Lecture 19: Some Star Basics

Don't Panic

Sections

1. Star Shape and Composition
2. Luminosity, Flux, Distance, Photometry
3. Stellar Parallax and Distance
4. Luminosity, Flux, and the Inverse-Square Law
5. The Distance Ladder

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Star Shape and Composition

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Before going into the detailed nature or physics of stars, some star basics must be introduced.
1. First, we should say that stars are spheres of hot gas.

   For ordinary stars (i.e., main sequence stars which we discuss in IAWL Lecture 20: The Nature of Stars) of order 1 M_Sun (i.e., one solar mass) or larger, the interior is a gas of totally ionized atoms (i.e., nuclei with no bound electrons), free electrons, and photons.

2. Second, ordinary stars (i.e., main sequence stars) have a composition that approximates that of the Sun's photosphere (which is often called its surface). The composition of the photosphere of Sun, is, in fact, close to the average cosmic composition.

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   Three quarters partial solar eclipse.

   There seems to be a sunspot in the upper part near the Moon's limb.

   Credit: Bill Livingston, NSO/AURA/NSF.

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   The solar surface composition by mass is:

   
            H, hydrogen  70.7 +/- 1.8 %,
   
            He, helium   27.4 +/- 2.1 %, and
   
            metals       1.89 +/- 0.17 % .
   
            Metals 
            in astro-jargon are everything which is not H or He.
   
            The deep interior (i.e., the core)
            of the Sun and other stars is richer in He because of ongoing 
            nuclear fusion
            which is discussed in 
   IAWL Lecture 22:  The Main Sequence Life of Stars. 
   
            The H and He abundances are approximately accurate throughout the 
   observable universe,
            except in those minor components:
            planets, asteroids, dust, etc.
            
            The abundances of metals 
            vary wildly from about 4 % down to 0.1 % or even much lower, but
            never 0
            (HI-414).
   
            The ratios of the metals among themselves often 
            vary LESS wildly. 
   
            The leading metals in decreasing order of solar surface abundance
            by number are oxygen (O),  carbon (C), neon (Ne),
            nitrogen (N), magnesium (Mg), silicon (Si), 
            iron (Fe), and sulfur (S)
            (Cox-28--29).
             
   

3. Third, we need to know certain basic parameters (i.e., controlling quantities) of stars.

   We will consider these in the following sections.

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Luminosity, Flux, Distance, Photometry

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Basic star parameters allow us to model stars and from the star models we know things about stars that we CANNOT observe directly.

Key parameters are a star's luminosity (short for total luminosity) and its luminosities in various wavelength bands.

Luminosity is a star's power output in electromagnetic radiation: i.e., what its ``wattage'' is.

Question: What is the luminosity of the Sun?
1. 60 W.
2. 100 W.
3. 3.845*10**26 W.



Answer 3 is right (Cox-340).

As far away as the Sun is, it is still brighter than a 100 W light bulb seen at ordinary room DISTANCES.

Thus, you ought to know that the biggest number is the only answer that could be right.

Of course, I could have provided no right answer.

Question: Brightness to the eye or any other light measuring device increases with light flux (i.e., power per unit area in the light beam) impinging on the eye or device.

The brightness or flux of a source:

1. decreases with DISTANCE from the source.
2. does not depend on DISTANCE from the source.
3. increases with DISTANCE from the source.



Answer 1 is right.

Below we will go into how flux decreases with DISTANCE.

A star's flux and DISTANCE are NOT an intrinsic star properties

A certain kind of star could have any flux or DISTANCE from the Earth.

But to know the INTRINSIC property luminosity, one must know flux and DISTANCE as it turns out.

We'll show how in the section Luminosity, Flux, and the Inverse-Square Law below.

Measuring flux is in principal easy. One just uses standard measuring devices.

One CANNOT measure flux over all wavelengths because no one device will do that and the Earth's atmosphere is opaque in many wavelength bands as we know from studying the solar spectrum in IAWL Lecture 7: Spectra.

Interstellar dust in space also causes extinction (i.e., opaqueness) in some wavelength bands.

We will NOT worry much about these problems in this course, but they are actually major difficulties that can be overcome to some degree.

One measures flux in particular wavelength bands and corrects for the problems as best one can.

One typically uses filters on telescopes to select particular broad wavelength bands which typically have an acceptance wavelength range of tens of nanometers.

Measurement with broad wavelength bands is called photometry.

