Chapter 17 Stars

1
Chapter 17 Stars

• Properties of Stars
• Classifying Stars
• Hertzsprung-Russel (H-R) Diagram

2
Properties of Stars

• Mass The single most important property that
  determines other properties of the star.
• Luminosity The total amount of energy (light)
  that a star emits into space.
• Temperature surface temperature, closely
  related to the luminosity and color of the star.
• Spectral type closely related to the surface
  temperature
• Size together with temperature determine the
  luminosity

3
What can we measure directly?

• The Easy Ones
• Apparent brightness a well-calibrated detector.
• Temperature spectroscopy
• Spectral type spectroscopy
• The Hard ones
• Distance stellar parallax, but the stars are so
  farrrrr away
• Size The stars are so far away. Their small
  angular size makes it really difficult to be
  measured directly.
• Mass Newtons version of Keplers Third Law
• Need to find the right targets

4
The Apparent Brightness

• Apparent brightness
• The brightness of the a star as it appears to our
  eyes (or detectors).
• It depends on both the luminosity AND distance
  between the star and the Earth.
• The apparent brightness of a star is related to
  its luminosity and distance by the formula

• The total energy in this cone is fixed
• At a larger distance from the star, the same
  amount of energy is spread into a larger area.
  Thus, the apparent brightness of a star is lower
  if we are further away from it.

5
The Magnitude System

• Apparent magnitude describes the relative
  brightness of objects as they appears in sky.
• A difference of 5 magnitudes is equivalent to a
  factor of 100 difference in apparent brightness.
• ? 1st magnitude star is 100 times brighter than
  a 6th magnitude star.
• A difference of one magnitude is a factor of 2.51
  difference in brightness.
• The larger the magnitude, the fainter the object
• Objects with negative magnitude appear brighter
  than objects with positive apparent magnitude.
• Apparent magnitude mv of selected objects
• The brightest star in the in night time sky,
  Sirius, is mv -1.4
• The Sun mv -27
• The full Moon is -13
• Maximum brightness of
• Venus mv -4.7
• Mars mv -2.9
• Jupiter mv -2.8
• Large Magellantic Cloud mv 0.9
• Andromeda galaxy mv 4.3
• Faintest star visible to human eyes mv 6

6
The Absolute Magnitude

• A stars absolute magnitude Mv is the apparent
  magnitude it would have if it were at a distance
  of 10 parsecs (32.6 light-years) from Earth.
• The Suns absolute magnitude is Mv 4.8
• Sirius Mv 1.4
• Betelgeuse Mv -5.1
• Apparent magnitude tells us nothing about the
  luminosity of the objects, but it tell us how
  difficult it is to see the objects in the sky.
• Absolute magnitude, on the other hand, is
  directly related to the luminosity of the object.
  But it does not tell us how bright they appear in
  the sky.

Astronomical Distance
7
Measuring the Temperature of Stars

• Everything with a temperature emit thermal
  radiation. We can measure the temperature of the
  stars or any object by studying the shape of
  their overall spectra.
• Black Body
• An idealized perfect light absorber that absorbs
  all the photons that strikes it (no reflection).
  It re-emits the absorbed energy through thermal
  radiation, with a spectrum characterized by the
  blackbody spectrum.

• The shape of the blackbody spectrum is always the
  same, independent of its temperature.
• The peak position (in wavelength) of the
  blackbody spectrum depends only on the
  temperature, independent of the blackbodys
  composition, or size, etc.

8
Spectral Type of Stars

• Spectral type is closely related to temperature

9
Spectral Type and Temperature

• The spectral features of the stars are closely
  related to the surface temperature of the star
  because the formation of ionized atoms, the
  excitation state of the atoms, and the existence
  of molecules in the stellar atmosphere strongly
  depends on the temperature
• High temperature
• ? Ionized atoms
• Medium temperature
• ? Neutral atoms
• Low temperature
• ? Molecules

10
Determination of Distance

• Stellar Parallax
• Knowledge of the distance to the stars is crucial
  for our determination of the luminosity of stars
• Current technology allows us to determine the
  distance accurately to within a few hundred
  light-years.
• Hipparcos mission (European Space Agency)
  measured the stellar parallax of roughly 100,000
  stars with precision of a few milli-arcseconds.
  So, it can measure distance of star up to 1,000
  light-years away

Simulation of Stellar Parallax
11
Astronomical Distance Units

• Light-year
• The distance light travels (in vacuum) in one
  year.
• one light-year is 10 trillion (1013) km
• Parsec parallax arcsecond
• One parsec the distance to an object with a
  parallax angle of 1 arcsecond.
• One parsec equals to 3.26 light-year.
• kiloparsecs 1,000 parsecs.
• megaparsecs 1,000,000 parsec.

