January 26, 2009, jeb

Outline

• The Distances to the Stars
  • (see also Extending the Cosmic Distance Scale)
• Stellar Motion
• Stellar Sizes
• Luminosity and Apparent Brightness
• Temperature and Color
• The Classification of Stars
• The Hertzsprung-Russell Diagram
• Extending the Cosmic Distance Scale
• Luminosity Classes
• Stellar Mass
• Stellar Lifetime
• Key Properties of Some Well-Known Stars
• Star Clusters
• Summary of Star Measures

The Distances to the Stars

Stellar Parallax:

1/2 angle through which a star's position shifts as earth orbits the sun.
(actually this only works in determining stellar distances for nearby stars.
Table of nearest stars )

Nearest Stars:
• Alpha Centauri complex (triple-star system)
  • Proxima Centauri at 1.3 pc (4.2 ly)
    • 0.77 arc-secs
• Barnard's Star
  • 1.8 pc (6.0 ly)
    • 0.55 arc-secs

(see also Extending the Cosmic Distance Scale below)

Stellar Motion

• Stars are moving and the motion across the sky, after correction for parallax, is called proper motion
• The largest known proper motion of any star is that of Barnard's star (227 arc-seconds in 22 years)
  
  
• The true space motion is the combination of the transverse (proper) motion and the radial motion, determined from the Doppler shift of the stellar lines.

Stellar Sizes



• Most stars appear as points of light, so their sizes cannot be directly measured
  • However, for a few we measure the size directly: Betelgeuse
  
  
• for most we must use indirect means (temperature and brightness)
  • Consider a "hot star" with surface temperature 30,000 K,
    and a "cool star" with surface temperature 3,000 K
    • identify these stars from the peak of their blackbody curves
  • These stars have names:
    • Type O : 30,000 K
    • Type M : 3,000 K
  • What are their luminosities?
    • We see this from a plot of the "main sequence" of the Hertzsprung-Russell Diagram
    • So the Type O star is 10¹⁰ times as bright as a Type M star
  • How does this compare to our expectations from Blackbody Theory?
    • Stefan-Boltzmann Law says the energy emitted per unit area is proportional to T⁴
      or emission per unit area from type O (30,000K) should exceed type M (3,000K) by:
      10⁴ = 10,000
      but not by 10¹⁰ = 10,000,000,000 that we observe
      we have an extra factor of 1,000,000 = 10⁶
  • So why are type O stars so much brighter?
    • Because type O are also much bigger
    Total brightness is proportional to
    Area x T⁴, not just T⁴
    
    
    or Luminosity ~ (Radius) ² x T⁴
    
    
  • So the extra factor of 1,000,000 = 10⁶
    represents an increased area, or a radius increase of 1,000
    that is, type O has a radius 1000 times the radius of type M
  
  The above equation can be rearranged to derive an equation for the radius: (Radius)² ~ Luminosity / T ⁴
  or
  
  

  Radius ~ SQRT(Luminosity) / T 2

  
  
• look again at the radius trends on the plot of the "main sequence" of the Hertzsprung-Russell Diagram
• Stellar Sizes
• Stars can be classified by their size
  • Giants (10-100 times the Sun's radius)
  • Supergiants (100-1000 times the Sun's radius)
  • Dwarf (comparable to or smaller than the Sun)

Luminosity and Apparent Brightness

Luminosity:

Total rate at which radiative energy is given off by a celestial body.

sometimes referred to as Absolute Brightness




Apparent Brightness

The brightness that the star appears to have to an observer on the Earth

• This depends on how far away the object is by the inverse-square law:
  
• Knowing Brightness and Distance, we can determine Luminosity
• Apparent brightness ~ luminosity / distance ²
• Often expressed relative to the sun's luminosity (L_SUN).



Magnitude:

Measure of Brightness



Apparent magnitude:

Greeks (Hipparchus) established scale called Apparent magnitude:

• Brightest stars visible to unaided eye = Magnitude 1
• Dimmest stars visible to unaided eye = Magnitude 6
  • This is a logrithmic scale

• Measurements show 1st magnitude stars are 100x as bright as 6th magnitude stars.
• So, a Magnitude difference of 1 corresponds to a factor of 2.5 in brightness.
  • or (2.51)⁵ = 100




Absolute Magnitude:

Apparent magnitude a star would have if it were exactly 10 parsecs from the Earth.

Absolute Magnitude = Luminosity, although in different units.


Luminosity and Brightness of the Sun
• Sun's Brightness = 1370 Watts/m²
• Sun's Distance (d) = 1.5 x 10¹¹m
• Therefore, Area of Sphere is 4 (pi) d²
  = 4 (pi) (1.5 x 10¹¹m)²
  = 3 x 10²³ m²
• Luminosity= (1370 Watts/m²)(3 x 10²³ m²)
  = 4 x 10²⁶ Watts
  
  
  
Another example: Sirius:

Brightest Star in the Sky

• Apparent Magnitude = -1.46
• Distance
  • parallax = 0.38 arcsec
  • distance (d) = 1/parallax = 2.5 parsec
• 3.26 l.y. in a parsec
• So, d = 8.6 l.y.
• Luminosity= 23 x L_sun

Temperature and Color

Star's Surface Temperature

Our understanding of Blackbody Radiation explains why star's color is related to surface temperature.

Colors of Betelgeuse and Rigel in Orion are clearly red and blue

B (blue) and V (visual) filters admit different amounts of light for different temperatures

The color index, relating the B and V intensities, is defined in two ways

• ratio of B to V (B/V)
• difference between B and V (B-V)
• relationship between color index and temperature

Star's absorption spectrum is also indicative of the surface temperature.

