| Jarak Bintang:Bacaan:
Schneider & Arny: Unit 52
|
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Untuk mempelajari komposisi, formasi, dan evolusi bintang-bintang, kita harus tentukan jarak terlebih dahulu. Sebagai contoh untuk menentukan luminositas mutlak sebuah bintang (jumlah energi dipancarkan perdetik) anda mengukur kecerlangan semunya (luminositas tampak dari bumi), kemudian dikali dengan kuadrat jaraknya dari Bumi.
Cara yang paling langsung dalam mengukur jarak dengan bintang terdekat adalah metode parallax. Gerak bumi mengitari matahari setiap tahun menghasilkan pergeseran kecil di posisi bintang terdekat jika dibandingkan dengan bintang latar belakang yang jauh. Pergeseran ini selalu kurang dari satu detik busur untuk tiap bintang, yang ukurannya sangat kecil (di mana satu lingkaran 360 derajat, satu derajat 60 menit busur, dan satu menit busur sama dengan 60 detik busur).
D = 1/θ
di sini jarak, D, diukur dalam parsek, di mana satu parsek sama dengan 3.26 tahun cahaya (jarak yang ditempuh cahaya dalam setahun).
Proper Motion:
Although the stars appear fixed in the sky, they are actually moving
through space at very high velocities. Their extremely large
distances makes this motion almost undetectable. This motion is
called proper motion,
and can also be used to judge the distance to stars.
The nearest stars should display the greatest proper motion, and the
star with the greatest proper motion is Barnard's star with a change
of 10 arcsecs per year. Since only a few stars have proper motions
greater than 1 arcsec per year, thus, the shape of the constellations
have remained mostly unchanged since the dawn of man. The diagram
below shows how the Big Dipper will look after 10,000 years.
Notice also that the Sun is moving through the Galaxy, so solar
motion will give the appearance of stars in front of our motion
moving towards an apex (like a car driving through snowflakes).
Those behind us will appear to move towards an antapex.
Stellar Magnitudes:
The measure of the brightness of a star is, for historical and
physiological reasons, called its apparent magnitude.
The human eye detects light in a logarithmic fashion, meaning
that changes occur in powers of 10 rather than in a linear manner. So
ancient astronomers divided stars into six classes or magnitudes
where the brightest are first magnitude, the faintest are sixth
magnitude. Later measurements showed that a change in 5 magnitudes
is equal to a 100 increase in brightness.
Here is a list of common objects in the sky and their magnitudes.
Note that the greater the magnitude (the more positive the number),
the fainter the star. Negative numbers are bright stars.
Object Apparent Mag
-----------------------------
Sun -26.5
Full Moon -12.5
Venus -4.0
Jupiter -3.0
Sirius -1.4
Polaris 2.0
eye limit 6.0
Pluto 15.0
limit for telescopes 25.0
----------------------------
From the table it is clear that the Sun is the brightest object in
the sky, but
it is not the brightest star in the Galaxy. It's apparent brightness
is high since it is relatively nearby.
We also distinguish between apparent magnitude and absolute
magnitude. Apparent magnitude is what we measure in the sky,
absolute magnitude is the real luminosity of the star, corrected for
distance.
The magnitude system works out such that a change in 1 in magnitude corresponds to a change in 2.512 in brightness.
The formula is as follows:
b1/b2 = 2.512(m2-m1)
where b1 and b2 are the brightnesses of two stars (ergs per sec) and m1 and
m2 are the magnitudes of the two stars. If you know the brightness of the stars, and want to
determine their relative magnitudes, this formula is the inverse of the one above:
m2-m1 = -2.5 log(b1/b2)
If the apparent magnitude of a star is m and its absolute magnitude is M (its real brightness), then
the distance to the star, d in parsecs, is given by:
m = M + 5 log (d/10)
Solar Neighborhood:
Stars have different absolute luminosities. So the brightest stars in the sky
are not necessarily the closest stars. Here is a list of the twenty brightest stars in
the sky. And here is a list of the twenty nearest stars.
