General Properties
Chapter 7:
• Average star
• Spectral type G2
• Only appears so bright because it is so close.
• Absolute visual magnitude = 4.83
(magnitude if it were at a distance of 32.6
light years)
The Sun – Our Star
• 109 times Earth’s diameter
• 333,000 times Earth’s mass
• Consists entirely of gas (av. density = 1.4 g/cm3)
• Central temperature = 15 million K
• Surface temperature = 5800 K
Very Important Warning:
The Photosphere
• Apparent surface layer of the sun
Never look directly
at the sun through
a telescope or
binoculars!!!
• Depth ≈ 500 km
• Temperature ≈ 5800 K
• Highly opaque (H- ions)
• Absorbs and re-emits radiation produced in the solar interior
This can cause permanent eye
damage – even blindness.
Use a projection technique or a special
sun viewing filter.
Energy Transport in the
Photosphere
The solar corona
Granulation
Energy generated in the sun’s center must be transported outward.
In the photosphere, this happens through
Convection:
Cool gas
sinking down
≈ 1000 km
Bubbles of hot
gas rising up
Bubbles last for
≈ 10 – 20 min.
… is the visible consequence of convection
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The Solar Atmosphere
The Chromosphere
• Region of sun’s atmosphere
just above the photosphere.
• Visible, UV, and X-ray lines
from highly ionized gases
Only visible
during solar
eclipses
Apparent surface
of the sun
Heat Flow
• Temperature increases gradually
from ≈ 4500 K to ≈ 10,000 K, then
jumps to ≈ 1 million K
Filaments
Transition region
Chromospheric structures visible
in Hα emission (filtergram)
Temp. incr.
inward
Solar interior
The Layers of the
Solar Atmosphere
The Chromosphere (II)
Spicules: Filaments
of cooler gas from
the photosphere,
rising up into the
chromosphere.
Visible in Hα
emission.
Visible
Sunspot
Regions
Ultraviolet
Photosphere
Corona
Chromosphere
Each one lasting
about 5 – 15 min.
Coronal activity,
seen in visible light
Helioseismology
The solar interior is opaque (i.e. it
absorbs light) out to the photosphere.
⇒ Only way to investigate
solar interior is through
helioseismology
= analysis of vibration
patterns visible on the
solar surface:
Approx. 10 million
wave patterns!
sunspots
Cooler regions of the
photosphere (T ≈ 4240 K).
Only appear dark against the bright sun.
Would still be brighter than the full moon
when placed on the night sky!
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The Solar Cycle
The Maunder Minimum
11-year cycle
After 11 years,
North/South order
of leading/trailing
sunspots is
reversed
Reversal of magnetic polarity
=> Total solar cycle
= 22 years
The Maunder
butterfly diagram
Period from ~ 1645 to 1715, with very few sunspots.
Sunspot cycle starts out with spots at higher latitudes on the sun
Evolve to lower latitudes (towards the equator) throughout the cycle.
Coincides with a period of colder-than-usual winters.
Sunspots and Magnetic Fields
Magnetic Field Lines
Magnetic North Poles
Magnetic
North Pole
Magnetic
South Poles
Magnetic
South Pole
Magnetic
Field Lines
Magnetic field in sunspots is about 1000 times stronger than average.
In sunspots, magnetic field lines emerge out of the photosphere.
Magnetic Fields in Sunspots
Solar Activity
Magnetic fields on the photosphere can be
measured through the Zeeman effect
→ Sunspots are related to magnetic
activity on the photosphere
Observations at ultraviolet and X-ray wavelengths reveal
that sunspots are regions of enhanced activity.
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The Sun’s Magnetic Dynamo
Magnetic Loops
Magnetic field lines
The sun rotates faster at the equator than near the poles.
This differential rotation might be responsible
for magnetic activity of the sun.
Prominences
The Sun’s
Magnetic Cycle
Relatively cool gas
(60,000 – 80,000 oK)
May be seen as dark
filaments against the
bright background of
the photosphere
After 11 years, the magnetic
field pattern becomes so
complex that the field
structure is re-arranged.
→ New magnetic field
structure is similar to the
original one, but reversed!
Looped prominences: gas ejected from the sun’s
photosphere, flowing along magnetic loops
Eruptive
Prominences
(Ultraviolet images)
Solar Aurora
Extreme events (solar
flares) can significantly
influence Earth’s
magnetic field structure
and cause northern lights
(aurora borealis).
Coronal mass ejections
~ 5 minutes
→ New 11-year cycle starts
with reversed magnetic-field
orientation
Sound
waves
produced
by a solar
flare
4
Energy Production
Coronal Holes
Energy generation in the sun
(and all other stars):
X-ray images of
the sun reveal
coronal holes.
These arise at
the foot points of
open field lines
and are the
origin of the
solar wind.
Energy generation in the Sun:
The Proton-Proton Chain
Basic reaction:
4 1H → 4He + energy
4 protons have
0.048*10-27 kg (= 0.7 %)
more mass than 4He.