[Measurement in narrow wavelength bands is spectroscopy which was covered in IAWL Lecture 7: Spectra.]

In the visual (400--700 nm), the most standard bands are called B (for blue light), V (for visual light, but green/yellow light would be more precise), and R (for red light).

The electromagnetic spectrum and visible light.

Modeling typically must be used to get from a star's photometry to its total flux.

So after a bit of story, total flux is in principal easy to measure.

But we still need DISTANCE.

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Stellar Parallax and Distance

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The most straightforward way to measure DISTANCE to an astro-body is to use parallax which we briefly mentioned in IAWL Lecture 2: The Sky.

This method is just that of a terrestrial surveyor.

Parallax is the shift in angular position of an object as one moves.

Parallax can also mean the angular shift itself.

To use parallax to measure distance we need a little angle lore and a little trigonometry.

First, recall that the Greek THETA is the customary physics symbol for angle.

The Greek Alphabet: alpha, beta, gamma, ...

Second, recall

      There are 360 degrees in a circle.

      1 degree = 60 arcminutes  = 3600 arcseconds
 
      1 arcminute = 60 arcseconds

Third, trigonometry---the only bit in this course.

Trigonometry and parsecs.

The formula

                        r(AU)
     d(parsecs) = ____________________

                    theta(arcseconds) 

         is a small angle formula that is valid for the specified units.

     1 parsec = 3.0856776*10**16 m

              = 3.2615638 lyr = approximately 3.26 lyr = approximately 3 lyr

              = 206265 AU

                  (Cox-12) .

Now ``d'' is the distance to be measured.

And ``r'' is the baseline.

Stars are very distant and so their parallaxes will be very small for any solar system baseline.

No parallax for small displacments: i.e., small baselines.

Question: In order to measure parallax for a star most easily, one wants to use:
1. largest baseline feasible.
2. smallest baseline feasible.
3. a zero baseline.



Answer 1 is right.

The largest baseline feasible today is twice the astronomical unit: one makes measurements half a year apart.

Conventionally, though the fiducial baseline is 1 AU and the parallaxes one measures with this baseline are conventionally called stellar parallaxes.

The stellar parallax formula is

     d(parsecs) = 1/theta(arcseconds) , where the

       parallax angles are always so small that we never need to
       worry about using a small angle approximation. 

Question: The stellar parallax formula is a/an:
1. linear relation with theta.
2. inverse-linear relation with theta.
3. inverse-square relation with theta.



Answer 2 is right.

Recall how an inverse-linear relation behaves when plotted.

Some simple function behaviors.

The smaller theta, the larger the distance.

Astro-bodies at very large distances will have immeasurably small parallaxes.

Stellar parallax for nearby stars is measured against the background of very remote stars which show NEGLIGIBLE stellar parallax.

Stellar parallax measurements.

It is NOT quite as simple in practice as the figure shows.

Even the closest stars are so far away that they have sub-arcsecond parallaxes: it required the degree of astronomical accuracy first achieved in the 19th century to measure such small angles.

Historically, the lack of observable stellar parallax was an argument against the moving Earth cosmologies since Aristotle (384--322 BCE)????, and thus an argument against Copernican solar system as Johannes Kepler (1571--1630), for example, well understood.

With stars pasted on a real celestial sphere centered on the Earth, there would be no stellar parallax.

A cartoon of the Aristotelian cosmos.

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Friedrich Wilhelm Bessel (1784--1846).

No its not Beethoven: all early 19th century guys looked like that.

In 1838, Bessel became the first to measure stellar parallax.

Bessel obtained 0.314 arcseconds for star 61 Cygni A (No-415,419).

The modern value for 61 Cygni A stellar parallax is 0.286 arcseconds (FK-A-6).

Thus,

     d(61 Cygni A) = 1/(0.286 arcseconds) = 3.50 pc  .

  

Stellar parallax had been sought since Greco-Roman Antiquity since it was understood that stellar parallax would be a disproof of Aristotelian cosmology and would strong evidence for a moving Earth.

Incidentally, Bessel is also famous for his elucidation of what we now call Bessel functions: they are the darndest things.

Credit: 19th century artist; modern credit: ?; download site St. Andrews Bessel biography

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We can look at the stellar parallaxes and other quantities for some of the nearest stars from a Table of Nearest Stars.

We do NOT need to remember the quantities, but we should contemplate them while we look at them.

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To digress for a moment on distances between stars, we note that inside galaxies, the distances to nearest-neighbor stars are typically of order 1 parsec.

