Download The Sun and Stars The Sun is a typical star with a mass of about 2

The Sun and Stars
The Sun is a typical star with a mass of about 2×1030 kilograms
or about 3 × 105 earth masses and a radius R ≈ 7 × 105 km or
about 109 earth radii. It is a huge ball of gas held together by
its own selfgravity.
• The spectrum of a star
The Sun and other stars emits a typical stellar continuous
spectrum with dark absorption lines (Fraunhoffer lines.) The
continuous spectrum follows closely the laws of ideal blackbody
radiation, or thermal radiation. The predominat color in the
continuous spectrum of the sun is yellow with a wavelength
λmax ≈ 500 nm. According to Wien’s law
λmax
2.9 × 10−3
=
T
this gives a temperature of about 5,800 K for the photosphere.
Other stars may be cooler (reddish) or hotter (bluish.) The
absorption lines are produced by the gases in the outer atmospheres
of the star.
• Luminosity of a star
The total energy output of a star is called its luminosity, L. The
luminosity of a star with absolute temperature T and radius R
obeys the law of a blackbody radiator
L = σT 4 × 4πR2 ,
where
σ = 5.67 × 10−8
Watt
.
m2 K
σ is called the Stefan - Boltzmann constant. The luminosity
increases rapidly with temperature; increasing the temperature
by a factor of 2 increases the luminosity by a factor of 16.
Clearly a luminous star is hot and/or large, while a faint one
is cool and/or small. For the Sun L = 3.8 × 1026 Watts.
• Apparent brightness of a star
The apparent brightness of a star is measured by the intensity
I of the light we receive from it. I is defined as the energy
received by a detector of area 1m2 . Since the energy emitted
by a star gets spread over a larger sphere as we get farther
away from the star, we see that the intensity at a distance d
from a star of luminosity L is
I=
L
.
4πd2
I is measured in Watt/m2 . So, a star that is 10 times farther
away appears 100 time less bright. The brightness is sometimes
expressed not in Watt/m2 but in magnitudes. The magnitude
of Sirius (the brightest star in our sky) is -1.46, of Canopus
-0.72, of Vega 0.04, of Deneb 1.26,. . . more or less according
to an ancient scale developed by Hipparchus. Each step in
magnitude amounts to a factor of about 2.5 in brightness.
The faintest stars visible to the naked eye are of about 6th
magnitude.
• The light year and the parsec
The Astronomical Unit (AU) is the distance from the Earth to
the Sun. It is 1 AU = 150 × 106 km. Stars are much farther
away than 1 AU Their distances are often given in light-years
(lty). One light year is the distance that light travels in one
year. The speed of light is c = 3×105 km/s. One year has about
3.2 × 107 seconds, so 1 lty = 3 × 105 km/s × 3.2 × 105 s = 9.6 ×
1012 km. 1 AU is only 8.3 light-minutes. Since the apparent
brightness or magnitude of a star depends on its distance from
us, we define absolute brightness (or absolute magnitude) as
the apparent brightness it would have if it were viewed from a
distance of 10 parsecs. A parsec is approximately 3.26 lightyears and it is the distance from which the average radius of the
earth’s orbit around the sun (1 AU) would be seen to subtend
an angle of 1 arcsecond. The parallax p of a star is the angle
that the AU would subtend as viewed from the star.
distant stars
1 AU = 150 million km
Earth
ap
Sun
1 AU
orbit of
Earth
p
nt
pare
posi
tion
1
Star
appa
rent
posi
tion
2
As the earth goes around the sun in a year a “nearby” star
shifts its position against the background of more distant stars
by an angle p to each side of its average position. The angle p,
is always very small so it is measured in arcseconds (arcs). If
we can measure the angle p, we can deduce that the star is at
a distance d given by
1 parsec · arcs
d=
, so a star with p = 1 arcs has d = parsec.
p
The nearest star, Proxima Centauri, is 1.3 parsecs (4.2 lty)
away
• Nuclear Fusion
The source of a star’s energy is the process of nuclear fusion
which takes small nuclei as a fuel and joins them to form larger
nuclei whose mass is less than the mass of the fuel. The
mass difference ∆m is transformed into energy according to
Einstein’s formula E = ∆mc2 . In a normal (main sequence)
star the fuel is hydrogen that is converted into helium. The
net nuclear reaction to produce one helium-4 nucleus is
4 11 H + 20−1 e →42 He + ∆mc2 .
For the fusion reaction to occur very high temperatures are
needed, on the order of 107 K, which are achieved in the core
of the star. The sun’s luminosity of L = 3.8 × 1026 Watts
is produced by transforming 4.5 × 109 kilograms of mass into
energy each second due to the fusion of 600 million metric tons
of hydrogen into helium each second. As a star gets older
it begins to run short of hydrogen in its core and begins to
fuse helium into carbon and so on with increasingly heavier
elements. The life of a star, from youth, to maturity, to old
age is the struggle between gravity that tends to contract the
star and the gas pressure that opposes the contraction. The
pressure is maintained by the fusion energy releases at the core
of the star. As the star begins to run out of hydrogen at the
core it begins to evolve.