The nearest stars make-up what is called the solar neighborhood, shown
below. Note that the nearest stars are mostly small dim stars. These
types of stars are hard to see at great distances. The twenty
brightest stars are mostly supergiant stars; which are rare, but very
bright.
Star Map Applet
Stellar Masses:
Since stars are so far away, it is impossible to measure their masses directly.
Instead, we look for binary star
systems and use Newton's law of gravity to measure their masses.
  
Two stars in a binary system are bound by gravity and revolve around a common center of mass. Kepler's 3rd law
of planetary motion can be used to determine the sum of the mass of the
binary stars if the distance between each other and their orbital period is
known.
Kepler's 3rd law states that the square of a planet's or star's orbital period is proportional to its mean distance
from each other such that:
r3 = k P2
where P is the orbital period in years and r is the distance between each other in Astronomical Units (the
distance from the Earth to the Sun). The constant, k, is derived from Newton's law of gravity to be the sum
of the masses of the stars, M1 + M2, in units of solar masses. So the full
equation becomes:
M1 + M2 = r3/P2
When you plot the mass of a star versus its absolute luminosity, one finds a correlation between the two
quantities shown below.
This relationship is called the mass-luminosity relation for stars, and it
indicates that the mass of a star controls the rate of energy production, which
is thermonuclear fusion in the star's core. The rate of energy generation, in
turn, uniquely determines the stars total luminosity. Note that this relation
only applies to stars before they evolve into giant stars (those stars which
burn hydrogen in their core).
Notice that stars range in mass from about 0.08 to 100 times the mass of the
Sun. The lower mass limit is set by the internal pressures and temperatures
needed to start thermonuclear fusion (protostars too low in mass never beginning
fusion and do not become stars). The upper limit is set by the fact that stars
of mass higher than 100 solar masses become unstable and explode. Notice also
that these range of masses corresponds to a luminosity range from 0.0001
to 105 solar luminosities.
This quiz uses information from Schneider & Arny: Unit 52
1) A star has a parallax of 0.2 arcsecs. What is the distance to the star in parsecs? (click for an example)
a) 0.2 parsecs
b) 1.0 parsecs
c) 5.0 parsecs
d) 20.0 parsecs
e) 100.0 parsecs
2) A star is 15 parsecs away, what is its parallax?
a) 0.01 arcsecs
b) 0.07 arcsecs
c) 0.15 arcsecs
d) 0.04 arcmins
e) 0.15 arcmins
3) If an alien civilization lives on a star that is 10 parsecs away, how
long will it take to get a radio signal from them? (hint: you need to
know the number of light-years in a parsec)
a) 1.53 years
b) 3.26 years
c) 32.6 years
d) 65.2 years
e) never
4) The Hubble Space Telescope can measure parallax angles down to 0.01
arcsecs. How many parsecs away can Hubble measure the distance to
stars?
a) 1 parsec
b) 2 parsecs
c) 50 parsecs
d) 100 parsecs
e) 1000 parsecs
5) For the region of space given by Hubble's range in the last problem,
is that a big or small piece of the whole Galaxy? (hint: you need to
look up the size of the Galaxy we live in)
a) big
b) small
c) cannot determine from information given
6) Star A displays a proper motion of 20 arcsecs a year, star B displays
a proper motion of 0.5 arcmins per year. Which star is closer?
a) star A
b) star B
c) they are both the same distance from Earth
7) Star A has an apparent magnitude of 3.5, star B has an apparent magnitude of 12.3. Which is brighter on the sky?
a) star A
b) star B
c) cannot determine from information given
8) Same stars as above, which star is brighter in intrinsic brightness, i.e. their true luminosity?
a) star A
b) star B
c) cannot determine from information given
9) Star A has an absolute magnitude of -10, star B has an absolute magnitude of -5. What is the ratio of their luminosities? (click for an example)
a) 0.25
b) 1.24
c) 25.66
d) 100.02
e) 453.88
10) Venus is around -1 in apparent magnitude in the evening sky. The
faintest stars you can see with your naked eye are 6. How much brighter
is Venus from those faint stars?
