Stellar Astrophysics. This is a summary of the the book "An Modern Introduction to Stellar Astrophysics" by Dale. A. Ostlie and Bradley W. Carroll. This summary contains the most important knowledge and equations from chapters 1, 2, 3, 5, 7, 8, 9, 10, 12, 13, 15 and 16. Included are the laws of mot...
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Summary An Introduction to Modern Astrophysics Lucas Lamping Utrecht University
This is a summary of chapters 1, 2, 3, 5, 7, 8, 9, 10, 12, 13, 15 and 16 of the book “An Introduction
to Modern Astrophysics” (2nd edition) of B.W. Carroll and D.A. Ostlie. I’m a bachelor student at
Utrecht University and made this summary while studying for my stellar astrophysics course.
Part 1: The Tools of Astronomy
1 The Celestial Sphere
Humans have been looking at the sky for thousands of years. Astronomy is the study and observation
of everything outside our atmosphere, especially the behaviour and characteristics of stars. The ancient
Greeks, Aztecs and Chinese all have tried to understand the motion the stars. The Greeks (and many
other civilizations) believed in a geocentric universe; they thought that the Earth is the center of the
universe and everything revolves around the Earth. In this model all the stars could be seen as points
on a (very big) celestial sphere which has the Earth in its center. The line through Earth’s North and
South Pole intersects the celestial sphere in the north celestial pole and south celestial pole. Some
observations like the retrogate motion of Mars couldn’t be explained by the geocentric model even
after adjusting it. It was only in the sixteenth century that Copernicus proposed a heliocentric model
of planetary motion where the Sun instead of Earth is the center of the universe.
Planets like Venus that are closer to the Sun than Earth are called inferior planets while planets like
Saturn that are farther from the Sun than Earth are called superior planets. When an inferior planet
aligns itself between the Earth and the Sun we speak of inferior conjunction. A superior planet is in
opposition when it’s distance to the Earth is minimal. The time between two oppositions or (inferior)
conjunctions is known as the synodic period S. The time it takes to make an orbit with respect to the
background stars is the sidereal period P . When P⊕ ≈ 365.25 d denotes the time it takes the Earth
to make one round around the Sun, then S1 = P1 − P1⊕ .
With the altitude-azimuth coordinate system we can describe the position of a star on the celestial
sphere. Your Horizon is defined as the (great) circle that you get when you intersect the the celestial
sphere with the plane through the Earth’s center that is parallell to plane tangent to the Earth at
your position. Note that the horizon is time and place dependent. The azimuth A is the angle
measured eastward along the horizon from north to the great circle used for the measurment for the
altitude h which is the angle from the horizon to the point of interest. Another way to indicate the
position of stars is the equatorial coordinate system. Here a coordinate is given by its right ascension
and declination. The right ascension α is the eastward angle along the celestial equator between the
vernal equinox (the intersection point of the celestial equator and the orbit of the Sun when the Sun
moves northward) and the hour circle (circle on the celestial equator through the north and south
celestial poles and the star of interest). The right ascension is usually measerd in hours instead of
degrees (1 h = 15◦ ). The declination is the angle δ along the hour circle from the celestial equator to
the star.
Stars and other objects in space don’t stay in one place but move constantly (relative to the Earth).
We can decompose the velocity v = vr + vθ of a star in two components: the radial velocity vr in
which the star moves away from the Earth (its direction is parallell to the line through the Earth and
the star) and the tangential velocity vθ that is perpendicular to the radial velocity.
2 Celestial Mechanics
The astronomer Johannes Kepler is famous for his three laws of planetary motion. To formulate these
laws he uses ellipses. An ellipse is the collection of all points such that the sum of the distances from
, that point to the two focal points equals some constant 2a also known as the major axis of the ellipse.
Formulas that describe an ellipse in cartesian respectively polar coordinates are
x2 y 2 a(1 − e2 )
+ 2 = 1, r= .
a2 b 1 + e cos(θ)
Here e denotes the eccentricity of the ellipse. We have 0 ≤ e < 1, and the ellipse is a circle for e = 0.
The constant b is known as the semiminor axis. We have the mathematical relation b2 = a2 (1 − e2 ).
The famous Isaac Newton stated the following laws. Newton’s First Law says if no force acts on
an object, then it will move in a straight line with constant speed (v = 0 for an object at rest).
d
Newton’s Second Law states that F = ma (if m is constant), or equivalently F = dt p. Here p = mv
is the impulse. According to Newton’s Third Law any action requires an equal but opposite reaction
(F12 = −F21 ). Newton also found the Law of Universal Gravitation F = G mM
r2
where G ≈ 6.67 · 10−11
2
Nm kg . −2
Kepler’s First Law: A planet orbits the Sun in an ellipse, with the Sun at one focus of the ellipse.
The formula of this ellipse is
L2
r= 2
µ GM (1 + e cos(θ))
mM
where µ = m+M is the reduced mass and L is the length of the angular momentum vector L = r × p =
µ(r × v).
Kepler’s Second Law: For a given time interval the area closed in by the planets orbit, the line from
the Sun to the planets initial position and the line form the Sun to the planets final position remains
the same. Mathematically, dA L
dt = 2µ .
Kepler’s Third Law (Harmonic Law): The orbital period of a Planet P satisfies P 2 = a3 if P is in
years and a in astronomical units (1 AU ≈ 1.50 · 1011 m, the average distance between the Earth and
the Sun). More general we have
P2 a3
= .
4π 2 G(m + M )
The formula for kinetic energy is K = 12 mv 2 and (potential) gravitational energy is given by U =
−G mM 1
r . The viral theorem says that the avergy total energy is ⟨E⟩ = 2 ⟨U ⟩ or equivalently that
−2⟨K⟩ = ⟨U ⟩.
3 The Continuous Spectrum of Light
1 AU
With the parallax angle p we can find the distance to a star d = tan(p) ≈ p1 AU. If p is in arcseconds,
then d = 1/p pc which leads to the definition of the parsec as 1 pc ≈ 3.26 ly.
The luminosity of a star L = − dE dt tells you how much energy a star losses so how bright it is. The
actual brightness of a star decreases when the star is farther away. Assuming no light is adsorbed
L
during the journey, the radiant flux is F = 4πr 2 . The apparent magnitude m measures how bright we
see a star looking from Earth. The difference between the apparant magnitudes of two stars is given
by m1 − m2 = −2.5 log10 (F1 /F2 ). How better we can see a star, how smaller it apparent magnitude
is. For the Sun we have m ≈ −26.83. When m > 6.5 it isn’t possible to see a star with the naked eye.
Another way to indicate the brightness of a star is its absolute magnitude. The absolute magnitude M
of a star is its apparent magnitude if the star would be 10 pc of Earth. So M = M⊙ − 2.5 log10 (L/L⊙ ).
The difference m − M between apparent and absolute magnitude is known as the distance modulus
and satisfies d = 10(m−M +5)/5 . Here d is the distance to the star in parsecs.
The speed of light (in vacuo) is c ≈ 3 · 108 m/s. A light wave with wavelength λ and frequency ν
satisfies c = λν. Stars and planets emit light but barely reflect it; they are known as blackbodies. The
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