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# Magnitude and Color in Astronomy

The stars we see in the sky have all variety of brightness and color.  We see in the constellation Orion the brilliant red star Betelgeuse (α Orionis) near the brilliant blue star Rigel (β Orionis), and around them we see numerous dimmer stars.

The ancient astronomers developed the original system of classifying stars by their brightness.  This system is attributed to Hipparcos, coming to us by way of Claudius Ptolemaeus, better known simply as Ptolemy.  In the Almagest, Ptolemy assigned values ranging from 1 to 6 to the stars based on their brightness on the night sky, with the brightest stars assigned a value of 1, and the dimmest stars a value of 6.  Under this system, the stars Betelgeuse and Rigel are magnitude 1, and stars like those in the belt of Orion are magnitude 2.

This early system is tied to the physiology of the human eye, and as anyone who has photographed a landscape of light and shadow knows, the human eye does not see brightness linearly, but logarithmically.  To our eyes, objects in the shadows are as distinct as the objects in the sunlight, but to a camera, which sees light more linearly, either the objects in shadows are indistinct when objects in sunlight are clearly visible, or the objects in sunlight are washed out when the objects in shadow are distinct.  We perceive large multiplicative changes in brightness rather than linear changes in brightness.

When modern astronomers gave the ancient magnitude system a modern, objective definition, they defined the magnitudes as the logarithms of the flux—the power received per unit area—from a star.  In this system, an arithmetic change in magnitude of 5 corresponds to a multiplicative change in flux of a factor of 100, so a magnitude 1 star is 100 times brighter than a magnitude 6 star.  This is expressed by the following formula:

ma - mb = -2.5 log( Fa/Fb ),

where ma and mb are the magnitudes of stars a and b, and Fa and Fb are the fluxes (power per unit area) from each of these stars.  The system is defined so that magnitude 6 stars are just visible to the unaided eye.  Stars invisible to the eye but visible in the telescope have magnitudes larger than 6, with the darkest nearby stars having magnitudes as large as 16.  On the other hand, several of the brightest stars under this modern system have negative magnitudes, with the brightest star in the sky, Sirius, having a visual magnitude of −1.43; it is magnitude 1 in Ptolemy's star catalog.  Under this quantitative system,  Betelgeuse has a visual magnitude of around 0.87 (the star is variable), Rigil of 0.10, and the stars in Orion's belt, running East to West, of 1.77, 3.35, and 2.25.

The total power per unit area arriving from a star can be measured using an instrument called a bolometer. The magnitude derived from such an instrument is called a bolometric magnitude.  Most magnitudes, however, are measured with telescopes that observe stars over a small frequency band, so a star's magnitude is generally tied to a frequency band.  For a given star, a magnitude measured in red light is generally very different from a magnitude measured in blue light.  This difference depends primarily on the star's photospheric temperature; at a high temperature, a photosphere predominately radiates blue light, but at a low temperature, it predominately radiates red light.  This effect of temperature on the color of a star has lead astronomers to develop systems for expressing a star's magnitude in terms of color filters.  Many such systems exist, literally dozens, each having its own set of frequency bands that are defined by the physics of the filters and the photon detector used by the observer.[1]

The most common photometry system for general observations over the infrared to ultraviolet range is the Johnson-Cousins system, which divides the spectrum into broad bands of ultraviolet (U), blue (B), visible (V), red (R), and infrared (I) light.[2,3]  Normally one uses these letters as the measures of magnitude at each wavelength, so one talks of Sirius having a visual magnitude of V = −1.43.  The responses of the filters in this system are given in the figure below.  The V filter was chosen to roughly match the response of the rods in the human eye to starlight.  The rods, which are responsible for sight at low light levels, are unable to see red; the cones in the eye responsible for red vision react to light with wavelengths shorter than 0.7 microns (μm, or 10−6 meters).  The I band, which is in the infrared band, covers a region invisible to human sight.  The same is true of the U band, which is in the ultraviolet band, as the eye is insensitive to light of wavelength shorter than 0.4 microns.