a) 0.02 times
b) 2.5 times
c) 631 times
d) 12,556 times
Stellar Color:
| Readings:
Schneider & Arny: Units 55, 57
|
Stars have a range of colors which represent their surface
temperatures due to Wien's law (which states that the peak emission
of light from an object goes as the inverse of temperature). The
color of a star is determined by that part of the visible spectrum
where the peak amount of radiation is emitted.
Blue stars are extremely hot, red stars are relatively cool.
Temperature here is a relative thing; cool means temperatures near
2,000 to 3,000K, about 15 times hotter than your oven. Blue stars
have temperatures near 20,000K. The Sun is an intermediate yellow
star with a surface temperature of 6,000K. The color of a star is
determined by measuring its color index.
It is important to remember temperature and luminosity for a star are
not strictly related. Stefan-Boltzmann's law states that the amount of energy
emitted goes as the temperature to the 4th power; but, this relation
is only strictly true for an object that is a point source (i.e. it
has no size). The temperature of a normal object is proportional to
its surface area (for example, things cool faster if you spread them
out = increase their surface area).
So, it is possible for a star to be very bright (emit alot of energy)
yet, be cool and red. We will see below that this means the star
must be very large to be both bright and cool.
Stellar Spectral Type:
Stars are divided into a series of spectral types based on
the appearance of their absorption spectra. Some stars have a strong
signature of hydrogen (O and B stars), others have weak hydrogen lines,
but strong lines of calcium and magnesium (G and K stars). After years
of cataloging stars, they were divided into 7 basic classes: O, B, A, F,
G, K and M. Note that the spectra classes are also divisions of
temperature such that O stars are hot, M stars are cool.
Between the classes there were 10 subdivisions numbered 0 to 9. For
example, our Sun is a G2 star. Sirius, a hot blue star, is type B3.
Why do some stars have strong lines of hydrogen, others strong lines of
calcium? The answer was not composition (all stars are 95% hydrogen)
but rather surface temperature.
As temperature increases, electrons are kicked up to higher levels
(remember the Bohr model) by collisions with other atoms. Large atoms
have more kinetic energy, and their electrons are excited first,
followed by lower mass atoms.
If the collision is strong enough (high temperatures) then the electron
is knocked off the atom and we say the atom is ionized.
So as we go from low temperatures in stars (couple 1,000K) we see
heavy atoms, like calcium and magnesium, in the stars spectrum. As
the temperature increases, we see lighter atoms, such as hydrogen
(the heavier atoms are all ionized by this point and have no
electrons to produce absorption lines).
As we will see later, hotter stars are also more massive stars (more
energy burned in the core). So the spectral classes of stars is
actually a range of masses, temperatures, sizes and luminosity. For
normal stars (called main sequence stars) the following table gives
their properties:
type Mass Temp Radius Lum (Sun=1)
-------------------------------------------
O 60.0 50,000 15.0 1,400,000
B 18.0 28,000 7.0 20,000
A 3.2 10,000 2.5 80
F 1.7 7,400 1.3 6
G 1.1 6,000 1.1 1.2
K 0.8 4,900 0.9 0.4
M 0.3 3,000 0.4 0.04
-------------------------------------------
So our Sun is a fairly middle-of-the-road G2 star:
A B star is much larger, brighter and hotter. An example is HD93129A
shown below:
Luminosity Classes:
Closer examination of the spectra of stars shows that there are small
changes in the patterns of the atoms that indicate that stars can be
separated by size called luminosity classes.
The strength of a spectra line is determined by what percentage of that
element is ionized. An atom that is ionized has had all its electrons
stripped off and can not absorb photons. At low
densities, collisions between atoms are rare and they are not ionized.
At higher densities, more and more of the atoms of a particular element
become ionized, and the spectral lines become weak.
