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Ast 4001, Fall 2015 -- file n020-magnitudes -- prepared by K. Davidson
PHOTOMETRY AND ASTRONOMICAL MAGNITUDES
We don't have time in this course for lectures on the magnitude scale, especially since you should
already be familiar with it from earlier courses. You'll be expected to know the basics outlined
here, and how to do calculations like the sample exercises. Please take a look at each internet
or wikipedia page that's cited below.
Why do astronomers use a peculiar logarithmic brightness scale? (Historical)
The word "magnitude" traditionally means something like "size" or "importance." When
Hipparchus made the first star catalog almost 2200 years ago, he listed each star's apparent
brightness as "first magnitude" (the 20 brightest stars), "second magnitude" (about 60 stars
that are fairly bright), "third," (easy to see but not bright), etc. -- a sequence like boxes in
a file cabinet. The faintest stars that he could see were "sixth magnitude." Hipparchus
had no way to actually measure brightness, but he tried to make the categories evenly
spaced according to normal human perception.
Two thousand years later, astronomers began to invent photometers: gadgets to compare
stellar brightnesses in a quantitative way. (Think about this! Without electronic detectors
or photography, how would you do it?) At about the same time, Ernst Weber and Gustav
Fechner discovered that physiological perceptions are basically logarithmic -- particularly
touch, hearing, and brightness. The decibel scale of loudness is the most famous example.
-- Look up Fechner's Law and decibel in Wikipedia or elsewhere --
Consequently, for any given star in Hipparchus' list
{ magnitude rating }  log F ,
where F is the energy flux of light entering the eye or photometer, watts per unit area.
This is an empirical relation, not a formal equation. Strictly speaking, we mean energy
flux of light in some given range of wavelengths, for example the range most visible to
the eye. Astronomers in 1850-1900 arbitrarily decided to set up a measurement system
that would match Hipparchus' categories as well as possible.
* Hipparchus: ordinal categories "first" "second" etc.
* Modern: continuous scale of numbers like 1.13 or 3.27 etc.
They discovered that Hipparchus' "sixth magnitude" stars were fainter than "first" by
factors around 100. Therefore it was decided that a difference of 5 magnitudes would
correspond to a factor of 100 in energy flux. The formula that satisfies these rules is
magnitude m = (constant)  2.5 log 10 F .
(eqn 1a)
Most textbooks don't comment on some important features of this. First, the zero-point
2
constant is not defined in terms of watts per square meter or some other physics measure;
that's hard to do for practical reasons. Instead the constant is arbitrarily defined by sets
of standard stars which can be observed for comparison. In other words this is a relative
measurement system, not absolute, a distinction that surprises some modern physicists.
Energy calibrations are available, but usually they're only valid to a percent or so. Next,
the negative sign and the coefficient 2.5 were chosen merely to be consistent with the crude
"magnitudes" that astronomers had been using since Hipparchus. Thirdly, it's a decimal
logarithm, because natural logarithms were hard to use in the days before electronic calculators.
Fourthly, today we use many different magnitude systems that sample light various wavelengths.
Equation 1a is almost always valid, but the zero-point constant varies from system to system.
Exercise problem: Deduce equation 1a from the definition that
"a difference of 5 magnitudes corresponds to a factor of 100."
How to assess relative brightnesses by just glancing at the magnitudes
Equation 1a can also be written as
F = (another constant) × 10  m .
(eqn. 1b)
This is a pretty convenient way to quantify stars, because huge factors in F can be quoted as
moderate differences in magnitude. The following examples look a lot like a table of e x :
Magnitude difference
m = m1  m2
Brightness factor
F2 / F1
0.1
0.2
0.5
0.75
1.0
1.5
2.0
2.5
3.0
4.0
5.0
10
15
20
close to 1.1
close to 1.2
1.585
almost 2.0
2.512
3.98
6.31
exactly 10
15.85
39.8
exactly 100
10 4
10 6
10 8
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Generally speaking, "n-th magnitude" means that m is in the range n  0.5 to n + 0.5. The
exception is first magnitude, because traditionally it includes the twenty brightest stars in the
sky (not counting Sun and planets), and several of them are so bright that that their visual
magnitudes are about zero or even negative. This sketch may help you visualize the scale.
