Download Lecture 8: The distance ladder

Lecture 8: The distance ladder
•  knowing the distances to galaxies is fundamental to a lot
of problems,
–  e.g. whether two galaxies may interact, or are just
coincidentally close on the sky
–  e.g. large-scale distribution of galaxies, and whether
the Universe has always expanded at the same rate
•  in this lecture, we compare various methods to estimate
galaxy distances
–  standard candles
–  velocity relations
–  using statistics of objects
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•  DISTANT GALAXIES & CLUSTERS
–  10-20%
•  VIRGO CLUSTER (NEAREST CLUSTER)
–  15%
•  LOCAL GROUP (And.+~50 DWARFS)
–  5-10%
•  LMC (NEAREST GALAXY)
–  10%
•  CEPHEIDS (1st STANDARD CANDLE)
–  10%
•  HYADAES & PLEIADES (NEAREST * CLUSTERS)
–  10%
•  PARALLAX (NEAREST STARS)
–  10%
•  SOLAR SYSTEM (AU, PARSEC)
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–  1%
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Primary and Secondary indicators
•  Primary DIs are single step methods
•  Secondary DIs rely on a primary DI
•  Tertiary Dis rely on a secondary DI etc
PRIMARY
PARALLAX
SUNYAEVZEOLDOVICH
LENSING TIME
DELAY
SECONDARY
CEPHEIDS
MS-FITTING
RR-LYRAE
SNIa
* cosmological
Stellar
based
*
*
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TERTIARY
TULLY_FISHER
FABER-JACKSON
SBF
GC LF
PN LF
Galaxy
based
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reminder of some scales
•  keep in mind some rough scales when considering
galaxies:
–  Sun’s distance from centre of Galaxy: ~ 8 kpc
–  diameter of Galaxy: ~ 30 kpc
–  nearest (non-satellite) galaxies: ~750 kpc
–  sizes of groups and clusters: 1-3 Mpc
–  nearest rich clusters: 20-100 Mpc
–  sizes of ‘walls’ and large-scale structure: 100’s Mpc
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methods for our Galaxy
•  these depend on things we can measure over quite
small scales
–  parallax (motion of nearby stars against fixed
background of more distant objects)
–  plotting Hertzsprung-Russell diagram for clusters
of stars
–  velocities of stars (Oort’s constants)
•  all of these require observations of individual stars, so
they won’t work for galaxies where the bulk of the stars
can’t be distinguished
–  e.g. two stars 1 pc apart would be separated by 0.01
arcsec in the Virgo cluster
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Hubble’s law and distances
•  the most fundamental method is to use the redshift to
estimate the distances to faint galaxies:
•  Hubble’s law is v = H0 d
–  so measure v and use H0 is ≈ 70 km s-1 Mpc-1
–  NB this only works if the motion is cosmological, not
within the Local Group, for example!
•  and to establish Hubble’s law required measurements of
d for some galaxies (as well as v)
–  hence need methods of distance estimation that work
out as far as other clusters (to tens of Mpc)
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from luminosities to distances
•  the general idea is to measure something that doesn’t
vary with the distance to a galaxy, e.g.
