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1.1 What is a star?
A star can be defined as a body that satisfies two conditions: (a) it is bound by self-gravity;
(b) it radiates energy supplied by an internal source. From the first condition it follows that the shape
of such a body must be a spherical, for gravity is a spherical symmetric force field. Or, it might be
spheroidal, if axisymmetric forces are also present. The source of radiation is usually nuclear energy
released by fusion reactions that take place in stellar interiors, and sometimes gravitational potential
energy released in contraction or collapse.
A direct implication of the definition is that stars evolve: as they release energy produced
internally, changes necessarily occur in their structure or composition, or both. This is precisely the
meaning of evolution. From the above definition we may also infer that the death of a star can occur
in two ways: violation of the first condition – self gravity – meaning breakup of the star and
scattering of its material into interstellar space, or violation of the second condition – internally
supplied radiation energy – that could result from exhaustion of the nuclear fuel. In the latter case,
the star fades slowly away, while it gradually cools off, radiating the energy accumulated during
earlier phases of evolution.
1.2 What can we learn from observations?
Astrophysics (the physics of stars) does not lend itself to experimental study, as do the other
fields of physical science. We cannot devise and conduct experiments in order to test and validate
theories or hypotheses. Validation of a theory is achieved by accumulating observational evidence
that supports it and its predictions or inferences. The information we can gather from an individual
star is quite restricted.
1.2.1 Magnitudes
The brightness of a star, as viewed from the Earth, is usually expressed as a magnitude. The
faintest stars visible to the unaided eye are assigned magnitude +6, while a very bright star has
magnitude about 0. Measurement of the energy received per second shows that a difference of 5
magnitudes corresponds to a ratio of 100 in the energy received per second. A step of one magnitude
100  2.512 . In general, the ratio of the energies F1and F2 received per
then represent a ratio
5
second
sources
from
two
of
magnitude
m1 and
log( F1 / F2 )  (m2  m1 ) / 5 log 100 
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m2
is
F1
 (100) ( m2 m1 ) / 5
F2

(m 2 - m1 ) = 2.5log (F1 / F2 ) .
(1.1)
The last equation defines apparent magnitude; note that m2 > m1 when F2 < F1, that is, brighter
objects have numerically smaller magnitude. Also note that when the brightnesses are observed at
the Earth, physically they are fluxes. Apparent magnitude is the astronomically peculiar way of
taking about fluxes.
The magnitude based on the observed energy fluxes, are called apparent magnitudes. In order to
compare intrinsic luminosities of stars, we define a system of absolute magnitudes. The absolute
magnitude of a star is that magnitude that it would appear to have as viewed from a standard
distance. This distance is chosen to be 10 pc. From this definition, you can see that if a star is
actually at a distance of 10 pc, the absolute and apparent magnitudes will be the same. By
convention, absolute magnitude is capitalized (M) and apparent magnitude is written lowercase (m).
The inverse-square law links the flux (f) of a star at a distance d to the luminosity, L, it would have it
if were at a distance D = 10 pc:
(1.2)
L/f = (d/D)2 = (d/10)2
If M corresponds to L and m corresponding to f, then Eq. (1.1) becomes
m – M = 2.5 log (L/f)
= 2.5 log (d/10)2 = 5 log (d/10)
Expanding this expression, we have the useful alternative forms
m – M = 5 log d – 5 log 10
= 5 log d – 5
(1.3)
M = m + 5 – 5 log d
(1.4)
M = m + 5 +5 log p"
(1.5)
Here d is in parsecs and p" is the parallax angle in arc seconds. The quantity (m - M ) is called the
distance modulus.
Problem (1): The apparent magnitude of the variable star RR Lyrae ranges from 7.1 to 7.8 (a
magnitude amplitude of 0.7). Find the relative increase in brightness from minimum to maximum.
Problem (2): A binary star consists of two stars A and B, with a brightness ratio of 2; however, we
see them unresolved as a point of magnitude +5.0. Find the magnitude of each star.
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Problem (3): Calculate the absolute magnitude of the Sun and the Sun’ distance modulus
( m = -26.81 and dSun-Earth = 1 AU).