⇒ Energy gain = ∆m*c2
= 0.43*10-11 J
per reaction.
Sun needs 1038 reactions,
transforming 5 million tons
of mass into energy every
second, to resist its own
gravity.
Need large proton speed (Ù high
temperature) to overcome
Coulomb barrier (electromagnetic
repulsion between protons).
T ≥ 107 K =
10 million K
nuclear fusion
= fusing together 2 or
more lighter nuclei to
produce heavier ones.
Nuclear fusion can
produce energy up to
the production of iron;
Binding energy
due to strong
force = on short
range, strongest
of the 4 known
forces:
electromagnetic,
weak, strong,
gravitational
For elements heavier than
iron, energy is gained by
nuclear fission.
The Solar Neutrino Problem
The solar interior can not be
observed directly because it
is highly opaque to radiation.
But neutrinos can penetrate
huge amounts of material
without being absorbed.
Early solar neutrino experiments
detected a much lower flux of
neutrinos than expected (→ the
“solar neutrino problem”).
Recent results have proven that
neutrinos change (“oscillate”)
between different types
(“flavors”), thus solving the solar
neutrino problem.
Davis solar neutrino
experiment
5
We already know how to determine a star’s
• surface temperature
Chapter 8:
• chemical composition
• surface density
The Family of Stars
In this chapter, we will learn how we can determine its
• distance
• luminosity
• radius
• mass
and how all the different types of stars
make up the big family of stars.
Distances to Stars
The Trigonometric Parallax
d in parsec (pc)
p in arc seconds
Example:
Nearest star, α Centauri, has a parallax of p = 0.76 arc seconds
d = 1/p = 1.3 pc = 4.3 LY
__
1
d= p
With ground-based telescopes, we can measure
parallaxes p ≥ 0.02 arc sec
=> d ≤ 50 pc
Trigonometric Parallax:
Star appears slightly shifted from different
positions of Earth on its orbit
This method does not work for stars
farther away than 50 pc.
1 pc = 3.26 LY
The farther away the star is (larger d),
the smaller the parallax angle p.
Intrinsic Brightness /
Absolute Visual Magnitude
The more distant a
light source is, the
fainter it appears.
The same amount of light
falls onto a smaller area at
distance 1 than at distance 2
=> smaller apparent
brightness.
Area increases as square of distance => apparent
brightness decreases as inverse of distance squared
Intrinsic Brightness /
Absolute Visual Magnitude(II)
The flux received from the light is proportional to its
intrinsic brightness or luminosity (L) and inversely
proportional to the square of the distance (d):
F~
Star A
L
__
d2
Star B
Earth
Both stars may appear equally bright, although
star A is intrinsically much brighter than star B.
1
Distance and
Intrinsic Brightness
Distance and Intrinsic Brightness
Example:
Recall:
Betelgeuse
Magn. Diff.
Intensity Ratio
1
2.512
2
2.512*2.512 = (2.512)2
= 6.31
…
…
5
(2.512)5 = 100
For a magnitude difference of 0.41
– 0.14 = 0.27, we find an intensity
ratio of (2.512)0.27 = 1.28
App. Magn. mV = 0.41
Rigel appears 1.28 times brighter
than Betelgeuse,
Betelgeuse
But Rigel is 1.6 times further
away than Betelgeuse
Rigel
App. Magn. mV = 0.14
Absolute Visual Magnitude
Thus, Rigel is actually
(intrinsically) 1.28*(1.6)2 = 3.3
times brighter than Betelgeuse.
Absolute Visual Magnitude(II)
Back to our example of
Betelgeuse and Rigel:
To characterize a star’s intrinsic
brightness, define absolute visual
magnitude (MV):
Apparent visual magnitude
that a star would have if it
were at a distance of 10 pc.
Rigel
Betelgeuse
Betelgeuse Rigel
mV
0.41
0.14
MV
-5.5
-6.8
d
152 pc
244 pc
Rigel
Difference in absolute magnitudes:
6.8 – 5.5 = 1.3
=> Luminosity ratio = (2.512)1.3 = 3.3
The Distance Modulus
If we know a star’s absolute magnitude,
we can infer its distance by comparing
absolute and apparent magnitudes:
The Size (Radius) of a Star
We already know: flux increases with surface
temperature (~ T4); hotter stars are brighter.
But brightness also increases with size:
Distance Modulus
= mV – MV
= -5 + 5 log10(d [pc])
Distance in units of parsec
Equivalent:
A
Star B will be
brighter than
star A.
B
Absolute brightness is proportional to radius squared, L ~ R2.
Quantitatively:
L = 4 π R2 σ T 4
d = 10(mV – MV + 5)/5 pc
Surface area of the star
Surface flux due to a
blackbody spectrum
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Organizing the Family of Stars:
The Hertzsprung-Russell Diagram
Example:
Polaris has just about the same spectral type
(and thus surface temperature) as our sun, but
it is 10,000 times brighter than our sun.
Stars have different temperatures,
different luminosities, and different sizes.
