## Stellar Magnitudes

### The Scale of Magnitudes

Magnitude was originally a way to classify stars by brightness to make identification easier, appearing in Claudius Ptolemy's catalog of stars. The brightest stars were of 1st magnitude, while the faintest that could be seen in a dark, clear sky were 6th magnitude. Those in between were classified in roughly equal steps of brightness. Since the eye judges equal differences in brightness to correspond to equal ratios of intensity, which we take as given by energy flux or its equivalent, in ergs/s, watt, or lumen, this turns out to be a logarithmic scale. If a star of the fifth magnitude is "a" times as bright as a star of the sixth magnitude, then a star of the fourth magnitude is "a" times as bright as one of the fifth, or a2 times as bright as one of the sixth. Continuing this scheme, a star of the first magnitude is a5 times as bright as one of the sixth magnitude.

Brightness is a perception, while the energy received as electromagnetic radiation, which we will call intensity or illumination, is a physical stimulus. Intensity is measured in energy per unit time per unit area, say in erg/s-cm2 or W/m2. Luminous intensity is a psychophysical quantity that weights energy according to its effect on the eye. The usual unit is the lumen, and at the frequency of maximum sensitivity of the eye, 680 lm = 1 W. For a spectrum of constant intensity, the ratio over the visual range from 400 nm to 700 nm is about 220 lm/W. The lumen is not a measure of brightness, but of visual stimulus. The brightness is a logarithmic function of the visual stimulus. If we define brightness as the common logarithm, then an increase in the energy by a factor of 10 increases brightness by 1 unit, and an increase by a factor of 100 increases brightness by 2 units. We can adjust the scale by a numerical factor multiplying the logarithm, or by an arbitrary zero. That is, we may define brightness by B = A log I + B, where A and B are constants, and I is the intensity, or visual stimulus per unit area. This is equivalent to the equivalent expression B = A log(I/I'), where I' is a reference intensity, and B = -A log I'. I can be measured in lumens/m2 or lux, if we are concerned with visual brightness. We now establish a scale of stellar brightness on this basis.

The logarithmic scale is an extension of the orginal scale of magnitudes, not restricted to the range 1 to 6, but capable of extension in both directions. In Ptolemy's catalog, all the brightest stars were of first magnitude, but with the logarithmic scale we can define a zero magnitude "a" times brighter than first magnitude, and -1 magnitude "a" times brighter than that. In fact, Betelgeuse is close to magnitude 1 (actually 0.9), Vega close to magnitude 0 (actually 0.1), and the brightest star of all, Sirius, is magnitude -1.6. In the 19th century, it was concluded that the ratio "a" was such that a5 = 100, or a first-magnitude star was 100 times as intense as a sixth-magnitude star. That is, it supplied 100 times as many lumens to the eye. I have avoided using "brightness" here so that that term can be used for the sensation, not for its physical stimulus. Therefore, 5 log a = 2, or log a = 0.4, or a = 2.5119. The logarithm is to base 10, which is convenient because of the ratio 100 that appears. The relation between intensity ratio and magnitudes is then I/Io = 100-m/5, where m is the magnitude difference. The (-) sign makes the magnitude difference negative for a ratio greater than 1. Taking the logarithm, -(2/5)m = log (I/Io), or m = -2.5 log(I/Io). For any intensity ratio, this will give us the magnitude difference. All this corresponds to choosing the constant A in the logarithmic relation.

If we use natural logarithms instead, m = -1.0857 ln(I/Io). Since the coefficient is close to unity, this is approximately I = Ioe-m, so that the factor "a" (2.5119) is close to "e" (2.7183). It is curious that magnitudes come out as corresponding roughly to powers of e. We won't use natural logarithms further.

Magnitudes are analogous to the decibel unit used for specifying power ratios. This relation is dB = 10 log (I/Io), so that dB is -4 times the magnitude difference. Decibels could be used in astronomy for stellar brightness, and then a large number would correspond to a bright star, a small number to a dim star. However, this is not done, and negative magnitudes correspond to bright stars, large positive ones to dim stars.