Absolute Magnitude
12
Determination of Stellar Mass

• Mass is the single most important property of a
  star. But it is also difficult to measure
• The most dependable method we have for measuring
  the mass of distant stars is Newtons version of
  Keplers Third Law of orbital motion
• Recall that
• So, if we can find
• two stars (binary star system) orbiting each
  other, and
• if we can measure their
• rotational period p, and
• semi-axis a of the orbit,
• then we can determine their masses.

13
Binary Star Systems

• Binary star systems are formed by two stars that
  are gravitationally bounded, and they orbit each
  other.
• About 50 of the stars are in binary star system.
  There are three categories of binary star
  systems
• Visual Binary a pair of stars that we can see
  distinctly (with a telescope) as the stars orbit
  each other.
• Eclipsing Binary is a pair of stars that orbit
  in the plane of our line of sight. The stars are
  not resolved, but we can see the effects of the
  stars blocking each other in their combined
  light-curve.
• Spectroscopic Binary in some binary system, we
  cannot see the two stars, nor can we see their
  light curve changes, but we can see the motion of
  the stars from Doppler effect measurement of the
  spectra.

14
Binary Star Systems
Two stars appearing close to each other in the
sky do not necessarily means that they are a
binary system.
15
Visual Binary Sirius
Sirius (in constellation Canis Major) is the
brightest star in the night-time sky (magnitude
-1.4). It is a visual binary system. Sirius A
(the larger of the two) is a main sequence star
with spectral type A0, and Sirius B is a white
dwarf.
Hubble Space Telescope image of Sirius
Sirius A B time sequence
White Dwarf
16
Eclipsing Binary

• About 50 of the stars are in binary star
  system. There are three categories of binary star
  systems
• Eclipsing Binary is a pair of stars that orbit
  in the plane of our line of sight, (measuring the
  time curve)

Animations source http//en.wikipedia.org/wiki/S
pectroscopic_binary
17
Algol Eclipsing Binary

• Algol (the demon star) is in the constellation
  of Perseus.
• Algol A main sequence star, more massive.
• Algol B subgiant, less massive.

18
Spectroscopic Binary

• Sometimes only the spectrum from one star is
  seen, the other star is too dim.
• Sometimes two sets of spectra can be seen at the
  same time
• Sometimes more than two sets of spectra can be
  seen
• Mizar is a visual binary system in the
  constellation of Big Dipper.
• Each star in the visual binary system is also a
  spectroscopic binary!

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Eclipsing Binary and Stellar Mass Measurements

• Among the three types of binary star systems,
  the eclipsing binary system is most important for
  the determination of stellar mass, because
• Determination of the stellar mass requires
  knowledge of the orbital period and distance (in
  real distance unit, not in angular separation).
• Orbital period is easy to measure, but distance
  between the stars is difficult to determine.
• For visual binary, we need to know the distance
  from Earth to the stars before we can determine
  the separation between the stars in the binary
  system.
• For spectroscopic binary, we can calculate the
  separation between the stars if we know their
  orbital speed. However, we can only determine the
  line-of-sight speed of the binary system from
  Doppler measurement. If the orbits are tilted
  with respect to our line-of-sight, then we under
  estimate the orbital speed.
• If an eclipsing binary is also a spectroscopic
  binary, then we know its true orbital speed, and
  can determine the separation between the two
  stars. Then, the masses of the stars can be
  determined!

20
Luminosity

• To directly measure the luminosity of a star
  (lets say, the Sun), we will need to surround
  the Sun completely with detectors, which is
  impossible.
• We can infer the luminosity of the Sun if we know
  
• the distance to the star, and
• the stars apparent brightness
• Further more, we need to assume that
• the star emits energy uniformly in all direction
• Then we can calculate its luminosity by the
  formula

d
The total area of the sphere with a radius of r
is 4?d2
21
Luminosity of Selected Stars
Star Distance ly Spectral Type Luminosity L/Lsun
Proxima Centauri 4.2 M5.5 0.0006
Bernards Star 6.0 M4 0.005
Gliese 725 A 11.4 M3 0.02
? Centauri B 4.4 K0 0.53
Sun 0.000016 G2 1.0
? Centauri A 4.4 G2 1.6
Sirius A 8.6 A1 26.0
Vega 25 A0 60
Achernar 144 B5 3,600
Betelgeuse 423 M2 38,000
Deneb 2500 A2 170,000
22
Luminosity and Distance Chicken and Egg
Most of the time, we need measurement of distance
to calculate the luminosity. Howver, if we can
determine the luminosity of an object with other
methods (independent of distance measurement,
such as the luminosity of supernovae), then we
can derive the distance to the object from
measurement of their apparent brightness.
23
Direct Measurement of the Size of the Stars