Thus, Polaris is 100 times larger than the sun.
To bring some order into that zoo of different
types of stars: organize them in a diagram of
Luminosity versus Temperature (or spectral type)
Absolute mag.
or
Luminosity
This causes its luminosity to be 1002 = 10,000
times more than our sun’s.
We know:
Hertzsprung-Russell Diagram
Temperature
Spectral type: O
The Hertzsprung Russell Diagram
B
A
F
G
K
The Hertzsprung-Russell Diagram (II)
St
lif ars
e
tim spe
e nd
on m
th os
e to
M
ai f t h
n
Sa
Se eir
m
fai e t
qu a c t
nte em
en ive
r → p.,
ce
Dw but
.
arf
s
Most stars are
found along the
main sequence
Radii of Stars in the
Hertzsprung-Russell Diagram
M
Ia Bright Supergiants
Ia
Same
temperature,
but much
brighter than
MS stars
→ Must be
much larger
→ Giant
Stars
Luminosity
Classes
Ib
Rigel
Betelgeuse
Polaris
Sun
10,
00
sun 0 tim
’s r es t
adi he
us
10
0
su time
n’s
s
rad the
ius
As
la r
ge
II
Ib Supergiants
II Bright Giants
III Giants
III
IV
IV Subgiants
V
V Main-Sequence
Stars
as
t
he
su
n
100 times smaller than the sun
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Luminosity effects on the width of
spectral lines
Examples:
• Our Sun: G2 star on the main sequence:
G2V
Same spectral type,
but different
luminosity
Lower gravity near the surfaces of giants
⇒ smaller pressure
• Polaris: G2 star with supergiant luminosity:
G2Ib
⇒ smaller effect of pressure broadening
⇒ narrower lines
The Center of Mass
Binary Stars
center of mass =
balance point of the
system.
More than 50% of all
stars in our Milky Way
are not single stars, but
belong to binaries:
Both masses equal
=> center of mass is
in the middle, rA = rB.
Pairs or multiple
systems of stars which
orbit their common
center of mass.
The more unequal the
masses are, the more
it shifts toward the
more massive star.
If we can measure and
understand their orbital
motion, we can
estimate the stellar
masses.
Estimating Stellar Masses
Recall Kepler’s 3rd Law:
Py2 = aAU3
Valid for the solar system: star with
1 solar mass in the center.
We find almost the same law for
binary stars with masses MA and
MB different from 1 solar mass:
aAU3
____
MA + MB =
Py2
Examples:
a) Binary system with period of P = 32 years
and separation of a = 16 AU:
163
____
MA + MB = 322 = 4 solar masses.
b) Any binary system with a combination of
period P and separation a that obeys Kepler’s
3. Law must have a total mass of 1 solar mass.
(MA and MB in units of solar masses)
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Visual Binaries
The ideal case:
Both stars can be
seen directly, and
their separation and
relative motion can
be followed directly.
Spectroscopic
Binaries (II)
Usually, binary separation a
can not be measured directly
because the stars are too
close to each other.
A limit on the separation
and thus the masses can
be inferred in the most
common case:
Spectroscopic
Binaries:
Spectroscopic
Binaries (III)
Typical sequence of spectra from a
spectroscopic binary system
Time
The approaching star produces
blueshifted lines; the receding
star produces redshifted lines in
the spectrum.
Spectroscopic Binaries
Doppler shift → Measurement
of radial velocities
→ Estimate of separation a
→ Estimate of masses
Eclipsing Binaries
Usually, inclination angle
of binary systems is
Eclipsing
Binaries (II)
Peculiar “double-dip” light curve
unknown → uncertainty
in mass estimates.
Special case:
Eclipsing Binaries
Here, we know that
we are looking at the
system edge-on!
Example: VW Cephei
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Eclipsing Binaries (III)
Example:
Algol in the
constellation of
Perseus
Masses of Stars
in the
HertzsprungRussell Diagram
The higher a star’s mass,
the more luminous
(brighter) it is:
L ~ M3.5
High-mass stars have
much shorter lives than
low-mass stars:
Hi
gh
m
as
s
es
M
as
s
Sun: ~ 10 billion yr.
10 Msun: ~ 30 million yr.
0.1 Msun: ~ 3 trillion yr.
s
ss e
ma
tlife ~ M-2.5
Low
From the light curve
of Algol, we can
infer that the
system contains
two stars of very
different surface
temperature,
orbiting in a slightly
inclined plane.
Masses in units
of solar masses
Surveys of Stars
The Mass-Luminosity Relation
Ideal situation:
More massive
stars are more
luminous.
Determine properties
of all stars within a
certain volume.
L ~ M3.5
Fainter stars are
hard to observe; we
might be biased
towards the more
luminous stars.
Problem:
A Census of the Stars
Faint, red dwarfs
(low mass) are
the most
common stars.
Bright, hot, blue
main-sequence
stars (highmass) are very
rare.
Giants and
supergiants
are extremely
rare.
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