- The color index
Band
U
B
V
R
I
/nm
365
445
551
658
806
W/nm
66
94
88
138
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Usually, magnitudes are measured in a restricted wavelength range. For example, the filters used
to obtain the magnitudes. Detectors of electromagnetic radiation are sensitive only over given
wavelength bands. Because the flux of star light varies with wavelength, the magnitude of a star
depends on the wavelength interval at which we observe. Originally, photographic plates were
sensitive only to blue light, and the term photographic magnitude (mpg) still refers to magnitude
centered at 420 nm (in the blue region of the spectrum). Similarly, because the human eye is most
sensitive to green and yellow, visual magnitude mv pertains to the wavelength region around 540
nm. Today we can measure magnitudes in the infrared and ultraviolet, by using filters in conjunction
with the wide spectral sensitivity of photoelectric photometers. A commonly used wide-band
magnitude system is the UBV system: a combination of ultraviolet (U), blue (B) and visual (V)
magnitudes. These three bands are centered at 365, 445, and 551 nm; each wavelength band is
roughly 100 nm wide.
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The apparent and absolute magnitudes measured over all wavelengths of light emitted by a star,
are known as bolometric magnitudes and are denoted by mbol and Mbol, respectively. A quantitative
measure of the color of a star is given by its color index (CI), which is defined as the difference
between magnitudes at two different effective wavelengths (e.g. (B-V) or (U-B)). A star’s (U-B)
color index is the difference between its ultraviolet and blue magnitudes, and a star’s (B-V) color
index is the difference between its blue and visual magnitudes:
U-B = MU-MB
(1.6)
B-V = MB-MV.
(1.7)
and
Stellar magnitudes decrease with increasing brightness; consequently, a star with a smaller (B-V)
color index is bluer than a star with a larger value of B-V. Because a color index is the difference
between two magnitudes, Eq. (1.3) shows that it is independent of the star’s distance. The difference
between a star’s bolometric magnitude and its visual magnitude is its bolometric correction (BC):
BC = mbol – V = Mbol-MV.
(1.8)
In practice, we use the bolometric correction BC, which is the difference between the bolometric and
visual magnitudes, to determine a star’s bolometric magnitude. Bolometric corrections are inferred
from ground-base observations by using theoretical stellar models; these corrections have been
checked and improved with ultraviolet data from orbiting satellites.
Problem (4): Use the data given in the appendix to answer the following questions.
(a) Calculate the absolute and apparent visual magnitudes, MV and V, for the Sun.
(b) Determine the magnitudes MB, B, MU, and U for the Sun.
1.2.2 Flux and luminosity
To obtain luminosity (total power output) of stars, one must add up flux measurements
overall wavelengths and measure
the distance of stars.
The
primary
characteristic
measured
is
that
the
can
be
apparent
brightness, which is the amount
of radiation from the star falling
per unit time on unit area of a collector (usually, a telescope). This radiation flux, which we shall
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denote F, is not however, an intrinsic property of the observed star, for it depends on the distance of
the star from the observer.
The stellar property is the luminosity, L, defined as the amount of energy radiated per unit time – the
power of the stellar engine. Since L is also the amount of energy crossing, per unit time, a spherical
surface area at the distance d of the observer from the star, the measured apparent brightness is
F
L
,
4d 2
(1.9)
and L may inferred from F if d is known. Best distance measurement is the trigonometric parallax.
The luminosity of a star is usually expressed relative to that of the Sun, the solar luminosity
L๏=3.85x1026 Js-1. Stellar luminosities range between less than 10-5 L๏ and over 105 L๏.
Problem (5): Find the radiant flux received by the Earth above its atmosphere from the Sun which is
called solar constant (L๏ =3.826x1033erg s-1, dEarth-Sun=1 AU). Calculate the same value for another
observer 10 pc away from the Sun.
- Trigonometric parallax
Relatively nearby stars appear to move back and forth with respect to much more distant stars as the
Earth travels in its orbit.
Since the Earth’s orbit is (nearly) circular, this works no matter what the direction to the star.
Since the size of the Earth’s orbit is known very accurately from radar measurements, measurement
of the parallax p determines the distance to the star.
Mean distance between centers of Earth and Sun: 1AU = 1.496 × 1013 cm. Thus
d = 1AU/tan p = 1AU/ p (for small angle tan p = p(radian))
Parallax is usually expressed in fractions of an arcsecond (radians = 180o = 10800′ = 648000"); the
distance to an object with a parallax of 1", called a parsec, is 3.087 ×1018 cm = 3.2616 ly.