...Another thing that most textbooks don't mention is that magnitudes are well adapted for
doing rough estimates in your head. For instance, suppose a bright star has magnitude 2.4,
a recent supernova in another galaxy has m = 18.5, and for some reason you want to get
some idea of their physical brightness ratio. After a little practice, in just a few seconds
one can go through the following thoughts:
(1) A magnitude difference m gives the brightness ratio F2 / F1 .
(2) In this case m = 18.5  2.4 = 16.1  16 magnitudes for a rough estimate.
(3) Each difference of 5 magnitudes implies a flux ratio of 100, and one magnitude
corresponds to a ratio of about 2.512, rounded to 2.5.
(4) Since 16 magnitudes = 5 + 5 + 5 + 1, the brightness ratio in this problem
is close to 100 × 100 × 100 × 2.5 = 2.5 × 10 6.
(5) And, if you want to be fancier, you might remember the 0.1 magnitude that we
dropped in step 2. Since 0.1 magnitude corresponds to about 10 percent in
flux ratio, an improved estimate for our comparison is about 2.75 × 10 6.
(But most experienced scientists automatically round off any quick estimate
like this and say "roughly 3 × 10 6 .")
After doing a few examples, you can go through reasoning like this in much less time than
it takes to read it -- and we didn't actually have to plug any numbers into the formula.
Practice problem (Doing things like this without a calculator is good mental exercise.)
The Sun has apparent magnitude 27 and the star Sirius has m = 1.4. (These are
negative values because they're both much brighter than a typical star.) Estimate the
approximate flux ratio F (Sun) / F (Sirius) without using the formula or a calculator.
Pencil and paper are OK, but see if you can do it in your head.
Remember, BTW, that a small magnitude difference immediately tells us the approximate
fractional difference in brightness, e.g. m = 0.1 magnitude indicates about a 10%
difference in flux, 0.15 magnitude is about 15%, etc. This rule of thumb is fairly
accurate if | m | < about 0.3 or 0.4. (Calculate a few examples.) Also remember
that higher magnitude numbers mean fainter, not brighter! That's Hipparchus' fault.
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Practice problems
The only way to really learn a technical subject is to do problems. Here are some.
A list of correct answers is given at the end of these notes.
--- An important generality: Magnitudes are seldom measured with precision better
than about ± 0.01 mag or ±1 percent in F. Therefore an answer like m = 7.834
is usually unrealistic and m = 7.8343 would be silly. --1. 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.
As an example, here's one way to think about it.
* The galaxy plus SN is 0.6 mag brighter than the galaxy alone (15.9 minus 16.5).
* According to eqn. 1b, 0.6 mag corresponds to a brightness factor of about 1.74,
so F (galaxy plus SN) = 1.74 F (galaxy).
* Therefore F (SN) = 0.74 F (galaxy).
* Using formula 1a, a ratio of 0.74 implies "about 0.33 magnitude fainter."
* Therefore m (SN) = m (galaxy) + 0.33 = 16.5 + 0.33 = about 16.8.
After a little practice you can do the above steps rapidly -- and some other approaches
are equally good. Now re-do this problem yourself without looking at these notes!
2. 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.
3. 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.
4. A particular double star system has total apparent magnitude m = 4.5. In terms
of intrinsic energy flux, component star A is twice as bright as its companion B.
Calculate the individual magnitudes of A and B.
5. 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 .
6. 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?
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Magnitudes of some real stars
Here are the apparent visual magnitudes of various famous objects.
The Sun ...
Full Moon ...
Venus when brightest
A 100-watt incandescent light bulb
seen from 1 kilometer away ...
Mars and Jupiter when brightest ...
Sirius (the brightest star) ...
Deneb (the twentieth-brightest star) ...
Stars of the Big Dipper ...
A 100-watt light bulb 10 km away ...
Naked-eye visibility limit ...
Proxima Centauri (nearest red dwarf star) ...
A supernova in a galaxy 10 Mpc away
3C 273 (brightest quasar)
Pluto ...
Pulsar in the Crab Nebula ...
A star like the Sun located in a
globular cluster 10 kpc away ...
Faint enough to be difficult with HST
For spectroscopy ...
For imaging ...
m = 26.7
roughly 12
 4.4
A star like the Sun located in the
Andromeda Galaxy ...
About 29
roughly 3 or 4
 2.5
 1.4
+ 1.3
around +2
roughly +2
around +6, depending on circumstances
11.1
Typically 11 to 13
About 13
About 14
About 17, but gradually fading
About 20
17 to 21, depending on goals
Roughly 23 to 25
An observational puzzle
Officially, apparent magnitude m is defined "as seen from above the Earth's atmosphere."
