# Magnitude (astronomy)

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Magnitude (astronomy)

Magnitude is the logarithmic measure of the brightness of an object, in astronomy, measured in a specific wavelength or passband, usually in optical or near-infrared wavelengths.

## Background

The magnitude system dates back roughly 2000 years to the Greek astronomer Hipparchus (or the Alexandrian astronomer Ptolemy—references vary) who classified stars by their apparent brightness, which they saw as size (“magnitude means bigness”[1]). To the unaided eye, a more prominent star such as Sirius or Arcturus appears larger than a less prominent star such as Mizar, which in turn appears larger than a truly faint star such as Alcor. The following quote from 1736 gives an excellent description of the ancient naked-eye magnitude system:

The fixed Stars appear to be of different Bignesses, not because they really are so, but because they are not all equally distant from us [Note—today astronomers know that the brightness of stars is a function of both their distance and their own luminosity]. Those that are nearest will excel in Lustre and Bigness; the more remote Stars will give a fainter Light, and appear smaller to the Eye. Hence arise the Distribution of Stars, according to their Order and Dignity, into Classes; the first Class containing those which are nearest to us, are called Stars of the first Magnitude; those that are next to them, are Stars of the second Magnitude ... and so forth, 'till we come to the Stars of the sixth Magnitude, which comprehend the smallest Stars that can be discerned with the bare Eye. For all the other Stars, which are only seen by the Help of a Telescope, and which are called Telescopical, are not reckoned among these six Orders. Altho' the Distinction of Stars into six Degrees of Magnitude is commonly received by Astronomers; yet we are not to judge, that every particular Star is exactly to be ranked according to a certain Bigness, which is one of the Six; but rather in reality there are almost as many Orders of Stars, as there are Stars, few of them being exactly of the fame Bigness and Lustre. And even among those Stars which are reckoned of the brightest Class, there appears a Variety of Magnitude; for Sirius or Arcturus are each of them brighter than Aldebaran or the Bull's Eye, or even than the Star in Spica; and yet all these Stars are reckoned among the Stars of the first Order: And there are some Stars of such an intermedial Order, that the Astronomers have differed in classing of them; some putting the same Stars in one Class, others in another. For Example: The little Dog was by Tycho placed among the Stars of the second Magnitude, which Ptolemy reckoned among the Stars of the first Class: And therefore it is not truly either of the first or second Order, but ought to be ranked in a Place between both.[2]

Note that the brighter the star, the smaller the magnitude: Bright "first magnitude" stars are "1st-class" stars, while stars barely visible to the naked eye are "sixth magnitude" or "6th-class".

Tycho Brahe attempted to directly measure the “bigness” of the stars in terms of angular size, which in theory meant that a star's magnitude could be determined by more than just the subjective judgment described in the above quote. He concluded that first magnitude stars measured 2 arc minutes (2’) in apparent diameter (1/30 of a degree, or 1/15 the diameter of the full moon), with second through sixth magnitude stars measuring 3/2’, 13/12’, 3/4’, 1/2’, and 1/3’, respectively.[3] The development of the telescope showed that these large sizes were illusory—stars appeared much smaller through the telescope. However, early telescopes produced a spurious disk-like image of a star (known today as an Airy disk) that was larger for brighter stars and smaller for fainter one. Astronomers from Galileo to Jaques Cassini mistook these spurious disks for the physical bodies of stars, and thus into the eighteenth century continued to think of magnitude in terms of the physical size of a star.[4] Johannes Hevelius produced a very precise table of star sizes measured telescopically, but now the measured diameters ranged from just over six seconds of arc for first magnitude down to just under 2 seconds for sixth magnitude.[5] By the time of William Herschel astronomers recognized that the telescopic disks of stars were spurious and a function of the telescope as well as the brightness of the stars, but still spoke in terms of a star's size more than its brightness.[6] Even well into the nineteenth century the magnitude system continued to be described in terms of six classes determined by apparent size, in which