One way to increase density at the surface of a star is by increasing
surface gravity. The strength of gravity at the surface of a star is
determined by its mass and its radius (remember escape velocity). For
two stars of the same mass, but different sizes, the larger star has a
lower surface gravity = lower density = less ionization = sharper
spectral lines (the opposite is true, higher gravity = higher density = broader spectral
lines, this is called pressure broadening).
This was applied to all stars and it was found that stars divide into
five luminosity classes: I, II, III, IV and V. Stars of type I and II
are called supergiants, being very large (low surface gravity), stars of
type III and IV are called giant stars. Stars of type V are called
dwarfs. The Sun is a G2 V type stars.
So now we have a range of stellar colors and sizes. For example,
Aldebaran is a red supergiant star:
Arcturus is an orange giant star:
HST imaging found that Betelgeuse is one of the largest stars, almost
the size of our whole solar system.
The other extreme was also found, that there exist a class of very small
stars called white and brown dwarfs, with sizes close to the size of the
Earth:
Red and blue supergiant stars, as well as giant stars exist. The
following is a comparison of these types.
Luminosity Function:
Surveying the skies for stars is a very biased method of doing
science since clearly the brightest stars are the easiest to
observe. But are the brightest stars typical of the stellar
population? To determine what a typical star is like we construct a
luminosity function, the number of stars as a function of absolute
magnitude in the form of a histogram.
A luminosity function is constructed by sampling a volume of space and
counting all the stars in that volume. The resulting plot will look
like the diagram below:
Notice that the most common type of star is actually small, low
luminosity stars. Bright stars are quite rare (although they can be
seen from great distances). Since luminosity is correlated with mass,
then this means that high mass stars are rare.
Russell-Vogt Theorem:
Despite the range of stellar luminosities, temperatures and
luminosities, there is one unifying physical parameter. And that is
the mass of the star. Hot, bright stars are typically high in mass.
Faint, cool stars are typically low in mass. This sole dependence
on mass is so strong that it is given a special name, the Russell-Vogt
Theorem.
The Russell-Vogt Theorem states that all the parameters of a star (its
spectral type, luminosity, size, radius and temperature) are determined
primarily by its mass. The emphasis on `primarily' is important
since we will soon see that this only applies during the `normal' or
hydrogen burning phase of a star's life. A star can evolve, and
change its size and temperature. But, for most of the lifetime of a
star, the Russell-Vogt Theorem is correct, mass determines
everything.
This quiz uses information from Schneider & Arny: Units 55, 57
1) How many light-years in a parsec?
a) 1.21
b) 3.26
c) 5.66
d) 10.44
e) 1993.35
2) A star is at a distance of 10 parsecs and has an absolute magnitude of +5. What is its apparent magnitude? (click for an example)
a) 0
b) +5
c) -5
d) +10
e) -10
3) A star has an absolute magnitude of -5 and an apparent magnitude of +5. What is its distance?
a) 1 parsec
b) 10 parsecs
c) 100 parsecs
d) 1000 parsecs
e) 12,888 parsecs
4) Your eye can see down to +6 mags. For a typical star of absolute mag of +4, what is the farthest you can see into space?
a) 1 parsec
b) 25 parsecs
c) 100 parsecs
d) 2040 parsecs
e) 16,443 parsecs
5) The Hubble Space Telescope can see down to +26 mags. How far away is
the typical star for HST, if the typical star has an absolute magnitude
of 1?
a) 1,344 parsecs
b) 25,000 parsecs
c) 251,189 parsecs
d) 500,322 parsecs
e) one million parsecs
6) How many kilometers is an A.U.?
a) 2.12x104 kms
b) 8.59x106 kms
c) 1.49x108 kms
d) 4.25x1011 kms
e) 1.83x1016 kms
7) How many kilograms in a solar mass?