As in the case of decibels, if magnitudes are quoted instead of simply magnitude differences, a reference magnitude must be given. Also, the powers appearing in the ratio must be specified. The quantity appropriate to visual stimulus is the lumen, which can be physically specified as an energy density spectrum times a sensitivity function. The sensitivity function has a peak of 1 at a wavelength of 555 nm, where 680 lumen = 1 watt. The corresponding magnitudes are called visual magnitudes. The eye will automatically make the spectral weighting, and brightnesses can be compared by adjusting a neutral filter in front of one source until the two sources are judged equally bright. A similar comparison can be made physically by measuring the energy spectrum, say w(f) in watt/s-cm2, so that the incident energy between frequencies f and f + df is w(f)df. Wavelength λ could be used as well as f, since λf = c, where c is the speed of light. We will generally use f as the independent variable. Some of our results will different from those using wavelengths, since equal frequency intervals are not equal wavelength intervals, and vice versa. Then, the total stimulus is ∫s(f)w(f)df, where s(f) is the sensitivity function.

The intensity of a light wave spreading spherically is given by I = Io/r2, where Io is the intensity at r = 1. Then m = -2.5 log(I/Io) = -2.5 (-2 log r) = 5 log r. We really should have used a distance ratio as well, (ro/r), so the argument of the logarithm would be dimensionless, but the simpler equation will do, since Io has disappeared. If M is the magnitude when r = 10, and m the magnitude at any distance, then m = M + 5 log (r/10) = M - 5 + 5 log r. This is a rather famous relation in astronomy, if r is the distance in parsecs. Note that it will still be true whatever unit we use for r, since M will always be the magnitude (with respect to any standard) when r = 10. This is the beauty of logarithmic relations. If r is in parsecs, then M is called the absolute magnitude, the apparent visual magnitude with the star at that distance. For each doubling of the distance, the magnitude increases by 5 log 2 = 1.505.

The parallax p of a star is the angle subtended at the star by the radius of the earth's orbit, a, or pd = a, where d is the distance of the star. Now, p" = 206,265 p (radians), so d = 206,265(a/p"). The distance in parsecs is just r = 1/p". Then, m = M - 5 - 5 log p", or log p" = -(m - M + 5)/5, from which the parallax of the star (and so its distance) can be found if the apparent and absolute magnitudes are known. Note that if m = M, we find log p" = -1, or p" = 0.1", a distance of 10 parsecs. The sun has an apparent visual magnitude of -26.8 at a distance of 1 AU, or p = 1 radian = 206265". From this, we find that M = m + 5 + 5 log p" = -26.8 + 5 + 5 log 206265 = 4.77. This is the absolute magnitude of the sun, or its visual magnitude when seen from a distance of 10 parsecs.

1 parsec is 206,265 AU = 206,265 x 1.496 x 1011 m = 3.086 x 1016 m. A year is 3.156 x 107 s, and light travels at 2.9979 x 108 m/s, or 9.460 x 1015 m in a year, a distance called a light-year, ly. Therefore, 1 psc = 3.2620 ly. Light-years are only useful for giving some idea of the immensity of space. Light takes 8.317 minutes to get to the earth from the sun, 1.28 s to get to the earth from the moon (centre to centre), 0.13 s to circle the globe, and 5.48 hours to get from the sun to Pluto.