There is no other rule for classing the stars but the estimation of the observer; and hence it is that some astronomers reckon those stars of the first magnitude which others esteem to be of the second.[7]

However, by the mid-nineteenth century astronomers had measured the distances to stars via stellar parallax, and so understood that stars are so far away as to essentially appear as point sources of light. Following advances in understanding the diffraction of light and Astronomical seeing, astronomers fully understood both that the apparent sizes of stars were spurious and how those sizes depended on the intensity of light coming from a star (this is the star's apparent brightness, which can be measured in units such as Watts/cm2) so that brighter stars appeared larger. Photometric measurements (made, for example, by using a light to project an artificial “star” into a telescope’s field of view and adjusting it to match real stars in brightness) had shown that that first magnitude stars are about 100 times brighter than sixth-magnitude stars. Thus in 1856 Norman R. Pogson of Oxford proposed that a standard ratio of 2.512 be adopted between magnitudes, so five magnitude steps corresponded precisely to a factor of 100 in brightness.[8] This is the modern magnitude system, which measures the brightness, not the apparent size, of stars. Using this logarithmic scale, it is possible for a star to be brighter than “first class”, so Arcturus is magnitude 0, and Sirius is magnitude -1.46.

## Apparent magnitude

Under the modern logarithmic magnitude scale, two objects whose intensities (brightnesses) measured from Earth in units of power per unit area (such as Watts per square centimeter or W/cm2) are I1 and I2 will have magnitudes m1 and m2 related by

$m_1-m_2=-2.5\log_{10} \left ( \frac{I_1}{I_2} \right )$

Using this formula, the magnitude scale can be extended beyond the ancient magnitude 1-6 range, and it becomes a precise measure of brightness rather than simply a classification system. Astronomers can now measure differences as small as one-hundredth of a magnitude. Stars between magnitudes 1.5 and 2.5 are called second-magnitude; there are some 20 stars brighter than 1.5, which are first-magnitude stars. (See List of brightest stars). To use the stars mentioned in the "Background" section of this article as examples, Sirius is magnitude -1.46, Arcturus is -0.04, Aldebaran is 0.85, Spica is 1.04, and Procyon (the little Dog) is 0.34. Under the ancient magnitude system, all of these stars might be classified as "stars of the first magnitude".

Magnitudes can also be calculated for objects far brighter than stars (such as the sun and moon), and for objects too faint for the human eye to see (such as Pluto). What follows is a table giving magnitudes for objects ranging from the sun to the faintest object visible with the Hubble Space Telescope:

Apparent
magnitude
Brightness
relative to
magnitude 0
Example Apparent
magnitude
Brightness
relative to
magnitude 0
Example Apparent
magnitude
Brightness
relative to
magnitude 0
Example
-27 6.3×1010 Sun -7 630 SN 1006 supernova 13 6.3×10-6 3C 273 quasar
-26 2.5×1010 -6 250 International Space Station (max) 14 2.5×10-6 Pluto (max)
-25 1.0×1010 -5 100 Venus (max) 15 1.0×10-6
-24 4.0×109 -4 40 16 4.0×10-7 Charon (max)
-23 1.6×109 -3 16 Jupiter (max) 17 1.6×10-7
-22 6.3×108 -2 6.3 Mercury (max) 18 6.3×10-8
-21 2.5×108 -1 2.5 Sirius 19 2.5×10-8
-20 1.0×108 0 1.0 Vega 20 1.0×10-8
-19 4.0×107 1 0.40 Antares 21 4.0×10-9 Callirrhoe (small satellite of Jupiter)
-18 1.6×107 2 0.16 Polaris 22 1.6×10-9
-17 6.3×106 3 0.063 Cor Caroli 23 6.3×10-10
-16 2.5×106 4 0.025 Acubens 24 2.5×10-10
-15 1.0×106 5 0.010 Vesta asteroid (max) 25 1.0×10-10 Fenrir (small satellite of Saturn)
-14 4.0×105 6 4.0×10-3 typical limit of naked eye[9] 26 4.0×10-11
-13 1.6×105 Full moon 7 1.6×10-3 Ceres (max) 27 1.6×10-11 visible light limit of 8m ground-based telescopes
-12 6.3×104 8 6.3×10-4 Neptune (max) 28 6.3×10-12
-11 2.5×104 9 2.5×10-4 29 2.5×10-12
-10 1.0×104 10 1.0×10-4 typical limit of 7x50 binoculars 30 1.0×10-12
-9 4.0×103 Iridium flare 11 4.0×10-5 31 4.0×10-13
-8 1.6×103 12 1.6×10-5 32 1.6×10-13 visible light limit of Hubble Space Telescope

