Apparent magnitude

History

The scale used to indicate magnitude originates in the Hellenistic practice of dividing stars visible to the naked eye into six magnitudes. The brightest stars in the night sky were said to be of first magnitude (m = 1), whereas the faintest were of sixth magnitude (m = 6), which is the limit of human visual perception (without the aid of a telescope). Each grade of magnitude was considered twice the brightness of the following grade (a logarithmic scale), although that ratio was subjective as no photodetectors existed. This rather crude scale for the brightness of stars was popularized by Ptolemy in his Almagest, and is generally believed to have originated with Hipparchus.

In 1856, Norman Robert Pogson formalized the system by defining a first magnitude star as a star that is 100 times as bright as a sixth-magnitude star, thereby establishing the logarithmic scale still in use today. This implies that a star of magnitude m is 2.512 times as bright as a star of magnitude m + 1. This figure, the fifth root of 100, became known as Pogson's Ratio. The zero point of Pogson's scale was originally defined by assigning Polaris a magnitude of exactly 2. Astronomers later discovered that Polaris is slightly variable, so they switched to Vega as the standard reference star, assigning the brightness of Vega as the definition of zero magnitude at any specified wavelength.

Apart from small corrections, the brightness of Vega still serves as the definition of zero magnitude for visible and near infrared wavelengths, where its spectral energy distribution (SED) closely approximates that of a black body for a temperature of 7004110000000000000♠11000 K. However, with the advent of infrared astronomy it was revealed that Vega's radiation includes an Infrared excess presumably due to a circumstellar disk consisting of dust at warm temperatures (but much cooler than the star's surface). At shorter (e.g. visible) wavelengths, there is negligible emission from dust at these temperatures. However, in order to properly extend the magnitude scale further into the infrared, this peculiarity of Vega should not affect the definition of the magnitude scale. Therefore, the magnitude scale was extrapolated to all wavelengths on the basis of the black body radiation curve for an ideal stellar surface at 7004110000000000000♠11000 K uncontaminated by circumstellar radiation. On this basis the spectral irradiance (usually expressed in janskys) for the zero magnitude point, as a function of wavelength can be computed. Small deviations are specified between systems using measurement apparatuses developed independently so that data obtained by different astronomers can be properly compared; of greater practical importance is the definition of magnitude not at a single wavelength but applying to the response of standard spectral filters used in photometry over various wavelength bands.

With the modern magnitude systems, brightness over a very wide range is specified according to the logarithmic definition detailed below, using this zero reference. In practice such apparent magnitudes do not exceed 30 (for detectable measurements). The brightness of Vega is exceeded by four stars in the night sky at visible wavelengths (and more at infrared wavelengths) as well as bright planets such as Venus, Mars, and Jupiter, and these must be described by negative magnitudes. For example, Sirius, the brightest star of the celestial sphere, has an apparent magnitude of −1.4 in the visible; negative magnitudes for other very bright astronomical objects can be found in the table below.

Astronomers have developed other photometric zeropoint systems as alternatives to the Vega system. The most widely used is the AB magnitude system, in which photometric zeropoints are based on a hypothetical reference spectrum having constant flux per unit frequency interval, rather than using a stellar spectrum or blackbody curve as the reference. The AB magnitude zeropoint is defined such that an object's AB and Vega-based magnitudes will be approximately equal in the V filter band.

Calculations

As the amount of light actually received by a telescope is reduced by transmission through the Earth's atmosphere, any measurement of apparent magnitude is corrected for what it would have been as seen from above the atmosphere. The dimmer an object appears, the higher the numerical value given to its apparent magnitude, with a difference of 5 magnitudes corresponding to a brightness factor of exactly 100. Therefore, the apparent magnitude m, in the spectral band x, would be given by:

which is more commonly expressed in terms of common (base 10) logarithms as:

where F_x is the observed flux using spectral filter x, and F_x,0 is the reference flux (zero-point) for that photometric filter. Since an increase of 5 magnitudes corresponds to a decrease in brightness by a factor of exactly 100, each magnitude increase implies a decrease in brightness by the factor ⁵√100 ≈ 2.512 (Pogson's Ratio). Inverting the above formula, a magnitude difference m₁ − m₂ = Δm implies a brightness factor of

Example: Sun and Moon

What is the ratio in brightness between the Sun and the full Moon?

The apparent magnitude of the Sun is −26.74 (brighter), and the mean apparent magnitude of the full moon is −12.74 (dimmer).

Difference in magnitude:

Brightness factor:

The Sun appears about 7005400000000000000♠400000 times brighter than the full moon.

Magnitude addition

Sometimes one might wish to add brightnesses. For example, photometry on closely separated double stars may only be able to produce a measurement of their combined light output. How would we reckon the combined magnitude of that double star knowing only the magnitudes of the individual components? This can be done by adding the brightnesses (in linear units) corresponding to each magnitude.

Solving for m f yields

where m_f is the resulting magnitude after adding the brightnesses referred to by m₁ and m₂.

Absolute magnitude

Since flux decreases with distance according to the inverse-square law, a particular apparent magnitude could equally well refer to a star at one distance, or a star four times brighter at twice that distance, and so on. When one is not interested in the brightness as viewed from Earth, but the intrinsic brightness of an astronomical object, then one refers not to the apparent magnitude but the absolute magnitude. The absolute magnitude M, of a star or astronomical object is defined as the apparent magnitude it would have as seen from a distance of 10 parsecs (about 32.6 light years). The absolute magnitude of the Sun is 4.83 in the V band (yellow) and 5.48 in the B band (blue). In the case of a planet or asteroid, the absolute magnitude H rather means the apparent magnitude it would have if it were 1 astronomical unit from both the observer and the Sun.

Table of notable celestial objects

Some of the above magnitudes are only approximate. Telescope sensitivity also depends on observing time, optical bandpass, and interfering light from scattering and airglow.