We can calibrate the scale of visual magnitudes, determining the constant B in the logarithmic function, by using two facts: first, the apparent magnitude of the sun m is -26.8, and second, its luminance L is 1.6 x 105 cd/cm2, from the Handbook of Chemistry and Physics. It may seem strange for the sun to have a luminance quoted this way, but the projected area of the sun increases as the square of the distance, at the same time that the illumination decreases at the same rate. The apparent area of the sun is A = πr2θ2, where θ is the angular semidiameter, 16' or 4.65 x 10-3 radians. Its intensity C in candles is then πr2θ2L. The illumination at a distance r is then I = C/r2 = πθ2L lm/cm2, where r has cancelled as promised. This gives I = 10.9 lm/cm2 or 109,000 lux. This is an estimate of the maximum illumination normal to the direction of the sun. [In illumination, the luminous intensity in candles is usually represented by I, and the illumination by E, but we use C and I instead to be consistent with our notation in defining magnitude.]

The definition of magnitude gives m = -2.5 log(I/I'), where I' is the illumination corresponding to m = 0. Therefore, -26.8 = -2.5 log(109000/I') = -12.59 + 2.5 log I', or log I' = -5.68, I' = 2.077 x 10-6 lux. Now we have an absolute scale for magnitudes: m = -2.5 log I - 14.2, where I is in lux. Now both constants A and B have been determined in the logarithmic function. For a check, the apparent magnitude of the full moon is -12.6, while its luminance is quoted as 0.25 cd/cm2. Since the moon subtends about the same angle as the sun, the illumination would be I = π(4.65 x 10-3)2(0.25)(104) = 0.17 lux. This gives m = -12.3, which is not too far off. We conclude that a good estimate of the illumination corresponding to a visual magnitude m is I = 2.1 x 10-6 10-0.4m lux.

The magnitude of Venus at her brightest is about -4.4. This corresponds to I = 1.2 x 10-4 lux. Sirius, at m = -1.6, gives I = 9.2 x 10-6 lux. When we look at Venus or Sirius, the light that enters the pupil is concentrated in the image, which allows it to be detected easily. If we look at an extended source, then the solid angles are magnified inversely as the size of imaged areas, so the illumination remains constant. A telescope does not brighten extended areas, but concentrates limited ones. It is also quite clear that the eye adapts to great differences in illumination, from the 100,000 lux of sunlight to the 0.2 lux of moonlight. This adaptation is necessary for the eye to be generally useful, and is wonderful evidence of the power of evolution. This adaptation is in the chemistry and physiology of vision, not in the minor degree of control exerted by the size of the pupil, which acts more to provide sharp vision when there is sufficient light by stopping down the aperture of the eye, at most a variation of about 16 in area, a long ways from the factor of 500,000 that actually exists. Mathematically, we express this by saying that brightness (the sensation) is a logarithmic function of the illumination (the physical stimulus).

The luminosity of a star is conventionally defined as the ratio of its output of visual flux to that of the sun. Since we are concerned with a ratio, it can be expressed as a magnitude difference Δm = -2.5 log (I/Is), and this magnitude difference can be the difference in the absolute magnitudes of the star and the sun. This is the same magnitude difference that would exist at any distance, of course. Then, M - Ms = -2.5 log (I/Is), or L = I/Is = 10(0.4)(M - 4.77). As usually defined, this is the ratio of visual outputs, which is by no means the same as the ratio of total outputs. For stars of spectral classes F5 to G5, it will probably not be too different that the ratio of total outputs, but will be very different for class M stars, which radiate mainly in the infrared, or class O stars, which radiate mainly in the ultraviolet. We shall now consider how to correct the luminosity for this circumstance.

The electromagnetic radiation from a star covers a wide band of frequencies. It is like the radiation that is emitted by a small aperture in the wall of a cavity resonator in which electromagnetic waves are excited thermally at a temperature T. Since all radiation falling on the small aperture will enter the cavity with very little probability of exiting again, the aperture appears "black," that is, will absorb all radiation falling on it. In thermal equilibrium, it will emit exactly as much radiation as it absorbs. In the absence of thermal equilibrium, when there is no radiation surrounding it, it will continue to emit exactly the radiation that it would if in equilibrium at the temperature T. This is, therefore, called "black-body" radiation.