But the atmosphere obviously affects any real ground-based observation, even on a high
mountain. At visual wavelengths the atmosphere blocks 15 to 40 percent of a star's light,
depending on circumstances. Astronomers long ago devised a simple way to measure this
effect, often with high accuracy. They did this before there were any space telescopes.
Can you think of a way to do it? Hints: (1) Imagine observing a star several times as it
rises higher in the sky and then sets. (2) For each observation it is possible to calculate the
star's current altitude, i.e., angle above the horizon. (3) Google "optical depth" and "Beers law".
Combining all these thoughts, invent a way to plot the data to show the atmosphere's effect.
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Intrinsic brightness and absolute magnitude
A star's apparent magnitude doesn't tell us much about its physical nature, since a feeble
nearby star looks as bright as a powerful distant one. After a few stellar distances became
available, astronomers adopted the concept of absolute magnitude, which expresses
intrinsic brightness or luminosity rather than appearance. Definition:
** Absolute magnitude M is the apparent magnitude m that the star would
have if it we could view it from a standard distance of 10 parsecs. **
10 pc was an arbitrary choice, selected only because many of the earliest measured star
distances were about that size. Here are some examples of M at visual wavelength.
Proxima Centauri, a feeble red dwarf star ...
+15.5
The Sun ...
+ 4.8
Vega, a star about twice the mass of the Sun...
+ 0.2
Betelgeuse, a colossal red supergiant ...
 5.0
The visually brightest normal stars ...
 10.0
A typical supernova at maximum ...
 18
In order to measure M , let's think about it backward: If a star has absolute magnitude M
and is located at distance D from us, what will be its apparent magnitude m ? First
consider the distance dependence. The apparent radiation flux obeys an inverse-square law,
F (apparent)  1 / D 2 . Putting this fact into equation 1a (page 1), we see that apparent
magnitude gets fainter with distance in the following way:
m = (a constant for this particular star)  2.5 log ( D)
= (same constant) + 5 log D ,
and we'll express D in parsecs.
But we also know that m = M if D = 10 pc, i.e., if log D = 1. In order to make
this true, the constant in the expression has to be M  5 . Thus our formula becomes
m = M  5 + 5 log D .
(eqn. 2)
If observations tell us a star's m and D , then obviously we can calculate M from eqn. 2.
There's no strong need to memorize equation 2, because we can convert between
m and M by using the definition of M and a little logic. Here's an example:
A star at D = 2000 pc has apparent magnitude 8.0; what's its absolute magnitude?
Imagine moving the star closer to us so it has the standard comparison distance of 10 pc.
Then it’ll be closer by a factor of 200 and brighter by a factor of 200 squared = 40,000.
But equation 1a on page 1 tells that a factor of 40,000 corresponds to a difference of 11.5
magnitudes. Evidently the star would become 11.5 mag brighter, i.e., m = 8.0  11.5
= 3.5 at a distance of 10 pc. By definition this is its absolute magnitude M.
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Magnitude systems that sample specific wavelengths
Historically, the first magnitudes were estimated by eye, which emphasizes yellow-green
wavelengths around 550 nm. When photography became available, the early photo
emulsions responded mainly to blue wavelengths and were almost blind to red light.
(You can see this in nineteenth-century photos of the U.S. flag, for example. The red stripes
look black, while the blue field often appears light gray in those old pictures.) If a blue
star and a red star had equal apparent magnitudes seen by the eye, then the blue one was
brighter in a photograph. Thus, for about 50 years astronomers used two parallel magnitude
systems: m vis (visual) and m ph (photographic). Differences between them were useful
for indicating a star's color and temperature. Like the eye, a photographic emulsion had a
roughly logarithmic response to radiation flux F .
Then, around 1950, electronic photon detectors became available, especially photomultiplier
tubes. They were very sensitive, they actually counted photons, and their sensitivity depended
on wavelength in a consistent way. By combining a standard photomultiplier tube with a
blue or yellow or red filter, it was fairly easy to sample well-defined wavelength intervals.