     Recall

     1 parsec = 3.0856776*10**16 m

              = 3.2615638 lyr = approximately 3.26 lyr = approximately 3 lyr

              = 206265 AU  
      
                  (Cox-12),

     Radius Sun = 6.95508*10**8 m = 4.6491837*10**-03 AU 

                  (Cox-12),

     Radius Large star = of order 1 AU .

     The actual record holders have radii of about 14 AU:  they
       are red supergiants:  one of
       these is also in constellation Cygnus:  KY Cygni
       (MacRobert, A. 2005, Sky & Telescope, April, 54).

     

Question: Compared to the interstellar distances, stars are:
1. huge.
2. of the same size scale.
3. pinpricks.



Answer 3 is right.

Because stars are so small compared to interstellar distances, we cannot usually CANNOT even nearby large stars.

The finite size they have to the unaided eye and in most images is an artifact of the observing technique and atmosphere.

With special techniques some very large stars can be resolved barely from the GROUND. Those special techniques are still limited to bright objects.

From space above the fluctuating atmosphere---which makes stars twinkle---one can resolve a few very close, large stars with the HST.

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Betelgeuse imagined by the HST.

Betelgeuse is an M1 Iab red supergiant star. It is 131 pc from Earth.

It is the eastern shoulder of Orion: i.e., the left shoulder on the image.

Orion is, of course, a giant hunter of Greek mythology: he pursued the Pleiades and was slain by Artemis (Ba-855).

The lines joining the stars are NOT present on the sky, of course.

Orion is one of the three constellations anyone can recognize: the other two are the Big Dipper (officially an asterism in Ursa Major) and Cassiopeia (the big W): bother are in the northern sky and are all-year constellations.

Credit: NASA/HST

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Question: Direct star-star collisions in a matter-striking-matter sense are:
1. frequent.
2. extremely rare.



Answer 2 is right.

In fact, because stars are so minute compared to interstellar distances, even when whole galaxies can collide, it is unlikely that there are any direct star-star collisions (FK-596; CK-398).

Question: Significant gravitational interactions between two stars are:
1. frequent because gravity is a long-range force.
2. extremely rare because stars are so small compared to interstellar distances.



Answer 1 is right.

Remember gravity is an inverse-square law force, and so falls off relatively slowly with distance unlike contact forces.

Some simple function behaviors.

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Now let us return to the subject of measuring stellar parallaxes.

Ground-based observations have difficulty measuring very small angles accurately because of the fluctuations in the Earth's atmosphere.

Without special techniques measuring angles much smaller than 0.3 arcseconds are very difficult.

Smaller stellar parallaxes, and thus greater distances, can be obtained from space.

1. The European Space Agency (ESA) launched the Hipparcos satellite in 1989 and it had the capability of measuring parallaxes as small as about 0.01 arcseconds with 10 % accuracy and and 0.005 arcseconds with 20 % accuracy ( Hipparcos summary):

   
          d_max = 1/theta_min = 100 pc  with 10 % accuracy for about 20,000 stars;
   
          d_max = 1/theta_min = 200 pc  with 20 % accuracy for about 50,000 stars. 
   
           

   Hipparcos distances out to about 500 pc were measured with lower accuracy (FK-587)???.

   Unfortunately, there seems to have been some calibration errors that left some distances very precise, but NOT accurate????.

2. NASA plans to launch the Space Interferometry Mission (SIM) in 2010. Among other things it will be able to measure stellar parallaxes as small as 4*10**-6 arcseconds ( SIM Stellar Distances).

   
          d_max = 1/theta_min = 1/4*10**-6 = 2.5*10**5 pc
   
                = 250 kpc
   
          

   The disk of the Milky Way has a diameter of about 30 kpc (CK-379) and nearby dwarf galaxies are within about 300 kpc (FK-593).

   Thus SIM will give us distances all over the Milky Way and to nearby dwarf galaxies.

   It will NOT be able to reach the nearest large galaxy, the Andromeda Galaxy (M31) at about 725 kpc (Cox-578).

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Luminosity, Flux, and the Inverse-Square Law

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For spherically symmetric light sources, luminosity, distance, and flux are related by an inverse-square law.

The inverse-square law in this case just follows from the conservation of energy principle.

The inverse-square law for light.

The inverse-square law can be used to determine luminosity if one can measure flux and distance. The expression again is:

          L=4*pi*r**2*F  , where

            r is distance and F is flux.