If U is the isotropic energy in the cavity of volume V, then the emission per unit time per unit area is J = cU/4V. The energy U is the integral over frequency from 0 to ∞ of the number of electromagnetic modes N(f)df times their photon energy hf times the probability that a mode of frequency f is excited at temperature T. N(f) is proportional to f3, while the probability of excitation is 1/(ehf/kT - 1). Let x = hf/kT, the dimensionless ratio of the photon energy hf to the average thermal exitation energy per mode kT. h is Planck's constant, 6.626 x 10-34 J-s, and k is Boltzmann's constant, 1.3807 x 10-23 J/K. The photon energy in eV for a wavelength λ nm is 1240/λ. The thermal energy in eV for a temperature of T K is 8.619 x 10-5T. Therefore, the photon energy for green light of 555 nm is 2.23 eV, and the thermal energy for 10,000K is 0.8619 eV, so x = 2.59 for 555 nm radiation and 10,000 K.

If we express N(f)df and the probability in terms of x, the number of available states increases as x3, while the probability of excitation varies as 1/(ex - 1). The product of these two factors gives the energy as a function of frequency, and its integral from 0 to ∞ is proportional to the total energy U in the volume V. This is illustrated in the figure at the right. The function varies as x2 for x << 1, and as x3e-x for x >> 1. Practically all the energy is included from x = 0 to x = 12. The integral of the function x3/(ex - 1) is π4/15 = 6.4939. The result for cU/4V is J = 5.67 x 10-12T4 W/m2, called Stefan's Law. This is the total rate of radiation, over all frequencies. The amount radiated in the frequency interval from f to f + df is the value of x3/(ex - 1) times df, divided by 6.4939, times the total J.

The maximum of the curve occurs at x = 2.82, or hf = 2.82kT, from which we can find that λT = 5.102 x 106 nm-K. This is the peak of the emission per unit frequency interval. Since frequency and wavelength intervals are related by df = - dλ/λ2, the energy density per unit wavelength interval is proportional to x5/(ex - 1), where x = hc/λkT. The maximum of this curve is at x = 4.96, which gives λmaxT = 2.901 x 106 nm-K. The rule that λmaxT = constant is called Wien's Law, in either case. The maximum depends on which expression for the spectrum that you are using, and really has no great significance by itself. Although the wavelength interval result is more often quoted, the frequency interval result may be more meaningful. For solar radiation at 5750 K, the maxima are at 505 nm and 887 nm, respectively.

An actual body may not emit as efficiently as a black body, and this is taken into account by a factor ε(f), called the emissivity. The emissivity is never greater than unity, and is generally a function of frequency. For lack of better information, the emissivity of a star is generally taken as 1, except for (usually) narrow dark lines in the emitted spectrum. That is, the emitted radiation is typical of a black body at the temperature T of the star's photosphere, while the cooler overlying chromosphere causes dark lines in the spectrum. The best example is the sun, which radiates typically of T = 5750K, and the spectrum is crossed by the dark, narrow Fraunhofer lines.

The spectral sensitivity of the eye extends in its farthest limit from 380 nm to 765 nm, but a range of 400 nm to 700 nm includes most of its sensitivity. Shorter wavelengths are ultraviolet, and longer are infrared. At the bottom of the earth's atmosphere, we can receive the visible, a considerable amount of infrared, and a little ultraviolet. Extraterrestrial spectroscopes now cover a wider range, from X-rays to radio, but the main study of stellar light has been in the visible range. It is remarkable that most stars form a continuous, one-dimensional sequence from hot, massive stars to cooler, lighter stars, expressed in the Herzsprung-Russell diagram of luminosity against spectral type.