This method led to "Johnson UBVRI magnitudes," a good color-dependent system that's still
used today with modern CCD detectors:
m U or just U: ultraviolet magnitude,  360 nm.
m B or B:
blue,  440 nm.
m V or V:
visual,  550 nm.
m R or R:
red,  700 nm. (or 650 nm in a revised system)
m I or I:
infrared,  900 nm. (or 800 nm in a revised system)
V was intended to match old-fashioned visual magnitudes, B mimicked the classical
photographic magnitudes, and the U wavelength had a special purpose that we won’t
explain here. Nowadays we regard 360 nm as violet rather than true UV, and a modern
astronomer’s “infrared” goes far beyond 900 nm; but the UBVRI labels continue to be
used. Remember that V denotes "visual" and not "violet." BTW, modern R and I
filters are somewhat different from the original Johnson system – see the list above.
UBVRI is a broad-band photometric system; for example, V actually samples wavelengths
from about 500 nm to 600 nm. There isn't any sharp cutoff, and strictly speaking B overlaps
V, etc. If f (  ) is flux from the star per unit wavelength, and s ( ) is an instrumental
sensitivity function which depends on the detector and filter combination, then
instrumental response =

s ( ) f ( ) d  .
(eqn. 3)
We use eqn. 1a with the response value in place of F . The main point to recognize here is
that s ( ) is usually a sort of bell-shaped function that peaks at some particular wavelength
and has a particular width. Again, the zero-point or calibration constant depends on which type
of magnitude we're measuring. This depends on some arbitrary choices. But once a magnitude
system is defined, then each constant remains constant.
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A bewildering assortment of magnitude systems exist, too many to list here; for a
professional-level account of some of them, see
http://www.astro.umd.edu/~ssm/ASTR620/mags.html .
A few other www sites, perhaps more readable, are listed at the end of these notes.
Some magnitude systems use narrow-band filters, which are more definite but are also
less sensitive for faint stars. Infrared astronomers have J, K, L ... magnitudes sampling
 ~ 1.25 m, 2.2 m, 3.4 m, and other wavelengths. Ultraviolet systems are used for
data from the Hubble Space Telescope (HST). A few magnitude systems try to employ
absolute fluxes observed at all wavelengths with spectrographs:
m (  ) = C 1  2.5 log 10 f ( )
or alternatively
m (  ) = C 2  2.5 log 10 f ( ) ,
where v is photon frequency and C 1 , C 2 are known constants; but doing this requires
a lot of careful calibration work. For the best-known example, google “AB magnitude”.
Photometric “colors”
Suppose we know the B and V magnitudes of a star. For simplicity, let’s pretend that
these sample the monochromatic brightness at wavelengths 440 nm and 550 nm; this
is a fair approximation even though broad-band filters were used. Then equation 1a
tells us that
B 
C B  2.5 log f (440 nm)
and
V  C V  2.5 log f (550 nm) ,
where C B and C V are calibration constants that we won’t need to evaluate. Take
the difference between these two magnitudes:
B – V  (C B  C B) + 2.5 log f (550 nm)  2.5 log f (440 nm)
 (constant)
+ 2.5 log { f (550 nm) / f (440 nm) }
(eqn. 4)
Evidently B  V indicates the ratio of brightness at 550 nm compared to 440 nm,
basically the slope of the spectrum. See figure on next page. Note that B  V doesn’t
depend on the star’s distance, so it’s an intrinsic attribute of the star. We can reasonably
call this quantity the star’s “color,” related to the temperature of the radiation.
Examples of B  V for real stars are
Very hot blue O- or B-type star …
Moderately blue star like Sirius or Vega …
The Sun …
Betelgeuse, a cool red supergiant star …
A very cool red/infrared dwarf star …
 0.3
0.0
+ 0.6
+ 1.5
+ 2.0
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(Above) B and V flux samples in Planckian blackbody spectra
at temperatures of 5000 and 15000 K. The ratio f (V) / f (B)
depends strongly on T in this temperature range.
Of course we can use other magnitude types to find other “colors” in the same way,
for instance U  B or V  R or J  K , etc. When we take differences like these,
we can use either apparent magnitudes or absolute magnitudes – they both give the
same answer.
Caution: Each of these color values becomes insensitive to T at high temperatures --- e.g., most stars above 30,000 K have approximately B  V  0.3. To see why,
look up “Rayleigh-Jeans law” and plug it into eqn. 4.
