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Astronomy
Toolkit
 Magnitudes
 Apparent magnitude
 Absolute magnitude
 The distance equation
 Luminosity and
intensity
Units and other basic
data
Logarithms
Hipparcos of Nicaea (c.190 – c.120 BC)
at work
Hipparcos, a Greek astronomer, invented
the first scale to rate the brightness of the
stars.
Magnitudes
• Hipparcos classified the stars visible to the naked eye into
six different brightness classes called magnitudes
• Hipparcos chose to categorize the brightest stars as
magnitude 1, and the faintest as magnitude 6 (smaller
numbers are brighter stars)
• The magnitude system of Hipparcos is still in use today in
a slightly revised form
Modern Magnitudes
• The magnitude scale is
logarithmic
• The difference in
magnitude between two
stars can be expressed as
a function of the ratio of
their brightness
mstar1  mstar2  2.5 log( Istar2 / Istar1 )
Istar2
Istar1
 10
mstar1 mstar2
2.5
 10
0.4 ( mstar1 mstar2 )
Apparent Magnitude
• Some stars appear bright and
others very faint in the sky
• The apparent magnitude “m” of a
star is a measure of how bright it
appears in the sky
– Some faint stars are intrinsically bright,
but are very distant
– Some bright stars are very faint but
happen to lie close to us
The apparent
magnitude of
the Sun is -26!
• A star’s apparent magnitude tells us little
about the star
• We need to know stars’ distances from
Earth
Absolute Magnitude
• The absolute magnitude
“M” of a star is defined as
the apparent magnitude a
star would have if it were
placed at a distance of 10
parsecs
• The absolute and apparent magnitudes are
related by the distance equation, where D is
the distance in parsecs
m  M  5 log ( pc)  5 log 10 (D)  5
D
10 10
Playing with Magnitudes
• The star α Orionis (Betelgeuse) has an apparent
magnitude of m = 0.45 and an absolute
magnitude of M = –5.14
• What is the distance to Betelgeuse?
m  M  5 log 10 (D)  5
0.45 – (-5.14)=5log10(D)-5
5.59/5 + 1 = log10(D)
D=102.12 =131 parsecs
Luminosity
• The total energy emitted by the
star each second is called its
luminosity, L
• Luminosity is measured in
watts (power = energy per
second)
• Knowing the apparent
magnitude and the distance of
a star, we can determine its
luminosity
• The star radiates light in all directions so that its emission is spread
over a sphere
• To find the intensity, I, of light from a star at the Earth (the intensity is
the emission per unit area), divide the star’s luminosity by the area of
a sphere, with the star at the centre and radius equal to the distance
of the star from Earth, D.
I = L/(4πD2)
Units and Other Basic Data
• Angle
– 1 arcminute = 1/60 of a degree = 2.9089 × 10-4 radians
– 1 arcsecond = 1/3600 of a degree = 4.8481 × 10–6 radians
– 1 milliarcsecond (mas) = 1/1000 arcsecond
• Speed of light (c) = 2.997 × 108 m/s
• Distance
– Astronomical Unit – 1.5 x 108 km
– Light Year ~ 1013 km
– Parsecs
• 1 parsec (pc) = 3.086 × 1013 km = 3.26 light-years
• 1 kiloparsec (kpc) = 1000 parsec = 3,260 light-years
• 1 Megaparsec (Mpc) = 106 parsec = 3,260,000 light-years
– 1 nanometer (nm) = 10–9 m
More Units
• Velocity – kilometers per second
• Mass – in units of the mass of the Sun
2 x 1030 kg
• Luminosity – in units of the solar luminosity
4 x 1026 watts = 4 x 1026 joules sec-1