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d(pc) = 1/p(arcsec)
(1.10)
Good parallax measurements: about 0.02" from the ground, about 0.002" from the Hipparcos satellite.
Problem (6): The parallax angle for Sirius is 0.377". (a) Find the distance to Sirius in units of (i)
parsecs, (ii) light years, (iii) astronomical units; (iv) centimeter. (b) Determine the distance modulus
for Sirius.
1.2.3 Masses and radii of stars
The mass of a star can be measured only by its gravitational effect. Under certain conditions,
the mass of star that is member of a binary system can calculate based on spectral line shifts. The
radii of a number of stars have been found
directly from measurement of their angular
radii by means of an interferometer. Very
seldom, in eclipsing binary systems, may
the radius of a star be directly derived; it
can, however, be estimated from the
independently derived luminosity (when
possible) and effective temperature. Stellar
masses and radii are measured in solar units. The solar mass, M๏=1.99x1030 kg, and the solar radius,
R๏=6.96x108 m. The mass range is quite narrow, between 0.1M๏ and a few tens M๏; stellar radii
very typically between less than 0.01R๏ to more than 1000R๏. Much more compact stars exist,
though, with radii of a few tens of kilometers. Angular diameter of sun at distance of 10pc:
  2R /10pc
4x10-9 radians 10-3arcsec .
Compare with Hubble resolution of ~0.05" which is very difficult to measure R directly. Radii of
~600 stars measured with techniques such as interferometry and eclipsing binaries.
1.2.4 Surface temperature
The surface temperature of a star may be obtained from the general shape of its spectrum, the
continuum, which is very similar to that of a blackbody. The effective temperature of a star Teff is
thus defined as the temperature of a blackbody that would emit the same radiation flux. If R is the
stellar radius, the surface flux is L / 4R 2 , and hence
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F  Teff4 
L
,
4R 2
(1.11)
where σ is the Stefan-Boltzmann constant. Thus
L  4R 2Teff4 .
(1.12)
The surface temperatures of stars range between a few thousands to a few hundred thousand degrees
Kelvin (K).
The wavelength of maximum radiation λmax shifting, according to Wein’s law,
max T  constant ,
(1.13)
from infrared to soft X-rays. The effective temperature of the Sun is 5780 K. We should bear in
mind, however, that conclusions regarding internal temperatures cannot be drawn from surface
temperatures without a theory.
Color temperature: Temperature inferred from color, usually by fitting a Planck function to the
continuous spectrum of a star at two wavelengths.
Effective temperature: The temperature a body would have if it were a blackbody of the same size
radiating the same luminosity.
Problem (7): (a) Estimate the effective temperature of the Sun’s surface (L๏ =3.826x1033erg s-1, R๏ =
6.96x1010 cm). (b) From the effective temperature find the radiate flux at the solar surface. (c) Using
Wein’s law to derive the wavelength of maximum continuous radiation of the Sun. At which band of
electromagnetic radiation the Sun emits most its radiation.
Problem (8): Calculate the wavelength of maximum radiation for the two stars: (a) Betegeuse (Orion
constellation), with surface temperature of 3400 K, and (b) Rigel (Orion constellation), with surface
temperature of 10100 K. At which band of electromagnetic radiation the two stars emit the maximum
radiation, and what is the expected color for each of them.
1.2.5 Surface gravity
The acceleration due to gravity near the surface of a star is one of the factors determining the
atmospheric structure. It also influences the finer details of the spectrum and, in fact, can be
estimated from spectroscopic analysis. If the mass M and radius R of a star are known, the surface
gravity g is calculated from its definition,
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g
GM
,
R2
(1.14)
where G is the constant of gravitation. For most stars, however, M and R cannot be determined
directly.
Problem (9): Calculate the gravitational acceleration (a) at the solar surface, (b) at the Earth’s
surface. Compare between the two values.
1.2.6 Chemical composition and age of a star
There are two other fundamental properties of stars that we can measure – age (t) and chemical
composition.
The
chemical
composition,
too,
can
be
inferred from the spectrum.
Each chemical element has its
characteristic set of spectral
lines. The elements that make
up the photosphere of a star,
which emits the observed
radiation,
may
thus
be
identified in the stellar spectrum. But since the photosphere is very thin, the deduced composition is
not representative of the bulk, opaque interior of the star. Most of the
chemical elements were found to be present in the solar spectrum.
