ASTR 3130, Majewski [SPRING 2015]. Lecture Notes
ASTR 3130 (Majewski) Lecture Notes
PHOTOMETRY: THE MAGNITUDE SYSTEM
REFERENCE: Birney et al. Chapter 5, 10, 14.
Photometry: Definition
 Photometry  measuring the amount of light received from an object
 Total light (energy) given off by a star
l = luminosity, in photons / sec, or in erg / sec
 Light flux received on Earth from the same star  i.e., a point source.
f = flux, in photons / sec / cm^{2}, or in
erg / sec / cm^{2}
 (Note: For light flux received on Earth from an extended source,
like a nebula or galaxy  measure flux coming from a
solid angle:
f = flux, in photons / sec / cm^{2} / steradian, or in
f = flux, in erg / sec / cm^{2} / steradian )
 Energy flux received at Earth really given by:
OR:
where the quantities in the first equation are functions of wavelength
(and similar quantities can be used as functions of frequency, as in
the second equation):
f_{λ}^{o} =
stellar flux incident on Earth atmosphere
T_{λ} = transmission of
atmosphere
R_{λ} = efficiency of
telescope + detector
S_{λ} = transmission
function of filter
 f_{λ}^{o} depends
not only on how bright something is  its luminosity 
but on its distance d:
Definition of Magnitudes
Magnitudes are a Brightness Scale, a measurement of received fluxes.
 Use system of brightness measurement with roots extending to second century
B.C. by Greek astronomer Hipparchus.
 Ptolemy  writes "The Almagest":
 catalog of 1000 stars
 6 "magnitude" classes:
 1 = brightest
 6 = faintest
 But revisions required and made in last few centuries. For
example, want to put Sun, Moon, bright planets on same scale  need
to extend scale to < 1 mags.
 Later, telescope invented  need to 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
Pogson proposes to formalize scale so that
ratios between successive magnitudes are
exactly 2.5119.
 Thus, two stars of fluxes f_{1} and
f_{2} have a magnitude difference
given by
 Note that the above equation also shows that fractions
of magnitudes are possible for stars with brightnesses
in between two integer magnitudes.
Thus, star brightnesses are quoted on a magnitude scale
which is:
 logarithmic  Each magnitude is a ratio of ~2.5X brightness.
Each 5 magnitude difference is 100X brightness ratio.
 "backwards"  Brighter objects have smaller magnitude values.
 openended  As we probe deeper, find fainter stars
of larger
magnitudes
NOTE:
 Brightest stars are about 0 magnitude (defined by summer star Vega in
constellation Lyra).
 Sun, Moon, Venus, Mars, Jupiter have negative magnitudes.
 The faintest visible with naked eye are about 56 magnitude. But
(as you have seen in Lab 1 and 2) the actual limiting magnitude depends very much on
prevailing conditions (clouds, moonlight, air and especially light pollution).
 A finding chart useful for determining the limiting magnitude is given in
Norton's Star Atlas.
 The "bowl" of the Little Dipper is a handy reference for the magnitude scale,
because it contains one star each of approximately magnitude 2, 3, 4 and 5.
 There are only 11 stars brighter than magnitude 1 visible from
Charlottesville but 1630 stars brighter than magnitude 5.
Faintest objects yet detected (Hubble Space Telescope):
31 magnitude = over 10 billion times fainter than anything visible to unaided eye.
Relative Photometry
Measuring the brightness of one star compared to another is called
relative photometry.
We have seen the Pogson magnitude system which gives the following
permutations of the same equations above:
 Δm = 5 be 100x
ratio in brightness
 In these equations we are always comparing pairs of stars and
are on a relative scale.
 When no consideration of 0 point of magnitude system,
only brightness ratios (magnitude differences) 
call this relative photometry.
Absolute Photometry
When we care about the magnitude of a star on a set (e.g., the Hipparchus)
magnitude scale, this is absolute photometry.
 To set up the absolute scale, astronomers need to decide on a
certain star to have a specific magnitude.
