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					 Distance
Measurement
            Cosmic Distances
Geometry
Standard candles are objects for which we are
 likely to know the true luminosity. Some
 astronomical objects make good standard
 candles, but never perfect.
 Standard rulers are objects for which we are
 likely to know the true size. Some astronomical
 objects make good standard rulers, but never
 perfect.
Creating a Cosmic Distance Ladder
   Radar (the replacement method for Kepler’s Law, P2 ~
   a3 )

   Parallax

   Moving clusters

   Cepheid Period/Luminosity Relationship

   Supernovae

   Redshift and Hubble’s Law
The Distance Chain
          Radar ranging: Solar-
           system.
          Parallax: Solar-
           neighborhood.
          MS fitting: Milky Way.
          Cepheids: Galaxies up to
           30 Mpc.
          WD supernovae and TF
           relation: Distant galaxies.
          Hubble’s law: Universe.
      Radar Measurements
• Beam travels at speed of light, c

• Measure the time it takes beam to leave Earth,
bounce off planet (or whatever), and return to Earth.
This represents the time for the beam, traveling at c,
to cover twice the distance between Earth and the
target object.


                                              2d = c t

                                              d = ct/2
                d
 Distances to Nearby Stars
Parallax : determined by the change of position of a nearby star
with respect to the distant stars, as seen from the Earth at two
different times separated by 6 months.
                        Parallax
Gold standard for astronomical distances. It is based on
         measuring two angles and the included side of a
triangle

The parallax of a star is one-half the angle



              parallax angle                      Approximation!
                                                D = Earth-Sun distance
                                                       parallax




Astronomers usually say the Earth-Sun distance is 1
astronomical unit, where 1 au = 1.5x1013 cm, and measure small
angles in arc-seconds. Parallax to Proxima Centauri is only
0.76”
           The Nearest Stars

Distance to Alpha
or Proxima
Centauri is
~4 x 1013 km
(~4.2 light-years)

Distance between
Alpha and
Proxima Centauri
is ~23 AU
       The Solar Neighborhood


Some stars are
   within
    about
 2 x 1014 km
(~ 20 light-years)
       The Solar Neighbourhood
Nearest neighbours
  66 stars within a radius of 5.18 pc
  Hence

                      66
         N    4           3
                                0.11 stars/pc3

                 3   (5.18 )
      So, average separation ~ 1.29 pc
                 Stellar Clusters
Stars are are often found to be moving through space in
 groups or clusters
   two major groups are recognized
Globular clusters
Galactic clusters
   loose (open) structures typically containing 100 to
    1000 stars
   irregular in shape and always found in the galactic
    plane
   sizes typically range from 5 to 20 pc
   typically 0.1 to 5 stars per cubic parsec
Moving (Galactic) Cluster Method
              (1)
a collection of stars that have a common space
 motion
Composed of stars that formed out of the same gas
 cloud and are moving through space along nearly
 parallel paths


                                VS
                VS
     Moving cluster method (2)
When seen from the Earth the stars in a moving
cluster all appear to be traveling towards the same
point in the sky (the convergence point)

The convergence point is due to a perspective effect
    e.g., the convergence of parallel lines effect

                                            Convergent
                                              point
      Moving cluster method (3)
 the projected tracks of the stars appear to converge
  to a point at an angle A away from the location of
  the cluster

                                   VR              VT
                                          tan(A) 
                                   VS              VR
                              VT


Sun          A                            To convergence
                                               point
     Moving cluster method (4)
The position of the convergence point is found by
 measuring the proper motions of the stars
   the tracks are projected forward on a star map
Once the convergence point has been found the
 distance to the cluster can be determined.
     Moving cluster method (5)
How the method works:
   find the convergence point from a proper motion study of
    the cluster (this entails a lot of hard work)
   determine the angle A for each star
   pick specific stars and measure their radial velocities (VR)
   key point of method is that: VT = VR tan(A)
   and we also know: VT = 4.74 d(pc )m(arc. sec)
   So:


                         VR
               d(pc)         tan(A)
                       4.74 m
                   The Hyades

The Hyades is an important cluster of stars
The cluster is used as a “standard cluster” for
 finding the distance to other clusters (more on this
 later) and for calibrating the properties of stars
It is an old galactic cluster (age ~ 6.6 x 108 yrs)
   galactic clusters will typically only survive for a few
    billion years


Various studies find a distance of 46 pc
Moving cluster method (6)
                    Cepheids
Cepheid variable stars are pulsating stars, named after
the brightest member of the class, Delta Cephei.

Cepheids are brightest when they are hottest, close to
the minimum size. Since all Cepheids are about the same
temperature, the size of a Cepheid determines its
luminosity.

Thus there is a period-brightness relationship for
Cepheids.

