# Distances (PowerPoint) by dffhrtcv3

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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.
 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
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.
• Beam travels at speed of light, c

• Measure the time it takes beam to leave 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
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 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
Absolute Brightness
Apparent Brightness =
4  distance2
Main-Sequence Fitting
Measure parallax to
nearby star cluster
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

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
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
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))
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