a) 3.57x1012 kgs
b) 7.18x1017 kgs
c) 2.83x1023 kgs
d) 4.35x1028 kgs
e) 1.99x1030 kgs
8) Two stars of equal mass are in orbit around each other with a period
of one year and a distance of 1 A.U. What are their masses? (i.e. their
sum)
a) 0.5 solar masses
b) 1.0 solar masses
c) 2.0 solar masses
d) 10.0 solar masses
e) cannot determine from information given
9) Two stars, A and B, mass 0.5 and 1.5 solar masses. They orbit each other with a period of 2 years. What is their distance?
a) 1 A.U.
b) 2 A.U.'s
c) 4 A.U.'s
d) 8 A.U.'s
e) 64 A.U.'s
10) Same stars as above, they are moved to only 1 A.U. apart, what is their orbital period?
a) 0.5 years
b) 0.7 years
c) 1.0 years
d) 2.0 years
e) 4.5 years
Binary Stars:
| Readings:
Schneider & Arny: Units 56, 57
|
Planet's revolve around stars because of gravity. However, gravity
is not restricted to only act between large and small bodies, stars can
revolve around stars as well. In fact, 85% of the stars in the Milky
Way galaxy are not single stars, like the Sun, but multiple star
systems, binaries or triplets.
If two stars orbit each other at large separations, they evolve
independently and are called a
wide pair. If the two stars are
close enough to transfer matter by tidal forces, then they are called a
close or contact pair.
Binary stars obey Kepler's Laws of Planetary Motion, of which there are
three.
- 1st law (law of elliptic orbits): Each star or planet moves in an
elliptical orbit with the center of mass at one focus.
Ellipses that are highly
flattened are called highly eccentric. Ellipses that are close to a
circle have low eccentricity.
- 2nd law (law of equal areas): a line between one star and the
other (called the radius vector) sweeps out equal areas in equal
times
This law means that objects travel fastest at the low point of their
orbits, and travel slowest at the high point of their orbits.
- 3rd law (law of harmonics): The square of a star or planet's
orbital period is proportional to its mean distance from the center
of mass cubed
It is this last law that allows us to determine the mass of the binary
star system (note only the sum of the two masses, see previous
lecture).
Visual Binaries:
Any two stars seen close to one another is a
double star, the
most famous being Mizar and Alcor in the Big Dipper. Odds are,
though, that a double star is probably a foreground and background
star pair that only looks near each other. With the invention of the
telescope may such pairs were found. Herschel, in 1780, measured the
separation and orientations of over 700 double stars and found that
only about 50 pairs changed orientation over 2 decades of
observation.
One such example is Sirius A and B shown below. Their motion through
the sky is a complex, twisted path which takes decades to map and
plot.
The observations made relative to center of mass of the two stars
shows their respective elliptical orbits.
Eclipsing Binaries:
In the late 1600's, Italian astronomers noticed that some stars
occasionally drop in their brightness up to 1/3 their peak luminosity.
Later measurements showed that these declines were periodic, ranging from
hours to days. It is now recognized that these brightness changes are
due to the eclipsing of one star by another (as they pass in front of
each other).
Eclipsing binaries are studied by monitoring their light curves
(shown below), the changes in brightness with time. When the
smaller, dimmer star passes in front of the brighter star, there is a
deep minimum. When the dimmer star passes behind the bright star
there is a second, less deep, minimum. Notice the transition zone at
the start and end of each eclipse.
Eclipsing binaries are very rare since the orbits of the stars must be
edge-on to our solar system. Notice that an eclipsing binary is the
only direct method to measure the radius of a star, both the primary and
the secondary from the time for the light curve to reach and rise from
minimum.
Eclipsing Binary Applet
Spectrum Binary:
Often a system of binary stars are too close (or too far away) to be
resolved into an optical pair. However, a spectrum of such an object
will display the spectral fingerprints of two different stellar
types (if the stars are different in spectral type).
Of course, the problem with this method is that since faint, cool
stars are more common than brighter stars, the odds are that the
companion is too faint to be detected in a spectrum. Also, just
detecting two spectrum will not determine their masses since relative
velocities are needed.