Since it is easy to measure the period of a variable star
and they can be very bright, Cepheids are wonderful for
determining distances to galaxies!
RR Lyrae stars
            Cepheid Variables
Pulsating variable bright stars that follow a
 simple period-luminosity relation.
The longer the time period between peaks in
 brightness, the greater the stellar luminosity.
Cepheids are primary standard candles to
 determine distances in the Milky Way and other
 galaxies.
              Cepheid Variables
Henrietta Leavitt studied variable stars that were all at
the same distance (in the LMC or SMC) and found that
their pulsation periods were related to their brightnesses




                                                              Polaris (the
                                                              North Star)
                                                                 is not
                                                              constant, it
                                                             is a Cepheid
                                                               variable!
 L =K P1.3
       Distances to Cepheids
Distance to closest Cepheid (Delta Cephei) in our Galaxy
 can be found using parallax measurements. This determines
 K in the period-luminosity relation (L = KP1. 3)


Since the period of a Cepheid is related to its absolute
  brightness, if you observe its period and the apparent
  brightness, you can then derive its distance
 (to within about 10%)
                                 Absolute Brightness
    Apparent Brightness =
                                      4  distance2
          Main-Sequence Fitting
Measure parallax to
 nearby star cluster
 (Hyades, Pleiades).
Compare MS of distant
 cluster to that of a nearby
 one.
Luminosity-distance
 formula (chapter 13):
 apparent
 brightness=L/4d2
       Distances to Supernovae
                                     Supernova 1987A in LMC
Brightest SN in modern times,
occurred at t0

Measure angular diameter of
ring, q

Measure times when top and
bottom of ring light up, t2 and t1

Ring radius is given by
   R = c(t1-t0 + t2-t0)/2

Distance = R / q


                                          D = 47 kpc
        Distances to Supernovae
Type Ia supernovae are “standard candles”

Occur in a binary system in which a white dwarf
 star accretes beyond the 1.4 Mo limit and
 collapses and explodes

 Decay time of light curve is correlated to
 absolute luminosity
White Dwarf Supernovae
            Cepheid distances are used to
             calibrate the distances to WD
             supernovae.
            HST has been used to
             determine the Cepheid
             distance to several historical
             WD supernovae.
            As expected WD supernovae
             are good standard candles.
             Standard Rulers
Size of HII regions
Size of planetary nebulae
Size of galaxies
             Galaxy Clusters
Calibrate galaxies in these
               Hubble’s Law
Edwin Hubble determined the distance to M11
 (1924) and other spiral galaxies using Cepheids.
In 1929, he announced that the more distant a
 galaxy is, the greater its redshift and hence the
 faster it is moving away from us.
Hubble’s law: v=H0d where v is the recession
 velocity, d stands for distance and H-naught is
 Hubble’s constant expressed in units of
 km/s/Mpc.
  Using Hubble’s Law to Measure
            Distances
We can use a galaxy’s recession velocity to
 determine its distance: d=v/H0
However, galaxies may have peculiar motions
 that change their velocity, particularly in the
 Local Group and nearby galaxy clusters.
The distances we find with Hubble’s law are
 only as accurate as our best knowledge of
 Hubble’s constant.
Hubble’s data
Hubble relation
    Measuring Hubble’s Constant
One of the main missions
 of HST.
Distant Cepheids can be
 measured with HST up to
 30 Mpc (108 l.y.)
 reaching the nearest
 galaxy clusters such as
 the one in Virgo.
 However, this is not
 enough to calibrate
 Hubble’s constant.
           Tully-Fisher Relation
Both the luminosity and
 the rotation speed of a
 spiral galaxy depend on
 the mass, and hence they
 are connected with a
 simple relation.
This relation allows to
 use large spiral galaxies
 as standard candles.
H0=65 +/-10 km/s/Mpc.
The Distance Chain
          Radar ranging: Solar-
           system.
          Parallax: Solar-
           neighborhood.
          MS fitting: Milky Way.
          Cepheids: Galaxies up to
           30 Mpc.
          WD supernovae and TF
           relation: Distant galaxies.
          Hubble’s law: Universe.
                 Hubble’s law
http://hyperphysics.phy-
 astr.gsu.edu/hbase/astro/distance.html
http://curious.astro.cornell.edu
http://imagine.gsfc.nasa.gov/docs/ask_astro/ask_an_astr
 onomer.html
http://imagine.gsfc.nasa.gov/index.html
http://www.astro.washington.edu/labs
http://www.astro.washington.edu/labs/parallax/java_par
 allax.html
           Brightness, Luminosity
Flux (density)
   Energy
   Wavelength/frequency interval
   Time
   Area
Intensity/surface brightness
   Flux density / unit solid angle
Magnitude - -2.5log10(Flux density/ reference flux
 density)
Inverse square law
Luminosity
Effects of Dust
                Magnitude
Nova seen by Hipparchos in 134BC
  prompted earliest known comprehensive star
  catalogue - 850 stars
Hipparchos characterised brightness by
  “Magnitude” in the range 1 to 6.
  Brightest 1, the just visible 6.
We still use magnitudes
Physiology of the eye means his magnitude
  difference of +1 implies, in modern units, a
  brightness ratio of 2.512 (10(2/5))
 38

				
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posted:3/22/2012
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