Composition parameterised with (X, Y, Z  mass fraction of H, He and
all other elements). For example X๏ = 0.747; Y๏ = 0.236; Z๏= 0.017.
Note that, Z is often referred to as metallicity. We would like to
studies stars of same age and chemical composition – to keep these
parameters constant and determine how models reproduce the other
observables. Stellar clusters very useful laboratories – all stars at same
distance, same t, and initial Z.
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In clusters, t and Z must be same for all stars. Hence
differences must be due to M. Stellar evolution assumes that the
differences in cluster stars are due only (or mainly) to initial M.
Cluster HR (or colour-magnitude) diagrams are quite similar – age
determines overall appearance.
1.3 The Hertzsprung-Russell digram.
H-R diagram is a plot of luminosity or absolute magnitude against spectral type. Most stars in
the vicinity of the Sun lie on the main sequence, a continuous
band that runs from the hot,
luminous stars at the upper left
to the cooler, fainter stars at
lower
right.
Main-sequence
stars are also called dwarfs to
distinguish them from the sub
giants, giants, and super giants
occupying the area above the
main sequence in the diagram.
The sub dwarfs and white
dwarfs fall below the main
sequence. It should noted that not all white dwarfs are actually white in color. The terms dwarfs and
giant represent the radii of the stars, as well as their luminosities. Equation (1.5) shows that, for a
given effective temperature (or, approximately, spectral type), a large luminosity requires a large
radius. Radii are largest at the upper right in the diagram, smallest at lower left. A Roman numeral
may be affixed to the spectral type to indicate the position of a star in the H-R diagram. Mainsequence stars are denoted by the number V, sub giants by IV, giants by III, bright giants by II, and
super giants by Ib and Ia, the last being brightest of all. Stars below main sequence are not usually
assigned luminosity classes.
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1.3.1 Temperature-luminosity relation
The most important empirical relation between stellar properties is provided by the main-sequence in
the HR diagram, where there exists a direct correlation between stellar luminosity and effective
temperature of the form
L  Teff
(1.15)
where on average  ~ 0.4
1.3.2
Mass-luminosity relation
The second most important empirical relation between stellar properties is provided by eclipsing
binary stars, for which a direct measurement of the stellar mass
can often be obtained. Plotting L against M for MS stars in a
log-log plot, we find a straight line which means that L follows
a relation
L M
(1.16)
where the average  = 3.8. Our theory of stellar structure must
reproduce both of these results.
1.4 Some definitions
In considering stellar structure, we will meet a number of quantities. The most important are
 Stellar mass (M)
often given in solar units (M/Mo)
 Stellar radius (R)
often given in solar units (R/Ro)
 Surface gravity (g)
g =GM/R2
 Effective temperature (Teff)
usually given in K.
 Stellar luminosity (L)
often given in solar units (L/L๏), L= 4R2Teff
 Composition (X,Y,Z)
mass-fractions of H (X), He (Y) and other elements (Z).
X+Y+Z=1
 Age (t)
usually given in years
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Example:
The Sun
M = 1 M๏ = 1.99 1030 kg
R = 1 R๏ = 6.96 108 m
g = 2.74 102 m s-2
Teff = 5780 K
L =1 L๏ = 3.861026 W
t ~ 4.6 109 years
X = 0.71
Y = 0.265
Z = 0.025
Problem (10): Consider a model star consisting of a spherical blackbody with surface temperature
of 28,000 K and a radius of 5.16x1011 cm. Let this model star be located at a distance of 180 pc from
Earth. Determine the following for the star: (a) Luminosity, (b) Absolute bolometric magnitude, (c)
apparent bolometric magnitude, (d) Distance modulus, (e) Radiant flux at the star’s surface, (f)
Radiant flux at Earth’s surface (compare this with the solar constant), (g) Peak wavelength λmax.
This is a model of the star Dschubba, the center star in the head of the constellation Scorpius.
1.5 Summary
•
Four fundamental observables used to parameterise stars and compare with models M, R, L, Te
•
M and R can be measured directly in small numbers of stars (will cover more of this later)
•
Age and chemical composition also dictate the position of stars in the HR diagram
•
Stellar clusters very useful laboratories – all stars at same distance, same t, and initial Z
•
We will develop models to attempt to reproduce the M, R, L, Te relationships and understand HR
diagrams
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