Spectroscopic Binary:
Another avenue to determine the masses of stars is to measure their
relative velocities via the Doppler shift of their spectral lines. This
is used when the pair can not be resolved as an visual binary, but
motion is seen in the spectra of one star.
Notice that you do not need to see two spectra, only the motion of
one of the stars is needed to deduce the existence of the binary
system (why would one star be moving on its own?). Most binary stars
are too close to separate the components, yet their existence can be
deduced from Doppler shifts.
Typical velocities between binaries are 3 to 5 km/sec, so very high
resolution, Coude spectra must be taken to observe this
phenomenon.
Spectroscopic Binary Applet
Contact Binaries:
When two stars are close in separation it is possible for tidal forces
to come into play. Since stars are not solid bodies, rather made of
gases, then gravity can strip material and transfer it from one star to
the other. Thus we say the binaries are in contact, even if their
surfaces are not touching directly.
How stars exchange material is similar to the way a ball rounds over
and down a hill. The ball must have enough kinetic energy to exceed
the potential energy of the hill. Around two stars there are lines
of equipotential. Imagine two nearby lakes. If the water rises it
takes on the shape of the contours of the land, the equipotential
contours. If the water level rises too high, the lakes merge.
In the same way, there exist lines of equipotential around stars, where
the gravitational pull from one star exceeds that of another. This line
where the forces or energies balance is called the Roche lobe. When
the star's radii exceed the Roche lobe, the gases are free to
transfer from one star to the other. Usually in the form of a tube
or stream.
In some binary stars, such as
Phi
Persei, one of the binary stars evolves and expands (see stellar
evolution lecture). Its surface exceeds the Roche lobe and material
is streamed from one star to the other.
Some contact systems, such as the Algol
system require sophisticated
supercomputer simulations to
understand the mass exchange.
This quiz uses information from Schneider & Arny: Units 56, 57
1) Click on this
figure to answer this question. What is the mass of a star that is 10 times the luminosity of the Sun?
a) 1 solar mass
b) 3 solar mass
c) 9 solar mass
d) 15 solar mass
e) 100 solar mass
2) Same diagram, how bright is a star that is 1/2 the mass of the Sun?
a) 0.1 solar luminosities
b) 0.05 solar luminosities
c) 0.01 solar luminosities
d) 0.001 solar luminosities
e) 0.0001 solar luminosities
3) Are stars of absolute mag +10 more common or less common than stars of absolute mag +5?
a) less common
b) more common
c) the same number
4) For two stars of the same mass, the one with the highest surface gravity has
a) a larger radius
b) a smaller radius
c) cannot determine from information given
5) A star that has ionized helium in its spectrum is
a) very hot
b) very blue
c) large in radius
d) all of the above
e) a and b
6) A visual binary is seen such that in 1980 star A was to the left of
star B. Then, in 1990, star B was to the left of star A. In 2000, they
are back to the positions seen in 1980. What is the orbital period of
the binary?
a) 5 years
b) 10 years
c) 15 years
d) 20 years
e) cannot determine from information given
7) In an eclipsing binary, a small star is orbiting a large star with a
velocity of 50 km/sec. First minimum takes 20 mins, what is the radius
of the primary star?
a) 15,000 km
b) 30,000 km
c) 60,000 km
d) 120,000 km
e) cannot determine from information given
8) Same binary, the time from 1st minimum to maximum light is 2 mins. What is the radius of the secondary star?
a) 1,500 km
b) 3,000 km
c) 6,000 km
d) 30,000 km
e) 60,000 km
9) One star has a radius of 10,000 km and the secondary star has a
radius of 5,000 km. Their orbit is 20,000 km apart. Are they in
contact?
a) yes
b) no
c) cannot determine from information given
10) Star A has a radius of 5,000 km, star B has a radius of 10,000 km. Second minimum is when
a) star A goes behind star B
b) star B goes behind star A
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