## Absolute scale based on Vega

The star Vega has been defined as having a magnitude of zero, or at least near. Modern instruments as bolometers and radiometers give Vega a brightness of about 0.03. The brightest star, Sirius, has a magnitude of −1.46. or -1.5. However, Vega has been found to vary in brightness, and other standards have been proposed.[10]

## Problems

The human eye is easily fooled, and Hipparchus's scale has had problems. For example, the human eye is more sensitive to yellow/red light than to blue, and photographic film more to blue than to yellow/red, giving different values of visual magnitude and photographic magnitude. Furthermore, many people find it counter-intuitive that a high magnitude star is dimmer than a low magnitude star.

## Apparent- and absolute-magnitude

Two specific types of magnitudes distinguished by astronomers are:

• Apparent magnitude, the apparent brightness of an object. For example, Alpha Centauri has higher apparent magnitude (i.e. lower value) than Betelgeuse, because it is much closer to the Earth.
• Absolute magnitude, which measures the luminosity of an object (or reflected light for non-luminous objects like asteroids); it is the object's apparent magnitude as seen from a certain distance. For stars it is 10 parsecs (10 x 3.26 light years). Betelgeuse has much higher absolute magnitude than Alpha Centauri, because it is much more luminous.

Usually only apparent magnitude is mentioned, because it can be measured directly; absolute magnitude can be calculated from apparent magnitude and distance using;

$m - M = 5 \left( log_{10}(d) - 1 \right)$

This is known as the distance modulus, where d is the distance to the star measured in parsecs.

## Notes

1. ^ Milton D. Heifetz/Wil Tirion, A walk through the heavens: a guide to stars and constellations and their legends, Cambridge, Cambridge U. Press, 2004, page 6.
2. ^ John Kiell, An introduction to the true astronomy, London, 1736, pages 47-48
3. ^ Victor Thoren, The Lord of Uraniborg, Cambridge University Press, 1990, on page 306
4. ^ Christopher M Graney, Timothy P Grayson, "On the telescopic disks of stars - a review and analysis of stellar observations from the early 17th through the middle 19th centuries", Annals of Science, Volume 68, Issue 3, 2011, DOI:10.1080/00033790.2010.507472, pages 351-358.
5. ^ Graney/Grayson 2011 page 355 and Graney, Christopher M., "17th Century Photometric Data in the Form of Telescopic Measurements of the Apparent Diameters of Stars by Johannes Hevelius", Baltic Astronomy, Vol. 18, 2009, p. 253-263.
6. ^ Graney/Grayson 2011 page 355-358.
7. ^ Alexander Ewing/John Gemmere, Practical astronomy, Allison & Co., Burlington N.J., 1812, page 41.
8. ^ Michael Hoskin, The Cambridge concise history of astronomy, Cambridge, Cambridge U. Press, 1999, page 258. Also Jean Louis Tassoul, Monique Tassoul, A concise history of solar and stellar physics, Princeton, Princeton U. Press, 2004, page 47.
9. ^ under very dark skies, such as are found in remote rural areas
10. ^ Astronomical Photometry: Past, Present, and Future by Eugene F. Milone (Springer 2011: ISBN 1441980490, 9781441980496), pages 182-184.

## References

Wikimedia Foundation. 2010.

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