Stars: Distances and Magnitudes

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Chapter 11

• Stars Distances and Magnitudes

• The Stellar Magnitude Scale
• Trigonometric Parallax
• Absolute Magnitude
• Magnitudes at Different Wavelengths
• Color Index
• Bolometric Magnitude and Stellar Luminosity

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Stellar Luminosity
Luminosity is the total amount of power (energy
per unit time) the star radiates into space. It
is measured in power units (Watts).
The apparent brightness (flux) is the amount of
light reaching us per unit area (W/m2).
Brightness of a star in the sky depends on the
distance towards a star and its luminosity.
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Measuring the Apparent Brightness
Stars emit radiation of all wavelengths. No
detector is sensitive to the entire spectrum.
Usually we measure apparent brightness in a small
range of the complete spectrum.
Eyes are sensitive to visible light. When we
measure the apparent brightness in the visible
region, we can calculate only the visible?light
luminosity.
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Stellar Magnitudes
Historically stellar brightness is described in
magnitudes suggested by Hipparchus.
The brightest stars received the designation
first magnitude, the next brightest second
magnitude, etc.
The faintest stars visible by the naked eye are
sixth magnitude.
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Definition of Stellar Magnitude
Star 1 has a flux fn and a magnitude n Star 2 has
a flux fm and a magnitude m
?2.5 log fn/fm n ? m fn/fm 100 (m?n)/5
m ? n 2.5 log fn/fm
Zero-point of the magnitude scale
Lets set m0, then ?2.5 log fn/f0 n A star
with a flux of f0 is a zero magnitude star.
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Stellar Magnitudes
The modern magnitudes system is more precisely
defined. Vega has a magnitude 0.00. Since a star
may have any brightness, fractional apparent
magnitudes are possible. Example a star of a
magnitude 1.00 is 2.5 times brighter than a star
of a magnitude 2.00.
The brightest star in the sky is Sirius with an
apparent brightness of 1.46. The faintest stars
observed with the Hubble Space Telescope are of
a 25th magnitude.
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Stellar (Trigonometric) Parallax
Parallax is the annual shift in a stars apparent
position in the sky due to the Earths orbital
motion.
The parallax angle is half the annual shift.
The parallax angle of the nearest star, Proxima
Centauri, is 0.77 arcseconds. 1 arcsecond 1??
1/3600 of a degree
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Parsec
An object with a parallax (p) of 1 arcsecond is
located at the distance of 1 parsec. This is the
distance from which the Earths orbit major axis
is seen at an angle of 1 arcsecond.
1 light-year 3.15 107 s 3 105 km s?1 9.45
1012 km 1 pc 3.26 light?years 3.09 1013 km
206265 AU
1 d (in
parsecs) p (in arcseconds)
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The Inverse Square Law for Light
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Apparent Brightness
Apparent brightness obeys an inverse square law
with distance.
At the distance of Jupiter is 5 A.U., the Sun is
25 times dimmer than on Earth.
Alpha Centauri radiates almost the same amount of
light as the Sun, but it is located 27,000 times
further away from Earth than the Sun. Thus, its
apparent brightness is 70 billion times less than
that of the Sun.
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Luminosity Distance Relation

Luminosity Apparent brightness
-------------------
4 p (distance)2
The units of apparent brightness are Watts per
square meter.
Luminosity is also measured in the units of solar
luminosity (LSun 3.8 1026 Watts).
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Absolute Magnitude
Absolute magnitude (M) is the magnitude that
would the star have at a distance of 10 pc from
the Sun. For example, M? 4.75.
For a star of an apparent magnitude m and a
luminosity L that is located at a distance D (pc)
f L/4?D2 ? log f log L ? 2 log D
m ?2.5 log f C ?2.5 log L 5 log D C M
?2.5 log L 5 log (10) C
Distance modulus
m ? M 5 log (D/10) 5 log D ? 5
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Surface Temperature
Surface temperature determines a star color.
The coolest stars are red, the hottest ones are
blue.
Only the brightest star colors can be recognized
by the naked eye. The color can be determined
better by comparing a stars brightness in
different filters.
Betelgeuse has a temperature of 3,400 K, Sirius
9,400 K, the hottest stars up to 100,000 K.
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Betelgeuse
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Magnitude at Different Wavelengths
Stars as complex objects emit radiation at all
wavelengths from ?-rays to radio, but not
equally. Therefore, response of a detector
depends on the shape of the stellar spectrum.
?(?) ? transmission function of a filter
(detector) B(?) ? stellar spectral energy
distribution ?0 ? effective wavelength of a filter
? ?(?) B(?) d? ?0 ?
-------------------- ?(?) B(?) d?
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Color Index
A color-index is a ratio of fluxes or difference
of magnitudes at 2 different wavelengths.
First color-index was mpg mvis for magnitudes
on 2 kinds of photographic plates with different
sensitivity. mpg were measured on blue-sensitive
plates, mvis ? on yellow-sensitive ones (close to
the human eye sensitivity).
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The UBV Photometric System
In 1953 H.L. Johnson and W.W. Morgan proposed a
3-color system with central wavelengths at 360 nm
(U), 440 nm (B), and 550 nm (V).
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Extension of the UBV System
In 1960s Johnson added longer-wavelength filters
at 700 nm (R), 900 nm (I), 1.25 ?m (J), 2.2 ?m
(K), and 3.5 ?m (L). A few more (e.g., H at 1.62
?m) were added later.
Photometry in this multicolor system allows to
measure the stars spectral energy distribution
(SED) in a large wavelength range and determine
various physical parameters (e.g., stellar
temperature, interstellar extinction).
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How to Use Color-Indices
CI m(?1) ? m(?2)
m(?1) Const ? 2.5 log (F(?1))
CI Const ? 2.5 log F(?1)/ F(?2)
For Vega, all CI 0 by definition.
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Atmospheric Extinction Correction
zenith
z
horizon
m(?) ? m0(?) ?2.5 log F(?)/F0(?) 1.086 ??
z ? zenith angle ?? ? atmospheric optical depth
F(?) F0(?) exp(???) ?? ??(0) sec z
m0(?) m(?) ? 1.086 ??(0) sec z
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Interstellar Extinction
A ? the value of the total interstellar
extinction ?m 1.086 ?? A(?) m ? M 5 log D ?
5 A
B?V (MB?MV) AB ? AV
B MB 5 log D ? 5 AB V MV 5 log D ? 5
AV
?
MB?MV (B?V)0
E(B?V) (B?V) ? (B?V)0
A color-excess, such as E(B?V), represents the
selective interstellar extinction. Observations
of many stars show that AV/E(B?V) R 3.13 for
hot stars ( R is higher for cooler stars).
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Bolometric Correction (BC)
fbol ? f(?) d?
mbol ? 2.5 log fbol const
fv ? f(?) S(?) d?
mv ? 2.5 log fv const
BC 2.5 log (fv/fbol)
Mbol(?) ? Mbol () 2.5 log (L/L?)
Mbol(?) 4.75 ? log (L/L?) 1.9 ? 0.4 Mbol ()
Mbol 4.75 ? 2.5 log (L/L?)
BC? ? 0.07
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The Surface Brightness Method
L 4? R2 ?T4
2.5 log (L/L?) M? ? Mbol
mbol Mbol ? 5 5 log D
mbol V BC
V BC M? ? 2.5 log (L/L?) ? 5 5 log D
L/L? (R/R?)2 (T/T?)4
log (L/L?) 2 log R4 log T ? C?
V BC C??? 10 log T ? 5 log (R/D)
log T 0.1 BC 4.2207 ? 0.5 log ? ? 0.1V