 By convention, we choose the star Vega ( Lyrae)
to be 0 mag in all filters. Then:
where we can use any measure of brightness for
f_{*}  e.g., different filters giving different
wavelength ranges observed 
and the constant only depends on measuring the same
set of wavelengths in the Vega spectrum.
NOTE: THE CONSTANT WILL BE DIFFERENT FOR EACH FILTERED SET
OF WAVELENGTHS USED, AND IS ALWAYS TIED TO THE SPECTRUM OF VEGA.
 In effect, the absolute photometry scale means we are always
doing relative photometry to the same star  Vega.
 ASIDE: An alternative magnitude scale gaining popularity is the
"AB magnitude system", which is not
based on Vega, but instead assumes the CONSTANT in the above equation
is the same for all filtered magnitudes.
 Simpler  don't always have to know what Vega is doing
to define the magnitudes.
 If flux is measured in units of ergs sec^{1} cm^{2} Hz^{1}, then
we have m_{AB} = 2.5 logf  48.60.
 Usually write "AB" subscript for AB magnitudes, e.g.,
"V_{AB}".
Magnitudes and Distances
 Now, assume m_{1} and m_{2} are for same star,
different d. Then:
 Example :
If the star with magnitude m_{2} is farther by 10x, then it is 100x fainter,
5 mags fainter (larger).
Absolute Magnitudes and the Distance Modulus
 Apparent magnitude (m)  magnitude of star as observed
 Note that the apparent magnitude of a star depends
not only on its luminosity, but its distance, because
its observed flux is given by:
f = l / (4 π d^{ 2})
where l is the luminosity of the star and d is the distance.
 THOUGHT PROBLEM: Where does the above relation come from?
 In some cases, we are interested in separating the luminosity and
the distance effects. For example, consider comparing the luminosities
of stars when placed at the same distance.
We can compare these luminosities as ratios of fluxes or as
differences on the logarithmic magnitude scale.
We define an absolute magnitude as the apparent
magnitude of star if placed at a distance of 10 parsecs.
 To distinguish apparent magnitudes from absolute
magnitudes, we write the latter as capital "M''.
 A parsec is 3.25 light years.
 The utility of the concept of absolute magnitudes comes from the fact that
if somehow we can guess the absolute magnitude, M, of a star or other object,
and we measure its apparent magnitude, m, we can determine the distance
to that star by:
The difference between the apparent and absolute magnitude of
an object, (mM ), is called the "distance modulus" of that object, and,
as we can see, is directly related to the distance of the object.
 We are generally working in a certain filter system, so
important to identify the filter; e.g.:
 One of the most important problems in astronomy has to do
with determining the absolute magnitudes, M, for
objects in order that we can estimate distances.
 A standard candle is a certain kind of object
(star, galaxy, or other object) that:
 makes its identity known by some easily
observable characteristic, and
 has a definable absolute magnitude.
Standard candles are extremely valuable in astronomy, because
if we find one, we can estimate its distance.
 Different ways of identifying "standard candles" that have
supposedly constant M:
 Variability
REFERENCE: Chapter 14 of Birney et al. discusses variable stars and how to find
their periods (useful to the RR Lyrae lab in this class).
E.g. certain kinds of stars like
RR Lyrae or Cepheids vary in brightness in easily identifiable
ways:
Supernovae explode with a reasonably welldefined light profile:
 Spectral type
E.g. We can take a spectrum of the star and see what
kind of star it is: Spectral type (OBAFGKM) and luminosity
class (I,II=supergiant, III=red giant, IV=subgiant, V=main sequence).
For example, Sun is a G2V star (G type star, 2/10's of the way
to being a K star, and of luminosity class V=main sequence type).
All stars of same spectral type + luminosity class should be of same
luminosity/absolute magnitude.
Getting distances in this way is called measuring a
spectroscopic parallax.
 Colors
Identify "spectral type" by photometry in different filters,
which is like very coarse spectroscopy.
Unfortunately, colors alone can give ambiguities.
 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
distance errors 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
distance errors off by factors of 10,000 or more.
 Morphology
E.g., Assume all globular clusters of certain concentration
are same absolute magnitude.
Or, assume all galaxies of one type (e.g., spirals with a certain
disk to bulge ration) have the same absolute magnitude.