Bolometric magnitude
A bolometer is a device that measures total radiation flux integrated over all wavelengths,
F =  f ( ) d  . Bolometers aren’t practical for most stars (can you guess why?),
but the concept is valuable. We can combine magnitudes at several wavelengths,
or spectroscopic data, to synthesize a “bolometric magnitude” m bol or M bol which
is supposed to include the star’s entire flux F . The Sun’s absolute bolometric
magnitude is + 4.75, so a star’s luminosity and bolometric magnitude are related by
M bol = 4.75  2.5 log 10 ( L / L sun )
( eqn. 5 )
where L sun  3.9 × 10 26 W. Usually, however, some part of the wavelength spectrum
is unobservable – particularly the far ultraviolet. We have to employ theoretical models
to extrapolate the data, always a questionable procedure. Therefore “observed”
bolometric magnitudes are not very reliable.
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The difference between bolometric magnitude and visual magnitude is called the bolometric
correction or B.C.: M bol = M V + B.C. We observe the visual magnitude and then
usually we get the B.C. from theoretical models of the star's atmosphere and temperature.
Here are a few examples -spectral type
surface temperature
B.C.
B0
30,000 K
 magnitudes
A0
10,000 K

G0
6,000 K

K0
5,100 K

M2
3,500 K

If the star is very hot, then most of its luminosity is invisible ultraviolet; and if it's very cool
then most of the luminosity is invisible infrared. That's why the B.C. is a big correction at
both ends of the scale. (Caveat: A few authors use the same numbers with positive signs;
they prefer to write M V = M bol + B.C. ) Note that the B.C. is near zero for middling
temperatures in the range 5500 to 9000 K, but becomes more serious for both lower and
higher temperatures. Can you visualize why?
Interstellar dust
Here’s a disagreeable complication. Gas between the stars is very transparent, but interstellar
dust grains block a substantial fraction of light from stars more than 300 pc away. “Dust grain”
is a misleading term, because the particles are much smaller than terrestrial dust; they’re more
like smoke particles in size. Anyway, because of dust, a distant star’s apparent magnitude at
wavelength  is really
m = m 0 + A (  ) ,
(eqn. 6)
where m 0 is the apparent magnitude that it would have if there were no dust, and A ( )
is called the “interstellar extinction.” Extinction is stronger at short wavelengths, so, like
the Earth's atmosphere, it makes distant objects appear red. A ( ) is roughly proportional to
the column density of material along the line of sight (atoms per cm 2). Whenever we
calculate an absolute magnitude, we should put m 0 into equation 2 instead of m .
Often the wavelength dependence can be used to estimate A ( ). As an example, let’s
consider only the B and V magnitudes of a star, with extinctions A B and A V . The shape
of the extinction curve A ( ) is approximately the same in most parts of interstellar space, and
usually A B  1.32 A V . Suppose we observe a star’s B  V color. According to eqn. 6,
B V
=
( B0  V0 ) + ( A B  A V )
 ( B0  V0 )
+
0.32 A V .
(eqn. 7)
The difference between observed color B  V and the star’s "intrinsic" color B0  V0 is
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called interstellar reddening and often denoted E B-V . If we know the star’s spectral
type, then we can estimate its true color B0  V0 from observations of many other stars of
the same type. Knowing both B  V and B0  V0 , we can calculate A V from eqn. 7.
In some places this technique fails because the A ( ) curve is abnormal; but it’s usually OK.
Extinction for other magnitudes such as U and R is treated in the same way.
For information about A ( ), see the internet and especially the wikipedia article about
“extinction (astronomy).” The simplest approximation that works fairly well for
350 nm <  < 800 nm is A ( ) / A V  { (700 nm) / }  0.27 . For stars in the
Galactic disk at distances around 1000 pc, A V typically ranges from 0.5 to 2 magnitudes,
and it tends to be roughly proportional to distance. But there are wide variations, and in a
few locales the extinction curve has a different shape because the dust grains have unusual
sizes or composition.
________________________________________________________________________________
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 “magnitude”
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 some
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)
Googling “magnitude converter” leads to several online routines for converting between
magnitudes and energy fluxes. Unfortunately, the bewildering variety of magnitude systems
makes these sites hard to use. Be careful with measurement units!
_________________________________________________________________________________________________________________
Answers for questions on page 4
#2. m (total ) = 4.80.
#3. m (average star) = about 11.3.
#4. m (A) = 4.94, m (B) = 5.69.
#5. The total of 6000 faint stars is roughly
6 or 7 times as bright as Sirius alone.
#6. Assuming that the pupil of the eye is 5 mm across, and ignoring various optical
inefficiencies, the binoculars give a limiting magnitude near 11. (In reality, for
most people this estimate is too optimistic. Can you think of reasons why?)