The spectral class of a star is decided by the nature of the dark lines that interrupt the black-body continuum. It is essentially a temperature series, as various atoms are thermally excited to different degrees. The spectral classes are O, B, A, F, G, K and M, with an index running from 0 to 9 in each class. Type O stars are extremely hot, with O5 at 50,000K. Temperatures are: B0, 21,000K; A0, 10,600K; F0, 7100K; G0, 5760K; K0, 4900K; M0, 3,400K. Giant stars are stars of much lower density than main sequence stars, so not quite as high a temperature is required to produce the same ionization. The temperatures of giant stars are: G0, 5300K; K0, 4000K; M0, 3000K; and M8, 2000K. Once the spectral type of a star is known, its absolute magnitude can be guessed fairly accurately. Then, since we know its apparent magnitude, its distance (parallax) can be found as we have shown above. This is a so-called spectroscopic parallax.

The type S4 photocathode has a response that extends from 300 nm to 600 nm with a peak at 400 nm. It is more responsive to blue light than the eye. Given an energy spectrum, for example that of a black-body, we can determine the response of a photodetector (usually a photomultiplier). These responses can be used to establish a magnitude scale exactly as in the visual case. However, there is no natural, predetermined relation between the two scales, since they correspond to distinct spectral sensitivities. To establish a relation, the radiation of a type A0 star at 10,600K is taken as standard, and the blue or photographic magnitude B is taken as equal to the photovisual magnitude V. The term "photographic" recalls the use of blue-sensitive photographic plates, while "photovisual" refers to some objective means of determining the visual magnitude, perhaps by the use of filters. In order to specify things definitely, the spectral sensitivities used to find the B and V magnitudes must be specified. The difference B - V is the color index, CI. For hotter stars than class A0, the CI is negative, for cooler stars it is positive. In fact, for class B0, the CI is -0.33, for class F0, +0.33; G0, 0.57; K0, 0.78; M0, 1.45. The CI is a clue to the spectral type, to the temperature, and to the spectral energy distribution of the emitted radiation. In fact, the surface temperature of a star is given approximately by T = 7200/(CI + 0.68) K.

The sun is spectral class G0, T=5750K, so the energy spectrum is as shown in the figure at the left. The maximum is at 890 nm, in the near infrared. The area corresponding to visual radiation is shown shaded, corresponding to 36.5% of the total. For any T, the value of x corresponding to 555 nm is x = 26,000/T, and the range of x corresponding to 400 nm to 700 nm is Δx = 14,300/T. Multiplying these gives an estimate of the area devoted to visual radiation. Tables, or better, a computer program using numerical integration, will give more accurate results (see References). We'll use the results of a program here, but the approximate method gives reasonable answers. The total solar radiation is 2.74 times the visible radiation. This corresponds to a magnitude difference of m = -2.5 log 2.74 = -1.09. Since the absolute visual magnitude of the sun is 4.77, the absolute bolometric magnitude will be 4.77 - 1.09 = 3.68. The term bolometric refers to the total radiation. This is one way of defining a bolometric magnitude, that may or may not be the one used by other authorities. However, comparison of bolometric magnitudes defined in this way will properly compare the total energy output of different objects.

The total solar energy reaching the earth's orbit is about 1360 W/m2. This corresponds to a total emission of 3.82 x 1026 W. The radius of the sun is 6.96 x 108 m, so its surface area is 6.09 x 1018 m2. The rate of emission is J = 6.27 x 107 W/m2 = 5.67 x 10-8T4, using Stefan's Law. From this, T = 5767 K. This is a good check that we have not strayed from the path.

The energy spectrum of the red giant Antares, α Scorpii, is shown at the right. Antares is of spectral class M1, with T = 3000K. The maximum occurs at 1700 nm, well in the infrared. Only 0.2% is radiated in the ultraviolet, 8.1% in the visible, and 91.7% in the infrared. The total radiation is 12.36 times the visible radiation, or Δm = -2.73. Since the visual magnitude of Antares is 0.96, its bolometric magnitude will be -1.77. The corresponding absolute magnitudes are -4.7 and -7.43. The distance of Antares is 41.7 psc, or 440 ly. The ratio of bolometric luminosities of Antares and the sun then corresponds to 11.2 magnitudes, or a ratio of 30,200. If we consider only the visual energy, the ratio is about 6100. In spite of its low temperature and the T4 effect, the star is bright because of its very large size. In fact, combining the total radiation and Stefan's Law allows the area of the surface to be found, and from it the diameter of the star. When using Stefan's Law, total radiation must be considered, not just visible. The exitance of Antares is 4.59 x 106 W/m2.