 Ensembles
E.g. Assume that the third brightest galaxy in cluster of galaxies
are typically all about name M; or the tenth
brightest giant star in globular cluster is of fixed M.
 How do we get the absolute magnitudes of different standard
candles??
 This is the basis of the Cosmic Distance Scale.
 The "bottom rung" of distance ladder is trigonometric
parallaxes.
The top part of the diagram above shows the Earth at two different times, and the triangle formed with a nearby star and these two positions of the Earth. The bottom part shows two pictures of the nearby star projected onto more distant stars taken from the two sides of the Earth's orbit. The moving platform of the Earth provides a kind of stereo view of the nearby star over the course of the year, Just like your two eyes allow you to have depth perception. If you stare at the lower images at the right distance and merge mentally the two pictures from the two sides of the Earth's orbit, you will see the nearby star standing in the foreground of the more distant stars in 3D (or else you will get a headache).
 This geometric method is perhaps the most reliable method
for estimating distances.
 But only good to 100 parsecs (McCormick Observatory),
1000 parsecs (HIPPARCOS Mission), but hope for
100,000 parsecs in the future from space astrometry missions.
(This was supposed to be done by NASA's
Space Interferometry Mission, but this project was canceled in 2010; ESA's
Gaia mission will go after parallaxes on ~10,000 parsec scales however.)
 To get distances of more distant objects, have to
use less direct means using the distance modulus equation.
 An example of how the "Ladder" builds is as follows:
 We obtain the parallax of a nearby RR Lyrae star and
obtain its absolute magnitude.
 Find an RR Lyrae in a globular cluster. Get the distance
to the globular cluster by apparent magnitude of
member RR Lyrae stars. Thereby derive cluster's
absolute magnitude.
 Find a similar globular cluster in the Andromeda
galaxy, measure its apparent magnitude and thereby get
the distance to Andromeda, and derive the absolute
magnitudes for Andromedalike spiral galaxies.
Alternatively, find an RR Lyrae in Andromeda Galaxy, etc.
 Find an Andromedalike galaxy for which we now
can estimate the distance, but that has a supernovae
explode in it. Gauge the absolute magnitude of the
peak of a supernova lightcurve.
 Use supernovae of similar type (light curve)
to get distances of even more distant things.
 A problem with this continuing extrapolation is that
it gets shakier and shakier with every step up the ladder.
 Good parallaxes are extremely valuable. McCormick
refractor responsible for about 1/3 of groundbased parallaxes
up to a decade or so ago.
 E.g., In 1980s the distance to the Hyades star cluster
was found to be 10% off. Since this cluster is used as
a low rung on the distance ladder, the entire universe
consequently expanded by this 10% systematic change.
An example of the use of a standard candle that resulted in a significant discovery
was the work by Trumpler to prove the existence of interstellar dust.
 Robert Trumpler (1930) showed existence of interstellar absorption from dust
by comparing distances of clusters from the brightnesses of their stars
to geometric distances from the cluster sizes (i.e., assuming a standard
linear size for open clusters). The latter method
always gave closer distances.
 Therefore, stars get dimmer both due to distance and because some light
gets absorbed, scattered by dust along lineofsight.
 Worse problem near the Galactic plane.
 To account for this dust extinction we can
write the distance modulus equation more accurately as:
THOUGHT PROBLEMS:
 Explain various ways that one might establish
the absolute magnitude of an entire galaxy.
The distance to a cluster of galaxies?
 Supernovae are extremely rare  they
happen on average once a century per galaxy.
How are their absolute magnitudes derived?
 Does it bother you that variable stars are called standard
candles? Why? How can this be?
Flux/distance figure modified from
http://zebu.uoregon.edu/~js/ast122/lectures/lec03.html. HR diagram
from http://www.jb.man.ac.uk/distance/life/sample/stars/. All
other material copyright © 2002,2006,2008,2012,2015 Steven R. Majewski. All
rights reserved. These notes are intended for the private,
noncommercial use of students enrolled in Astronomy 313 and Astronomy 3130 at the
University of Virginia.
