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Astronomers say a star is "magnitude 1" or "magnitude 6", and the bigger number is the fainter star. That odd backwards scale is ancient, but the exact rule behind it, that five magnitudes make a factor of 100 in brightness, was fixed in 1856 by a young English astronomer, Norman Pogson. He later spent thirty years running the Madras Observatory. This post covers his life, how he chose 2.512, a worked example with logarithms, and lessons from his way of working.
Who was Norman Pogson?
His obituary in the Monthly Notices of the Royal Astronomical Society (1892) says he was born in Nottingham on 23 March 1829, the son of a hosiery manufacturer. He was meant for business and had "only an ordinary education", but showed scientific tastes early, and his mother found him access to an optician's workshop. Hind's father, a Nottingham acquaintance, suggested sending him to London, to J. R. Hind at George Bishop's private observatory in Regent's Park, where he learned practical astronomy and paid his way by teaching mathematics.
At the end of 1851 he became an assistant at the Radcliffe Observatory, Oxford. There he discovered the variability of the star R Cygni in 1852 and, in 1856, the minor planet Isis, which earned him the Lalande Medal. In January 1859 he took charge of the private Hartwell Observatory.
How did he fix the magnitude scale?
Pogson's paper of November 1856, Magnitudes of Thirty-six of the Minor Planets, was written to predict how bright each asteroid would look month by month. To do that he needed a rule linking magnitude to light.
He explains that the Rev. W. R. Dawes had proposed a ratio of 4 between one magnitude and the next, while M. J. Johnson, the Radcliffe Observer at Oxford, had measured 2.43. Pogson tested it himself by finding the telescope aperture at which stars of known magnitude just vanished. His results, using magnitudes from several catalogues, were "remarkably accordant": a mean of 2.4.
Then he made a practical choice. In his words, "I selected 2·512 for convenience of calculation", because with that ratio the constant that kept appearing in his formulas came out as exactly 5. He also noted that "nearly all the great catalogue-makers have involuntarily fallen into this ratio". The obituary adds that the number, "of which the logarithm is 0·4, became by degrees universally adopted."
Worked example: using the scale with logarithms
The rule is: a difference of m magnitudes means a brightness ratio of 100^(m/5), or 10^(0.4m). Equivalently, m = 2.5 × log10(brightness ratio).
Five magnitudes. 2.512 × 2.512 = 6.310. Squaring again gives about 39.82. Multiply by 2.512 once more: about 100.0. So a magnitude 1 star is 100 times brighter than a magnitude 6 star.
A 2.5 magnitude gap. 10^(0.4 × 2.5) = 10^1 = 10. A star of magnitude 1 is ten times brighter than one of magnitude 3.5.
Pogson's own telescope formula. A telescope's light grasp grows with the area of its lens, so an aperture of A inches collects A² times the light of a 1-inch lens. In magnitudes that is 2.5 × log10(A²) = 5 log10 A. So the faintest star you can see is l = L + 5 log10 A, where L is your limit with a 1-inch lens. Pogson found L = 9.2 for his own eyes. For a 7-inch telescope: 5 × log10 7 = 5 × 0.845 = 4.2, so l = 9.2 + 4.2 = 13.4. He wrote exactly this, "l = 9·2 + 5 log 7 = 13·4 magnitude", and checked it against Hind's report that a 7-inch telescope showed stars of about magnitude 13.
Common mistakes:
Wrong direction. A bigger magnitude means a fainter star. If your "brighter" star has the larger number, swap them.
Natural log instead of log10. The scale uses base 10.
Forgetting the square. Light grasp depends on area, so it goes as aperture squared. That is where the 5 in Pogson's formula comes from.
Magnitudes are not in the JEE Main or CBSE physics syllabus, but the logarithms are Class 11 maths. Our profile of John Napier and logarithms covers where they came from.
What did he do at Madras?
The obituary records that in October 1860 he was appointed Government Astronomer at Madras, succeeding Captain Jacob, and arrived early in 1861. Systematic observations began in 1862 with three Indian assistants, and from 1862 to 1887 more than 50,000 observations of stars were made. He found five minor planets there, recovered the lost asteroid Freia, and continued his atlas of variable stars.
The Indian Institute of Astrophysics history lists the asteroids Asia, Sappho, Sylvia, Camilla and Vera among the discoveries at Madras, along with several variable stars. It notes that the discovery in 1867 of the light variation of R Reticuli by C. Raghunathachary "is perhaps the first astronomical discovery by an Indian in recent history". At the total solar eclipse of 18 August 1868, Pogson's team at Masulipatam saw the bright yellow line later traced to helium.
His later years were hard. The obituary says he was left "without proper assistance against an overwhelming amount of routine work", so much of his work was never published. He was made a Companion of the Indian Empire in 1878, never took a holiday in thirty years in India, and died on 23 June 1891. Madras's work later moved to Kodaikanal; see our profile of Vainu Bappu.
What can a student take from his way of working?
Test a claim yourself. He began, he admits, with "a little bias" toward Dawes's ratio of 4, and let his own measurements change his mind.
Choose conventions for convenience, and say why. 2.512 sat within the measured range and made the arithmetic clean.
Check a formula against an independent observation. His telescope formula matched Hind's report.
If logarithm rules still trip you up, the Fermi Concept Library covers every concept in physics, chemistry, biology, mathematics and computing.
Next step: work out how many times brighter a magnitude 2 star is than a magnitude 7 star, then check your answer against step 1.
