The magnitude system ASTR320 Wednesday January 30, 2019
What we measure: apparent brightness How bright a star appears to be in the sky depends on: How bright it actually is Luminosity and its distance from us, d We call this the apparent brightness : b = L 4πd 2 This is a measure of the radiant flux produced by the star This is known as the inverse square law for light
What we want to know: luminosity Luminosity, the total light given off by a star Units of energy: photons / sec, or in erg / sec Intrinsic property of the star (does not depend on distance) WARNING: The definitions of things like "luminosity", "flux", "flux density", etc. used by astronomers are not the same as those used in other disciplines (e.g., our "luminosity" is called "flux" in other sciences)
Luminosity Note that we have not yet said which wavelengths or energies of the individual photons or light For now we re discussing bolometric quantities i.e., summed over all frequencies Practically this is a very hard thing to measure, since it requires measuring the entire electromagnetic spectrum of a source, which is impossible with a single type of detector Bolometric fluxes have to be inferred by knowledge of the physics producing the luminous source for which you have partial information, or pieced together from observations at all parts of the EM spectrum In astronomy we usually use fluxes at specific energies
Luminosity Because f ν δν = f λ δλ, and ν = c / λ, ν f ν = λ f λ The functions f ν (ν) and f λ (λ) are referred to as the spectral energy distribution (SED) of the source When referring to bolometric fluxes, it is common to use units of Janskys (Jy) (borrowed from radio astronomy): 1 Jy = 10-26 W m -2 Hz -1 1 Jy = 10-23 erg sec -1 cm -2 Hz -1
The magnitude system Astronomers quantify the intensity of light produced by a source with the unit magnitudes Magnitudes are a logarithmic representation of the spectral flux density of a source Allows for easy comparison of sources with immense ranges in flux density The magnitude system, let s be honest, is not readily intuitive
History of the magnitude system The system was devised by Greek astronomer Hipparchus, ca. 150 BC, to catalog the brightness of stars Brightest stars were placed in the first magnitude class, next brightest were second magnitude, etc Based on how bright a star appears to the unaided eye Ptolemy also used them in his Almagest Catalog of ~1000 naked eye stars 6 "magnitude" classes: 1 = brightest 6 = faintest
Magnitudes of some familiar objects Here s where it gets messy: Bright stars have smaller magnitudes than faint stars! This has confused/frustrated/enraged many an astronomer
History of the magnitude system Revisions have been made in last few centuries. Extend scale to < 1 mag to place Sun, Moon, bright planets on same scale Once the telescope was invented, extend scale to > 6 mag 1850: N. R. Pogson (British astronomer) notices that, because eyes work logarithmically, the classical magnitude scale corresponds roughly to set ratios of brightness between successive magnitudes Also notes that mag 6 is about 100x fainter than mag 1 Since Δm = 5 appears to be 100x ratio in brightness, and 100 1 5 = 2.5119, Pogson formalized scale so that ratios between successive magnitudes are exactly 2.5119
Magnitude: definition Take two stars, one is 100x (5 magnitudes) brighter than the other Remember: the brighter star has a smaller magnitude Compare their fluxes: F 1 F 2 = 100 (m 2 m 1) 5 = 2.5119 (m 2 m 1 ) Note that the above equation also shows that fractions of magnitudes are possible for stars with brightnesses in between two integer magnitudes
Magnitude: definition Take the logarithm of F 1 = 100 (m 2 m 1) 5 F 2 To derive a more useful equation: m 2 m 1 = 2.5 log 10 ( F 2 F 1 ) That can be used to compare the apparent magnitude, m, of two sources
Apparent magnitude Remember: the apparent brightness of a star observed from the Earth is called the apparent magnitude. The apparent magnitude is a measure of the star's flux received by us. This is the quantity we actually measure with a telescope
Absolute magnitude By definition: if a star is 10 parsecs from Earth, then its apparent magnitude would be equal to its absolute magnitude, M The absolute magnitude is a measure of the star's luminosity---the total amount of energy radiated by the star every second
Absolute magnitude If you measure a star's apparent magnitude and know its absolute magnitude, you can find the star's distance (using the inverse square law of brightness) If you know a star's apparent magnitude and distance, you can find the star's luminosity The luminosity is an intrinsic property of the star, not based on how far away it is A star's luminosity tells you about the internal physics of the star and is a much more important quantity than the apparent brightness
Absolute magnitude We can relate the absolute and apparent magnitude by using the definition of absolute magnitude (the magnitude of a star if it were at a distance of 10 pc) 100 (m M) 5 = F 10 F = d 10 pc Where F is the flux measured at d, in parsecs And F 10 is the flux measured at d=10 pc Take the logarithm to get: M = m 5 log 10 d + 5 (If d is in pc) 2
Distance modulus Rewrite this equation: M = m 5 log 10 d + 5 As the distance modulus, µ: μ = m M = 5 log 10 d 5 The distance is the difference between the apparent, m, and the absolute magnitude, M, of a source This is a very useful quantity!