The figure shows the response functions for the filters of the Johnson-Cousins UBVRI photometry system.  The graph shows the transmittance of each filter as a function of wavelength, where the wavelength is in units of microns.  The filter responses given in this diagram are take from data obtained through the ADPS online database. The U, B, and V filters are those described by Johnson and Morgan, and the R and I filters are those described by Cousins.[2,3] The data used for the U, B, and V responses are the reconstructions of Matthews and Sandage (called the USA system).[4]

A star's magnitude is a measure of how that star's brightness compares to the brightness of other stars, and that is how one measure a star's magnitude: by comparing it to the magnitude of another star.  It is measured this way because of the difficulty of measuring the power received from a star.  As we all know from looking at the night sky, the weather causes the brightness of a star to change dramatically from night to night.  This difficulty extends to the instruments used to measure the brightness of a star.  As light travels through a telescope, some of it is absorbed or scattered by the optics of the instrument.  The detectors also do not detect every photon, and the efficiency of detecting photons can change over time as conditions—detector temperature, for instance—change.  For these reasons, it is easier to derive a star's brightness by comparing it to a standard reference star.  A photometry system, therefore, includes a catalog of reference stars uniformly distributed across the sky.

The magnitudes in the Johnson-Cousins system are defined so that stars classified by their spectra as A0 V stars—main-sequence stars of about 2.5 solar masses—have the same magnitude in each wave band.  Originally six A0 V stars were chosen for the calibration of the system; the most familiar of these stars is Vega (α Lyrae).  This star has a magnitude of 0.03 in each of the UBVRI channels.  The visual magnitude scale is defined to match the photographic magnitude system developed in the 19th and early 20th centuries.

The measured magnitude—called the apparent magnitude—of a star expresses both the power radiated by the star in a given waveband and the distance of the star from Earth.  The energy flux that reaches Earth is therefore proportional to the luminosity (L) of the star divided by the distance (R) squared (L/R2).  If the distance of the star is known, the effect of distance can be removed, giving a magnitude that is a measure of luminosity alone.  Such a magnitude is called an absolute magnitude, and formally it is the magnitude a star would have on the sky if the star were 10 parsecs away.  Because a factor of 100 change in flux corresponds to a change of 5 in magnitude, a factor of 10 change in distance also corresponds to a change of 5 in magnitude, increasing as distance increases.  The absolute magnitude M is related to the apparent magnitude m by

M = m − 5 log( 10 pc/R ),

where R is the distance of the star in parsecs.

A star's color is measured by taking the difference in magnitude between two color bands.  A commonly-used color index is B − V, which is a measure of the flux in the blue band to the flux in the visual band.  This works well for stars similar to the Sun, which is why it is the standard measure for color in Hertzsprung-Russell diagrams.  Often the color index used for darker, less massive stars than the Sun is V − I.  Stars with negative color indexes are bluer than A0 V stars such as Vega, and stars with positive color indexes are redder than A0 V stars.  A star's color index is  much easier to measure than its magnitude, because the loss of light in the atmospheric, telescope, and detector is proportionally the same for all band.  This means that the color index can be unchanging despite large daily changes in magnitude because of weather.

## Color and Magnitude Definitions

The characteristics of the Johnson-Cousins color bands are given in the following table. The band width is defined as the full-width half-maximum of each curve (half the distance between the two points that have a value that is half of the peak value).  The center frequency is half-way between the half-maximum points.  The 0 magnitude flux is the power per unit area per unit wavelength received in the wave band from a 0 magnitude star.  In other words, this flux is almost precisely the flux one measures at each wave band for the star Vega.  The data in this table is from Lamla (1982) via the ADPS.[5,6]

Wave Band

Center Frequency
μm

Band Width
μm

0 Magnitude Flux
(ergs s−1 cm−2 μm−1)

U

0.3500

0.0700

3.980 10−5

B

0.4380

0.0985

6.950 10−5

V

0.5465

0.8700

3.630 10−5

R

0.6470

0.1515

2.254 10−5

I

0.7865

0.1090

1.196 10−5

[1]Bessell, Michael S.  “Standard Photometric Systems,” in Annual Review of Astronomy and Astrophysics, vol. 43.  Palo Alto: Annual Reviews, 2005: 293–336.

[2]Johnson, H.L., and Morgan, W.W.  “Fundamental Stellar Photometry for Standards of Spectral Type on the Revised System of the Yerkes Spectral Atlas.”  The Astrophysical Journal 117 (May 1953): 313–352.

[3]Cousins, A.W.J.  “VRI Standards in the E Regions.”  Memoirs of the Royal Astronomical Society 81 (1976): 25–36.

[4]Matthews, T.A., and Sandage, A.R.  “Optical Identification of 3C 48, 3C 196, and 3C 286 with Stellar Objects.”  The Astrophysical Journal 138 (1963): 30.

[5]Lamla, E.  In Landolt-Bornstein Series, vol. 2b Stars and Star Clusters, K. Schaifers and H.H. Voight eds. ( Springer-Verlag, Berlin, 1982).

[6] Munari, U., Fiorucci, M., and Moro, D. “The Asiago Database on Photometric Systems (ADPS).”  Version 2.1 (August 24, 2002).