[ kd ] hwk01 Ast 4001, 2013 Sept 3

Homework set 1 -- Practice exercises with astronomical magnitudes

The only way to become really comfortable with magnitudes is by practicing with them. These exercises have two goals: (1) to visualize what a quoted magnitude signifies, and (2) to become able to calculate with magnitudes easily, quickly, and reliably. After some practice you can do a typical magnitude problem in only a few seconds. Don’t turn your results in, but try to be confident that you got the right answers. Try to devise the easiest or quickest method for each type of problem. Invent other problems yourself.

Three facts to remember ...

* The relation between an apparent magnitude m and the corresponding radiation flux F :

m = 2.5 log 10 ( F / F 0 ) , F = 10 – 0.4 m F 0 . ( smaller m implies brighter )

Usually (but not always) F is an energy flux with units such as Watts per square meter. Magnitude m is a dimensionless quantity, a pure number with no measurement units.

* F and m usually refer to some limited range of wavelengths or photon frequencies . For example we might be using a blue or red filter. Wavelengths and frequencies aren’t mentioned in the problems below, because they don’t affect the magnitude calculations.

* Usually we don’t need to know the quantitative value of the normalization constant F 0 . _________________________________________________________________________________________________________________

A few www sites to read or skim (this list may be a little out of date) Wikipedia “magnitude (astronomy)”, “photometry (astronomy)”, ”UBV photometric system”, etc. Google has trouble with this topic, because keywords such as “magnitudes” or “photometry” or “UBVR” may lead to a bunch of specialized research papers that merely have these words in their titles. But if you look around, you may find various course notes from other universities. www.astrophysicsspectator.com/topics/observation/MagnitudesAndColors.html www.sizes.com/units/magnitude_stellar.htm (historical notes) aas.org/archives/BAAS/v33n4/aas199/738.htm (trivial fussing) www.badastronomy.com/bad/misc/badstarlight.html (popular-level blurb) _________________________________________________________________________________________________________________

Practice problems

1. 61 Cygni is a famous double star. When separated with a telescope, its component stars have apparent visual magnitudes m = 5.22 and 6.03 respectively. Calculate m for the total of the two, as seen with the unaided eye.

2. M 67 is an important cluster of about 80 stars located 830 parsecs from us. Its total brightness amounts to apparent visual magnitude m 6.5. Estimate the average magnitude for a typical individual star in M 67.

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3. A particular double star system has total absolute magnitude m = 4.5. In terms of intrinsic energy flux, component A is twice as bright as its companion B. Calculate the individual magnitudes of A and B.

4. UGC 4591 is a distant galaxy that normally has apparent visual magnitude m = 16.5. Briefly, however, this magnitude brightens to 15.9 because a supernova explosion has occurred there. Deduce the apparent magnitude of the supernova.

5. We can get decent spectra of stars as faint as m = 20. The Sun has m – 27. Calculate the ratio of corresponding brightnesses F . Try to do this in your head! (How? – Recall that a 2.5-magnitude difference implies a factor of 10 in F . Remember that 1 magnitude is a factor of about 2.512, 2 magnitudes is a factor of about (2.512)

2 6.3, etc.)

6. Apropos problems like 4, it’s useful to memorize log 10 (2) 0.301, log 10 (3) 0.477, log 10 (4) 0.602, and log 10 (5) 0.699. ( Historical comment: Decades ago, most technical people automatically knew these values through experience with slide rules and tables of logarithms.)

7. At visual wavelengths, there are about 6000 sixth-magnitude stars in the sky, i.e., in the magnitude range 5.5 < m < 6.5. These are the faintest stars that can be seen with the unaided eye in favorable conditions. Compare the total radiation flux F from all of them to the brightest star, Sirius, which has m 1.4 .

8. Suppose, in a given set of circumstances, you can see stars as faint as m = 6 without optical aid. How faint a star can you see with a 7 x 50 binocular? (“50” means it has objective lenses with diameter 50 mm.) With a 1-meter telescope?

9. A star’s apparent magnitude obviously depends on its distance as well as its intrinsic luminosity. Absolute magnitude M , a measure of just the luminosity, is defined as the apparent magnitude that a star would have if it we could view it from a standard distance of 10 parsecs. Use this definition to deduce the relation between M, m, and distance D . Verify that it gives reasonable answers for one or two easy test cases. (This formula can be found in most elementary astronomy textbooks. The point here, however, is that you can work it out yourself, so quickly that you don’t really need to memorize the formula or look it up. Just recall that F 1 / D 2 .) 10. If one star is 0.1 magnitude brighter than another, then they differ by about 10% in brightness ( measured by F ). Likewise m = 0.05 implies a 5% difference, etc. This convenient approximation works for differences smaller than, say, 25% or so. In terms of math, figure out why it’s roughly true. ( Hint: consider e –m . )

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