Magnitudes and distances We can use relative magnitudes to estimate relative distances, if the objects being compared have the same absolute magnitude: For example: m 1 m 2 = 2.5 log 10 f 1 = 2.5 log f 10 ( d 2 2 2 m 1 m 2 = 5 log 10 d 2 d 1 = 5 log 10 ( d 1 d 2 ) d 2 d 1 = 10 m 1 m 2 = 5 log 10 = 5 m 1 m 2 = 5 So a star with 10x greater distance is 5 mags fainter (100x fainter in flux) d 1 2 )
The color index It can be useful to measure the amount of energy or photons detected over a discrete bandpass of the EM spectrum Also this is a much easier measurement to make! A relative measure of the brightness of an object two wavelengths (or frequencies) is called its color An object s color index is the difference in the magnitudes in two bandpasses E.g. U-B, B-V The UBV filter system, one of the first that enabled the measurement of the color index
The color index A star or galaxy s color can be used to deduce its temperature: Obviously now we re no longer dealing with bolometric luminosities but we can still use the magnitude system
Photometric parallax You can use the color of a star to gauge its distance Identify "spectral type" of the star by its color, gauged by photometry in different filters (which is like very coarse spectroscopy) Once we have the spectral type, and an assumed luminosity class of the star, in principle you know the absolute magnitude, and then can get the distance through the distance modulus This method of deriving a distance is called measuring a photometric parallax
Photometric parallax Obviously this is not as good as trigonometric parallaxes, due to the color ambiguities we ve discussed For example, red stars can be either very luminous red giants or very dim red dwarfs. Making a mistake in confusing the two can lead to distances off by factors of 100 or more There are many kinds of blue stars, from blue supergiants to white dwarfs. Errors in proper identification can lead to distances off by factors of 10,000 or more The hope is that such large errors in distance can be readily identified through other means (or by "sanity checking" that these erroneous results make sense)
Color-magnitude diagram Even if we don t know the distance, we can use the relative magnitudes of stars to study their properties CMD: plot of color versus magnitude Measure color using the color index Measure apparent magnitude This is most easily done in a group or cluster of stars, since they should all lie at the same distance We ll talk about open and globular clusters here
Axes of the color-magnitude diagram CMD: plot of color index versus apparent magnitude Or absolute magnitude if you know the distance, as shown here
Open clusters Young clusters of stars loosely bound by gravity Recent star formation (notice the many blue stars) High metallicity, not much stellar evolution Pleiades, a young, nearby open cluster The Jewel Box cluster
Relative ages Compare CMDs of multiple clusters to estimate their ages
Globular clusters Old, dense clusters No blue stars No recent star formation CMD for globular cluster M55
Relative ages Compare CMDs of multiple clusters to estimate their ages Open cluster Globular cluster
Open vs. Globular Clusters Open cluster: 1000 s of stars of a wide range of temperatures (young stellar population) Globular cluster: 100,000s of stars, only cool red stars present (old stellar population)
Stellar evolution CMD of the triple main sequence of the globular cluster NGC 2808, taken from Piotto et al. (2007). The stars are all proper-motion members of the cluster and a correction has been made for differential reddening along the line of sight. Inset: theoretical isochrones for an age of 12.5 Gyr, with different helium content ((m M) 0 =15, E(B V)=0.18). From Kalirai & Richer 2010