Download Lecture 12

The magnitude system
ASTR320
Monday October 10, 2016
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”:
𝐿
𝑏=
4𝜋𝑑 2
• This is a measure of
the radiant flux
produced by the star.
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 today 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 EM spectrum of a source,
which is impossible with a single type 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 fνδν = fλδλ, and ν = c / λ,
𝜈 𝑓𝜈 = 𝜆 𝑓 𝜆
• The functions fν(ν) and fλ(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
1
5
100 = 2.5119, Pogson formalized scale so that ratios
between successive magnitudes are exactly 2.5119.
Magnitudes
in context:
ColorMagnitude
Diagram
(CMD)
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:
(𝑚2 −𝑚1 )
𝐹1
= 100 5
= 2.5119(𝑚2−𝑚1)
𝐹2
• Note that the above equation also shows that fractions of
magnitudes are possible for stars with brightnesses in
between two integer magnitudes.
Apparent Magnitude
• 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.
Magnitude: definition
• Take the logarithm of
(𝑚2 −𝑚1 )
𝐹1
= 100 5
𝐹2
• To derive a more useful equation:
𝐹2
𝑚2 − 𝑚1 = −2.5 log10 ( )
𝐹1
• That can be used to compare the apparent magnitude,
m, of two sources.
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)
2
(𝑚−𝑀)
𝐹10
𝑑
5
100
=
=
𝐹
10 𝑝𝑐
• Where F is the flux measured at d, in parsecs.
• And F10 is the flux measured at d=10 pc.
• Take the logarithm to get:
𝑀 = 𝑚 − 5 log10 𝑑 + 5
• (If d is in pc)
Distance modulus
• Rewrite this equation:
𝑀 = 𝑚 − 5 log10 𝑑 + 5
• As the distance modulus, µ:
𝜇 = 𝑚 − 𝑀 = 5 log10 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
in context:
ColorMagnitude
Diagram
(CMD)
Example:
Distance modulus
𝜇 = 𝑚 − 𝑀 = 5 log10 𝑑 − 5