the flux ratio is given by:

$${\frac{f_{1}}{f_{2}}}$$ = 100^{(m₂ - m₁)/5}

Since 100 = 10², it is also true that:

$${\frac{f_{1}}{f_{2}}}$$ = 10^{2(m₂ - m₁)/5} = 10^{0.4(m₂ - m₁)}

A difference of 1 magnitude corresponds to a flux ratio of 10^2/5 = 2.512. Notice that if m₂ is bigger than m₁, then f₂ is smaller than f₁ (the ``backwards'' character of the magnitude system). Another way of expressing this is to take the logarithm of both sides of the equation:

log$$\left(\vphantom{ \frac{f_{1}}{f_{2}} }\right.$$$${\frac{f_{1}}{f_{2}}}$$ $$\left.\vphantom{ \frac{f_{1}}{f_{2}} }\right)$$ = log$$\left(\vphantom{ 10^{0.4 (m_{2} - m_{1})}}\right.$$10^{0.4(m₂ - m₁)}$$\left.\vphantom{ 10^{0.4 (m_{2} - m_{1})}}\right)$$ = 0.4(m₂ - m₁)

which is usually written as,

(m₂ - m₁) = 2.5log$$\left(\vphantom{ \frac{f_{1}}{f_{2}} }\right.$$$${\frac{f_{1}}{f_{2}}}$$ $$\left.\vphantom{ \frac{f_{1}}{f_{2}} }\right)$$

With logarithmic scales, it is possible to ``compress'' a vast range of values onto a manageable scale. For example, the faintest stars detected by the largest telescopes are 25 billion times fainter than 1^st magnitude stars, but they have a magnitude of only 27.

The magnitudes discussed so far are called apparent magnitudes because they indicate how bright an object appears as measured from Earth. The apparent brightness depends on the intrinsic brightness (how much energy the object is emitting) and how far away the object is. Often it is the intrinsic brightness which is of interest. One way to measure this is to consider placing all objects at the same distance. Conventionally, astronomers use a distance of 10 parsecs (1 parsec = 32.5 light years or 3.1 x 10¹⁴ km). The absolute magnitude is the magnitude a star would have if it were placed at this distance. Therefore, it measures the intrinsic brightness of the star. For example, the Sun and Antares have absolute magnitudes of +4.8 and -5.2 respectively. If both were placed at a distance of 10 parsecs, the Sun would appear as a rather faint star (+4.8) while Antares would dominate the night sky (-5.2). The difference of 10 magnitudes implies Antares radiates 10,000 times more light energy per second than the Sun. By ``placing'' the stars at the same distance, we have removed the bias due to the Sun's proximity and Antares' great distance, allowing us to compare the two stars' intrinsic brightnesses.