The case of Sirius, α Canis Majoris, is illustrated at the left. Sirius is spectral class A0, and its temperature is 10,600K. Here, the visual radiation corresponds to the peak of the curve, with λmax = 480 nm. The total radiation is about 3.25 times the visible radiation, so the magnitude difference is 1.28 magnitudes. The apparent visual magnitude of Sirius is -1.46, absolute visual magnitude +1.42. The absolute bolometric magnitude will then be 1.42 - 1.28 = 0.14. The magnitude difference relative to the sun is 3.54 magnitudes, so Sirius radiates 26 times as much energy as the sun. Sirius radiates 52.3% in the ultraviolet, 30.8% in the visible, and 17.0% in the infrared.

If the ratio Itot/Ivis is plotted against spectral class, it reaches a minmum for class F5, increasing for later and earlier classes. The additional energy for later classes is mainly in the infrared, while earlier classes radiate more ultraviolet. A very hot class O star at 50,000K radiates 98.6% in the ultraviolet, while a cool M8 giant at 2000 K radiates 99.2% in the infrared. The class O star is still bright, because of its stupendous exitance of 3.54 x 1011 W/m2, of which 1.1% is still 3.89 x 109 W/m2, about 63 times more than the sun. The M8 giant is, on the other hand, invisible unless it is very large indeed, since its exitance is only 9.07 x 105 W/m2, only 0.0146 that of the sun. To radiate 100 times as much visual energy, its radius must be 83 times the sun's radius, or 5.8 x 1010 m, or about 0.39 AU, the radius of Mercury's orbit. Giants of this size are relatively common, so we can see both class O stars and class M8 stars. Mira, omicron Ceti, is class M7 and its absolute magnitude is M = -0.5 at maximum. A star 100 times as luminous as the sun has an absolute magnitude of -0.23, so we have an idea of Mira's size.

Class O stars, on the other hand, are quite rare; only about 19 are visible to the naked eye, and most are O8 or later. The easiest to see is probably ζ Orionis, mag. 2.05, the easternmost star in the belt, or ζ Ophiuchi, mag. 2.56, in the middle of the line of stars across the centre of the constellation. Both are O9.5, but should still have temperatures a little above 30,000 K. The earliest is ζ Puppis, mag. 2.25, class O5, at declination -40°, probably the hottest naked-eye star. Not far behind is λ Cephei, mag. 5.04, just north of ζ, class O6. All these stars are very bright, and usually very distant. The absolute magnitudes cover a wide range, and all are negative.

### Appendix: Photometry

We have had to use concepts from photometry in the above. Brief explanations of these concepts, which may be unfamiliar to the reader, are presented here. We have already defined the lumen, which is energy flux corrected for the sensitivity of the eye. This is a measure of luminous flux F, and is the basic variable of photometry. The illumination E is luminous flux received per unit area, E = F/A. An isotropic point source of luminous intensity I emits 4πI lumens equally in all directions. I is measured in candles, cd [actually candela]. The normal illumination at a distance r from a point source of intensity I is then E = 4πI/4πr2 = I/r2. If r is in metres, then E will be in lumens/m2 or lux. If r is in feet, then E will be in foot-candles, or lumens/ft2. It is easy to work out that 1 ft-cd is 10.764 lux.

Analogous definitions can be made for radiant power as they are for luminous flux. They are generally distinguished by the words "radiant" and "luminous" and the fields are called radiometry and photometry, respectively. The analogue of watts is lumens. What we usually call intensity is strictly termed exitance, power per unit area.

