Transcript The Family of Stars - Montgomery College

Chapter 8:
The Family of Stars
We already know how to determine a star’s
• surface temperature
• chemical composition
• surface density
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
d in parsec (pc)
p in arc seconds
1
d = __
p
Trigonometric Parallax:
Star appears slightly shifted from different
positions of Earth on its orbit
The farther away the star is (larger d),
the smaller the parallax angle p.
1 pc = 3.26 LY
The Trigonometric Parallax
Example:
Nearest star, a Centauri, has a parallax of p = 0.76 arc seconds
d = 1/p = 1.3 pc = 4.3 LY
With ground-based telescopes, we can measure
parallaxes p ≥ 0.02 arc sec
=> d ≤ 50 pc
This method does not work for stars
farther away than 50 pc.
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):
L
__
F~ 2
d
Star A
Star B
Earth
Both stars may appear equally bright, although
star A is intrinsically much brighter than star B.
Distance and
Intrinsic Brightness
Example:
Recall:
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
Betelgeuse
App. Magn. mV = 0.41
Rigel
App. Magn. mV = 0.14
Distance and Intrinsic Brightness
Rigel appears 1.28 times brighter
than Betelgeuse,
Betelgeuse
But Rigel is 1.6 times further
away than Betelgeuse
Thus, Rigel is actually
(intrinsically) 1.28*(1.6)2 = 3.3
times brighter than Betelgeuse.
Rigel
Absolute Visual Magnitude
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.
Absolute Visual Magnitude(II)
Back to our example of
Betelgeuse and Rigel:
Betelgeuse
Betelgeuse Rigel
mV
0.41
0.14
MV
-5.5
-6.8
d
152 pc
244 pc
Difference in absolute magnitudes:
6.8 – 5.5 = 1.3
=> Luminosity ratio = (2.512)1.3 = 3.3
Rigel
The Distance Modulus
If we know a star’s absolute magnitude,
we can infer its distance by comparing
absolute and apparent magnitudes:
Distance Modulus
= mV – MV
= -5 + 5 log10(d [pc])
Distance in units of parsec
Equivalent:
d = 10(mV – MV + 5)/5 pc
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:
A
Star B will be
brighter than
star A.
B
Absolute brightness is proportional to radius squared, L ~ R2.
Quantitatively:
L = 4 p R2 s T4
Surface area of the star
Surface flux due to a
blackbody spectrum
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.
Thus, Polaris is 100 times larger than the sun.
This causes its luminosity to be 1002 = 10,000
times more than our sun’s.
Organizing the Family of Stars:
The Hertzsprung-Russell Diagram
We know:
Stars have different temperatures,
different luminosities, and different sizes.
Absolute mag.
or
Luminosity
To bring some order into that zoo of different
types of stars: organize them in a diagram of
Luminosity versus Temperature (or spectral type)
Hertzsprung-Russell Diagram
Spectral type: O
Temperature
B
A
F
G
K
M
The Hertzsprung Russell Diagram
Most stars are
found along the
main sequence
The Hertzsprung-Russell Diagram (II)
Same
temperature,
but much
brighter than
MS stars
 Must be
much larger
 Giant
Stars
Radii of Stars in the
Hertzsprung-Russell Diagram
Rigel
Betelgeuse
Polaris
Sun
100 times smaller than the sun
Ia Bright Supergiants
Ia
Luminosity
Classes
Ib
II
Ib Supergiants
II Bright Giants
III Giants
III
IV
IV Subgiants
V
V Main-Sequence
Stars
Luminosity effects on the width of
spectral lines
Same spectral type,
but different
luminosity
Lower gravity near the surfaces of giants
 smaller pressure
 smaller effect of pressure broadening
 narrower lines
Examples:
• Our Sun: G2 star on the main sequence:
G2V
• Polaris: G2 star with supergiant luminosity:
G2Ib
Binary Stars
More than 50% of all
stars in our Milky Way
are not single stars, but
belong to binaries:
Pairs or multiple
systems of stars which
orbit their common
center of mass.
If we can measure and
understand their orbital
motion, we can
estimate the stellar
masses.
The Center of Mass
center of mass =
balance point of the
system.
Both masses equal
=> center of mass is
in the middle, rA = rB.
The more unequal the
masses are, the more
it shifts toward the
more massive star.
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:
3
a
____
AU
MA + MB =
Py2
(MA and MB in units of solar masses)
Examples:
a) Binary system with period of P = 32 years
and separation of a = 16 AU:
3
16
____
MA + MB =
= 4 solar masses.
2
32
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.
Visual Binaries
The ideal case:
Both stars can be
seen directly, and
their separation and
relative motion can
be followed directly.
Spectroscopic Binaries
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 (II)
The approaching star produces
blueshifted lines; the receding
star produces redshifted lines in
the spectrum.
Doppler shift  Measurement
of radial velocities
 Estimate of separation a
 Estimate of masses
Spectroscopic
Binaries (III)
Typical sequence of spectra from a
spectroscopic binary system
Time
Eclipsing Binaries
Usually, inclination angle
of binary systems is
unknown  uncertainty
in mass estimates.
Special case:
Eclipsing Binaries
Here, we know that
we are looking at the
system edge-on!
Eclipsing
Binaries (II)
Peculiar “double-dip” light curve
Example: VW Cephei
Eclipsing Binaries (III)
Example:
Algol in the
constellation of
Perseus
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 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:
tlife ~ M-2.5
Sun: ~ 10 billion yr.
10 Msun: ~ 30 million yr.
0.1 Msun: ~ 3 trillion yr.
Masses in units
of solar masses
The Mass-Luminosity Relation
More massive
stars are more
luminous.
L ~ M3.5
Surveys of Stars
Ideal situation:
Determine properties
of all stars within a
certain volume.
Problem:
Fainter stars are
hard to observe; we
might be biased
towards the more
luminous stars.
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.