For nearby stars one can use stellar parallax to determine distances and measure flux correcting for Earth's atmosphere and, when possible, interstellar dust which causes deviations from the inverse-square law.

For farther stars one can turn the procedure around to find distances using known luminosities and measured fluxes.

One classifies stars by spectral types and these types have known luminosities determined from nearby examples of the types for which stellar parallax can be used.

We will discuss spectral types in IAWL Lecture 20: The Nature of Stars.

For a distant star, one determines the spectral type from its SPECTRUM and the type determines the luminosity.

[Stellar spectra can be measured to much larger distances than stellar parallaxes.]

Then the distance follows from the expression:

      d = sqrt{ L/(4*pi*F) }  ,  where L is luminosity and F is flux.

This method of determining distance is called spectroscopic parallax: this is another one of the great misnomers of astronomy since the method does NOT use parallax in any direct sense.

Distances from spectroscopic parallax are less accurate than the best distances from stellar parallax, but they can extend much farther.

This is because spectroscopic parallax distances depend on stellar parallax distances and on the spectral type.

Both of these dependencies introduce errors which are ``ADDED'': ``added'' is in quotes because the error propagation in analysis is not simple addition.

Typically spectroscopic parallax distances have uncertainties of order 10 % at best (CK-289; FK-430).

We can look at the distances, fluxes, luminosities, and other quantities for the brightest stars from a Table of Brightest Stars as Seen from Earth.

Question: Why are the nearest stars not necessarily the brightest stars on the sky?
1. Brightness depends on luminosity as well as distance. Dimmer stars farther away can be brighter on the sky than nearby bright stars.

2. Brightness depends on luminosity as well as distance. Luminous stars farther away can be brighter on the sky than nearby dim stars.

3. The nearest stars are mainly hidden in clouds of interstellar dust.



Answer 2 is right.

Interstellar dust is also a factor since affects the observed brightness or flux of star, but we won't consider this problem in depth.

Let's see how many brightest stars we can find in the northern winter sky.

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The northern constellations: a mid-winter night-time view judging from the position of old man Orion.

The Milky Way is not displayed, but it passes through Cassiopeia and over the Betelgeuse (eastern) shoulder of Orion.

Credit: Mount Wilson Observatory StarMap program by Bob Donahue. StarMap is fortran program, but it's been broke since 2000jan03. Download site: Univ. of Tennesse, Knoxville Astro course; more precisely here.

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Stellar luminosities vary tremendously.

The Sun's luminosity is

 

      L_Sun = 3.845*10**26 W 

          (Cox-12).

      L_Sun is often used as a unit itself.

      For stellar luminosity, L_Sun is a CONVENIENT UNIT.

      The range of stellar luminosity is

          about 10**-4 to 10**6 L_Sun

          (FK-414).

      The frequency distribution of stars with luminosity is
    
          is fairly flat for star's less luminous than the Sun.

      It decreases rapidly with luminosity for L > L_Sun

          (FK-414).

      Thus, very luminous stars are relatively rare, but 

          their luminosity tends to make them very prominent. 

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The Distance Ladder

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Stellar parallax and spectroscopic parallax are two ways of finding distances beyond the solar system.

Spectroscopic parallax can reach to farther distances, because we can determine a star's spectral type to distances beyond where we can determine its stellar parallax---at least in the pre-SIM epoch.

But spectroscopic parallax depends on stellar parallax to determine the luminosities of the spectral type.

Thus, it is not such a basic means of determining distances: it is in fact calibrated using stellar parallax distances.

Stellar parallax and spectroscopic parallax are, respectively, RUNGS 1 AND 2 of what is called the distance ladder.

The distance ladder (at least as yours truly uses the term) is a series of methods for determining distances: the methods constitute the RUNGS: each method allows us to take another step outward.

Each RUNG determines farther distances that than the next LOWER RUNG, but is usually calibrated by the next LOWER RUNG or other LOWER RUNGS, and thus has lower accuracy than the next LOWER RUNG.

Thus, the farther out we go in the observable universe, the less accurate our distance determinations become.

There are ways of skipping RUNGS, but they have their own uncertainties.

Improving overall cosmic distance scale is a key reason for using SIM to extend to stellar parallax distances to outside the Milky Way.

It is beyond the scope of these lectures to discuss the distance ladder in depth.

We can at least present a cartoon it.

A cartoon of the distance ladder (FK-587).

Actually, the distance ladder has many more RUNGS and is very complex with many uncertainties.