• Ionization state of atoms depends on temperature
• Energy of light (and therefore absorption) depends on temperature

The Classification of Stars

Star's Absorption Spectrum
Used to classify Stars (according to Spectral Class)




Mnemonic for O B A F G K M :
Oh, Be A Fine Guy (Girl), Kiss Me



Types of Spectra

Hydrogen Lines Strongest in A spectra

Molecular Lines Strongest in M spectra

Neutral Metals Strongest in G,K, and M

Neutral Helium Strongest in B

Ionized Helium Strongest in O


The spectral class matches with surface temperature

Each lettered spectral class is further sub-divided in 10 subdivisions, denoted by 0-9
• so, for example:
  • The Sun is G2,
  • Betelgeuse is M2,
  • and Barnard's star is M5

The Hertzsprung-Russell Diagram

Distribution of "Main Sequence" H-R Diagram is plot of Luminosity vs. Spectral Class
• Some of the well-known stars
• Nearby stars (within 5 pc of Sun)
• 100 Brightest stars in our sky

Different classes of stars populate different regions of the H-R diagram
• On the main sequence
  • Blue giants
  • Blue supergiants
  • Red dwarfs
• Off the main sequence
  • Red giants
  • White dwarfs

In our Solar neighborhood
• 90% of the stars are on the main-sequence
  • this is probably also true elsewhere
• 9% of the stars are white dwarfs
• 1% of the stars are red giants

Extending the Cosmic Distance Scale

Parallax only works for the closest stars



Spectroscopic Parallax

(not really a parallax - just a method for finding distance)

Consider "Main-Sequence" Star:

1. Find Spectral Type
2. Measure Brightness
3. Look at H-R diagram for expected Luminosity
   And use inverse-square law to deduce distance

Luminosity Classes


Only distances to stars on the main-sequence can be correctly determined by the simple procedure outlined above under spectroscopic parallax
since 9 out of 10 stars are on the main-sequence, this is already pretty good

Can we distinguish stars that are off the main-sequence by their spectra?
Yes

Technical Difference in Spectra of Same Spectral Class based on the width of lines

width of lines determine luminosity classes Ia-V


So now we have a third label for a star:
• The Sun is G2V, V being the stellar luminosity class for the main-sequence.
• Betelgeuse is M2Ia

An example of the discrimination between K2-type stars is illustrated in the following table:

Stellar Mass

Stellar Masses and Binary Stars


Most stars are part of a multiple star system


• Visual Binary - See Orbiting Stars - double star Kruger 60
• Spectroscopic Binary - Spectrum reveals binary nature thru Doppler Effect
• Eclipsing Binary - Light curve shape

From these measurements we can learn about masses:

• For visual binary use Kepler's law to deduce masses:
  • M₁ + M₂ ~ a³ /p²
    a = semimajor axis = radius for circular orbit
    p = period of orbit
    • Note: I don't expect you to know this formula, just that there is a relationship between the mass, the period, and the size (semimajor axis) of the orbit
• For other binaries, some information on orbits and masses can be derived
• Most of our knowledge of the masses of stars is based on these binary measurements


Example binary star: Sirius (the brightest star in the sky)
• bright Sirius A and faint companion Sirius B
• orbital period = 50 years
• semi-major axis = 20 AU
• M_A + M_B = 3.2 M_sun
• further study reveals:
  • M_A = 2.1 M_sun
  • M_B = 1.1 M_sun

Stellar properites depend on the mass
• variation of mass along the main sequence

The mass of the star at the time of formation determines its location on the main sequence.

Main sequence stars range in mass from 0.1 to 20 times the mass of the Sun (with a few exceptions)

Most main-sequence stars are low-mass stars, and only a small fraction are much more massive than the Sun

The main-sequence star's radius and luminosity depend on its mass
• mass-radius relation for main sequence stars
• mass-luminosity relation for main sequence stars
  • luminosity ~ mass³ or mass⁴ (approximate)

Stellar Lifetimes

• Luminosity increases as (Mass) ³ for massive main-sequence stars
  and (Mass) ⁴ for more common main-sequence stars
  
  reference to details by Nick Strobel
• Total fuel to burn in star is the mass
• Therefore:
  ............. More massive stars burn up fastest and have shortest lives
  
  
  • 
    
    and since the luminosity increases as the cube of the mass for the most massive stars:
    
    
  • 
    
    (these are only approximate relationships)
    

Key Properties of Some Well-Known Stars





H-R diagram of some well-known stars

Star Clusters

Star clusters are a collection of stars at approximately the same distance from us, so they can be compared without correcting the brightness to absolute brightness (or we don't need to know how far away they are to compare them)



Open (or Galactic) Clusters

a loose association of young stars - metal-rich

Example: The Pleiades:

• we assume these stars were formed around the same time
• O type stars are still on the main-sequence, therefore this cluster is younger than lifetime of O type stars: 25 million years
• heavy elements are abundant in these stars
  • metal-rich
  • created by earlier stars




Globular Clusters

large spherical clusters of stars- metal-poor

Example: Omega Centauri

• again, we assume these stars were formed around the same time
• no O or B type stars
• no main-sequence stars with masses greater than 0.8 solar masses
  • A type stars are passing back through the main-sequence
• few heavy elements are found in these stars
  • metal-poor
  • so stars were created in distant past, when heavy elements were less abundant
• We can conclude that this (and other) globular cluster was formed over 10 billion years ago
  • recall lifetime of the Sun will be 10 billion years, and if it had formed with this cluster is would have been extinguished by now, as seen on the H-R diagram



• Open clusters are of recent (last billion years) origin
• Globular clusters are very old (more than 10 billion years)

Summary of Star Measures