• Except for the Sun, all the stars in the sky
  are very far away, and their angular sizes (the
  size of the star as it appears to observers on
  Earth, not the physical size) are all very small.
  Although the theoretical resolving power of
  modern large telescopes (such as the Keck
  telescope with 10-meter aperture) is about 0.01
  arcseconds in the visible wavelength, it is
  difficult to realize the full resolution of the
  large telescopes due to atmospheric seeing
  effects.
• Interferometry have directly measure the angular
  size of stars. Direct measurement by
  interferometry can achieve about 0.01 arcseconds
  angular resolution.
• The angular size of Betelgeuse was first observed
  using interferometry in 19210.051 arcseconds.
• R Doradus (in constellation Dorado in the
  southern hemisphere) is the star with the largest
  observed angular size 0.057 arcseconds.
• 0.057 arcseconds is equal to 0.000016 degrees!
• If we know the angular size and the distance of
  a star, we can derive its physical size
• Size of star angular size radian ? distance

24
Betelgeuse and R Doradus

• The physical size of Betelgeuse (a red
  supergiant) is roughly 500 times the size of the
  Sun, or 4.6 AU (radius of 2.3 AU, or 345 million
  km).
• The size of R Doradus (a red giant) is 370 times
  the size of the Sun, or 3.4 AU (radius of 1.7
  AU).
• If R Doradus or Betelgeuse are placed at the
  center of our solar system, then their surface
  would extends beyond the orbit of Mars (1.5 AU,
  or 225 million km).

Image of hot spots on Betelgeuse from
http//www.mrao.cam.ac.uk/telescopes/coast/betel.
html using interferometric technique.
Giants and Supergiants
25
Indirect Determination of the Size of Stars

• Since the stars are so far away, we can only
  directly measure the angular size of just about
  10 stars by interferometric technique so far.
  However,
• if we know the luminosity (from apparent
  brightness and distance measurements) and the
  temperature of the stars, then we can calculate
  their physical size
• Assuming that stars are blackbody
• The energy output of a unit surface area on the
  surface of the star is determined by its
  temperature (Stefan-Boltzman Law)
• The total energy output (luminosity) therefore
  depends on the temperature and its total surface
  area, which is related to its size.
• where r is the radius of the star.
• We can then calculate the size of the star by

26
Clues to Relationships Between the Properties of
Stars

• General trends of the stars
• Most of the very brightest stars are reddish in
  color.
• If we ignore those relatively few bright red
  stars, theres a general trend to the
  luminosities and colors among all the rest of the
  stars
• The brighter ones are white with a little bit of
  blue tint,
• the more modest ones are similar to our Sun in
  color with a yellowish white tint, and
• the dimmest ones are barely visible specks of
  red.

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Hertzsprung-Russell Diagram

• Since there appears to be a strong correlation
  between luminosity and color (temperature), we
  put all the stars on a Luminosity Temperature
  plot, and this is what it looks like
• Properties of Stars shown in the H-R Diagram
• Luminosity (log scale).
• Temperature and spectral type
• Size
• Mass of the main sequence
• Lifetime

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Hertzsprung-Russell Diagram

• Notice that
• Temperature scale decreases from left to right.
• The scale of luminosity is in power of 10 (log
  scale).
• Mass increases from lower right to upper left
• Size increases from lower left to upper right.

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Classification of Stars in H-R Diagram

• The Main Sequence stars
• healthy stars, fusing hydrogen in the core.
• High-mass, high-luminosity, high-temperature, and
  short-lived stars on the upper-left-hand corner
• Low-mass, low-luminosity, low-temperature, and
  long-lived stars on the lower-right-hand corner
• The Supergiants,
• The Giants,
• Supergiants and giants are dying stars, fusing
  helium and heavier elements.
• The White Dwarfs.
• dead stars, exposed core of dead main-sequence
  stars.

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Classification of Stars

• Full classification of stars includes both
  spectral type and luminosity class
• Spectral type OBAFGKM (hottest to coolest)
• Luminosity Class in descending order
• I Supergiants
• II Bright giants
• III Giants
• IV Subgiants
• V Main-sequence stars
• The full classification of a star includes both a
  spectral type and a luminosity class Each
  spectral class is subdivided on a 0-9 scale, with
  0 being the hottest, 9 being coolest.
• The Sun is a G2 V
• Proxima Centauri is M5 V
• Betelgeuse is M2 I
• Sirius A A1 V
• Sirius B DA2 V