If the source is non-isotropic, then I is the flux per unit solid angle in the direction considered, dF/dΩ. The total flux emitted is then F = ∫IdΩ. The total solid angle in all directions is 4π steradians. If the normal to an area dA makes an angle of θ with a certain direction, then dA cos θ is the projected area in that direction. The flux falling on dA from a source of intensity I at a distance r is then dF = (I dA cos θ)/r2, so the illumination is E = dF/dA = (I cos θ) /r2.

Sources may be distributed over an area dA with a surface density of B cd/m2. If we assume the sources to emit isotropically, then the flux emitted in a cone between angles θ and dθ from the normal to dA will be dF = 2πB sin θ cos θ dA, since the flux must be proportional to the projected area of dA in this direction. Integrating from 0 to 90°, we find F = πB. That is, a plane source of B cd/m2 emits π lumens per unit area. A source does not have to emit isotropically, but if it does, this is the result. This means that the flux is proportional to the projected area (we have assumed this), so an area appears equally bright however its normal is inclined to the direction of view. A uniformly radiating sphere will then appear equally bright in the centre and at the limbs. Such a sphere will radiate like a point source of intensity I = 4πr2B, if r is the radius of the sphere. Limb darkening is observed with the sun; therefore, star photospheres do not radiate isotropically, but preferentially in a radial direction.

Suppose a flux F falls on unit area of a diffusely reflecting plane surface. Some of this flux will be absorbed, and some re-emitted. If a flux F is re-emitted as if the surface were covered with an isotropic surface density of sources, then the surface is called Lambertian. Of course, not all diffuse surfaces are Lambertian. An obvious departure is the specularly reflecting surface. If B is the surface density of intensity, then we have F = πB, or B = F/π. A surface is said to have a brightness of 1 lambert if it emits 1 lm/cm2 uniformly. This corresponds to a brightness 1/π cd/cm2. Considerable confusion exists with respect to this factor of π because of the double definition of brightness. A lambertian surface illuminated with 1 lm/cm2 will have a brightness of 1 lambert (L). It will be as bright as a surface with a brightness of 0.318 cd/cm2. There are also foot-lamberts (a surface of brightness 1 ft-L emits 1 ft-cd per ft2, for example). Photometrists have given silly names to various photometric quantities. A metre-lambert is also known as an apostilb. A cd/cm2 is called a stilb, while a cd/m2 is a nit. 1 lm/cm2 is called a phot. These silly names do not have to be remembered; just use units consistently derived from the lumen and candle.

### References

I thank Sarah Ashton for finding a numerical error and correcting it.

R. H. Baker, Astronomy, 6th ed. (New York: D. Van Nostrand Company, 1955). pp. 318-331. This is a very readable classic general astronomy text with excellent basic information, illustrating the state of astronomy in 1955, at the beginning of radio astronomy and before planetary probes.

C. Kittel, Thermal Physics (New York: John Wiley & Sons, 1969). pp. 255-260.

Handbook of Chemistry and Physics 56th edition (Cleveland, OH: Chemical Rubber Company, 1975). pp. E-204 to E-208 and E-247.

R. Dibon-Smith, StarList 2000 (New York: John Wiley & Sons, 1992). A good catalogue of bright stars including apparent and absolute magnitudes, and spectral types.

E. Shulman and C. V. Cox, American Journal of Physics, 65, 1003-1007 (1997) discusses magnitudes with reference to a more accurate relation of intensity to visual stimulus.

It is not difficult to write a computer program that integrates the spectral distribution and finds the exitance (energy emitted per unit area) for any desired spectral band. The extended trapezoidal rule, presented on pp. 136-138 of W. H. Press, et. al., Numerical Recipes in C, 2nd ed. (Cambridge: Cambridge University Press, 1992), works very well and can be the basis of a serviceable program.