The Hertzsprung-Russell Diagram
The spectrum of a star yields a lot of information about the
star. When examined closely, the discrete absorption lines in
the spectrum of a star allow us to classify the stars into spectral
classes A, B, C, .... Here class A stars have the strongest lines in
the visible part (Balmer series) of the hydrogen (H) spectrum,
stars with somewhat weaker H lines are in class B, and so on.
The order O, B, A F, G, K, M (remembered as ”Oh be a fine girl
kiss me”) is arranged from hottest (bluest), (O about 30,000
K) to coolest (reddest) (M about 3,000 K). The HertzsprungRussell diagram is a plot of individual stars where the vertical
axis is the absolute luminosity and the horizontal axis is the
temperature.
It reveals that stars that are ”mature” lie on a diagonal line
called the Main Sequence. These are stars like the Sun, that
are burning hydrogen in the core. As they start to run out
of hydrogen in the core, gravity wins and the core contracts.
Then they start to burn helium to produce carbon at the core.
Hydrogen is still available outside the core so hydrogen burning
begins in a shell. This causes the star to expand to become a
red giant (bright, cool, and large) as it evolves toward the upper
right in the H-R diagram . This will happen to the Sun in some
5 billion years, when it will swallow the Earth. Then they start
to burn oxygen at the core, and helium in a shell, and hydrogen
in another shell.
An onion like pattern of layers forms with heavier and heavier
elements fusing and the star may begin to expand and contract
becoming a variable star, specially a Cepheid variable. These
variable stars have a period-luminosity relation that makes
them useful for measuring far away distances,
Death of Stars
Eventually, the core becomes mostly iron. Beyond iron fusion
does not produce energy and the star enters its final stages and
collapses under its gravity.
• White Dwarfs
If the star is not to massive, Mstar < 1.3Msun the star collapses
to a white dwarf, where the iron core is supported by electron
gas pressure. The density is very large, a solar mass compressed
to the size of the earth. A teaspoon of white dwarf material
has the mass of some 6 tons. The collapse of a white dwarf
produces an explosion, a nova, that ejects gas that goes to form
a planetary nebula, so called because it appears as a fuzzy ball
in small telescope.
• Neutron Stars
If 1.3Msun < Mstar the collapse is a supernova. This is one
of the most violent events known in the universe. The crab
nebula below is the remnant of a supernova explosion observed
by the chinese in the year 1054 AD.
If Mstar < 4 or 5Msun the final stage after the supernova explosion
is a neutron star. Every electron is packed into the nucleus and
the star is a ball of neutrons with a few kilometers diameter
and a tremendous density It is a solar mass compressed to a
few kilometers diameter. One cc would have a mass of some
6 × 1011 kg, or about 6 × 108 tons.
Neutron stars spin very rapidly because of conservation of angular
momentum and have a fantastically strong magnetic field.
Their radiation can only escape in a beam along the magnetic
axis. As the star spins it acts like the rotating beam of a
lighthouse and the star appears as a pulsating star, called a
pulsar. If the beam does not pass by the Earth, then we do not
receive the pulses. At the center of the carb nebula there is a
neutron star pulsating so that it seems to blink on and off as
the light beam sweeps around, as shown in the figure below.
• Black Holes
If Mstar > 5Msun the final stage may be a black hole. If
a mass M is so concentrated that its size is less than the
Schwarzschild radius Rs = 2GM/c2 , where G is Newton’s
gravitational constant and c is the sped of light, a black hole
is formed. This is a solution of Einstein’s equations of general
relativity Rs is the radius of the event horizon. Anything that
gets closer than the event horizon will never escape back out of
the event horizon, and will eventually be crushed at the center
singularity.
Astronomers have found at the core of many galaxies, including
ours, a rapidly spinning accretion disk with radiation coming
out perpendicular to the disk, as shown below.
Calculations show that there is an enormous mass at the center,
millions of solar masses, in a fairly small area. Is it a black hole?
Strange things happen to space and time as one approaches the
event horizon. Imagine a mother ship in orbit far away from
the black hole. A volunteer is sent in a spaceship to approach
the black hole. As the volunteer approaches the event horizon
his time slows down. As he speaks by radio to the mother
ship his speech sslllooooowwwwsssss dddooooowwwwwnnnnnn
more and more, although the volunteer does not particularly
notice this. When he nears the event horizon he might say
IIII ammmmm nearrrrrrrrrrrr..... And we would never hear
the end of it. He might survive into the black hole, but to
us he would seem to be slowing down more and more as he
approaches the event horizon, and it would take an infinite time
by the mother ship clock for him to reach the event horizon.
However the volunteer enters the blackhole in a finite time by
his measure, but we would never receive any message from him.
THIS IS WEIRD. Actually, he would probably be spaghettized
by gravitational forces as he nears the event horizon.