–  flux of specific phenomenon
–  period of a regular phenomenon
•  relate this velocity or period to the luminosity, using local
objects (with distances known from other methods)
•  then we use F = L / 4 π d2
–  measure F, know L....work out d
•  or use the distance modulus:
–  calculate the absolute magnitude M for the luminosity
L, measure m... use m - M = 5 log10 (d / 10 pc)
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three methods used already
•  standard candle
–  Cepheid variable stars, that have longer periods when
more luminous
•  in elliptical galaxies:
–  luminosity versus velocity dispersion
(Faber-Jackson relation)
•  in spiral galaxies:
–  luminosity versus rotation velocity
(Tully-Fisher relation)
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Standard candles
•  a standard candle is anything that has a predictable
brightness
–  usually related to some change with time
•  Cepheid variable stars are very useful for nearby galaxies
–  very luminous
–  several-day periods
–  distance record is
~ 30 Mpc
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NGC 4603 (HST)
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Faber-Jackson relation
•  derived this for elliptical galaxies
(L / 2 x 1010 Lsolar) ≈ (σ / 200 km/s )4
–  the velocity dispersion σ is independent of distance
–  so work out L from the F-J relation, measure flux F
and get d
•  physical basis for this relation:
–  from the Virial Theorem and the definition of L, and
assuming M/L = constant, then σ2 ∝ Ie Re
–  if surface brightness Ie is constant, L ∝ Re2
–  so L ∝ σ4
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Tully-Fisher relation
•  similarly for spiral galaxies
(L / 3 × 1010 Lsolar) ≈ (vmax / 200 km/s)4
–  where the velocity quantity is now the rotational speed
at the maxima of the gas spectrum
–  so work out L from the T-F relation, measure flux F
and get d
•  again this has a physical basis:
–  from the circular velocity formula, M ∝ vmax2 Rd
–  again assuming M/L is constant and using an
expression for L, we get vmax2 ∝ Rd
–  and as L ∝ Rd2 if the central brightnesses I0 are the
same.... then L ∝ vmax4
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general problems
•  get the velocities from spectra, which are difficult to
measure for faint galaxies
–  spectral lines occupy much less wavelength range
than e.g. the broad bands used for magnitudes
•  for standard candles, need to measure the light from
single objects in a galaxy
–  may be rare, or different types that can be confused
–  not many stars vary systematically with time
•  so now consider two new methods for distances
–  based on lots of objects per galaxy
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Globular clusters Luminosity fn
•  Based purely on optimistic assumption that globular
clusters are drawn from a fundamental distribution, I.e.,
–  the observed distribution is roughly a Gaussian, with
a peak at MV = -7.5 ± 0.2 magnitudes
N
PEAK INVARIANT FROM GALAXY TO GALAXY
OBSERVED DISTRIBUTIONS OF GCs
MV
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•  plot the apparent magnitudes mV for globular clusters
around a galaxy of unknown distance
•  work out the distance needed to shift the magnitude of the
centre of the distribution to -7.5
–  i.e. the distance modulus m – M = -5 log (d / 10 pc)
where M is -7.5
N
-7.5
↓
shift by distance d
mV
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Problems with GCLF
•  No real physical basis
•  Substantial variation seen in LF dispersion
•  GC’s more readily seen for E/Sos but not local
calibrator
•  Easy to mistake background galaxies for GCs in low
quality data
•  Disagreement in local calibration, I.e., M=-7.7 to -7.41
Note: The Planetary Nebulae Luminosity Function is
essentially the same except using PNe
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Surface brightness fluctuations
•  Uses the idea of groups of objects,
but in a more statistical way
•  if we count the number of stars in a
box of fixed angular size, e.g. 1
arcsec2, the box would contain
more stars in a more distant galaxy
•  the random variation in the star
count, N, is ±√N (Poisson statistics
for counting objects)
•  so the fluctuation in the count will
be larger if N is small
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•  if an average star has a flux f, and there are N stars in a
detector pixel, the signal per pixel is
S = Nf
•  signal does not change with d, because more but fainter
stars per pixel at larger distance:
N ∝ d2
f ∝ 1/d2
•  but number of star fluctuates by ±√N, so noise on S is
δS = √N f, i.e. δS ∝ 1/d
•  so the fractional error per pixel is
δS/S ∝ N-1/2, or ∝ 1/d
–  e.g. 10x further away means 1/10th as noisy, much
smoother flux distribution
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example
•  an elliptical galaxy has old stars of a type with typical MV
≈ +1, observed brightness of 20 mag per arcsec2 and 1
pixel=1 arcsecond:
–  the count per pixel of our CCD camera is 1440, and
the fluctuation between nearby pixels is 120
•  I.e., N.f=1440, √(N).f=120 which gives N=144
•  therefore f = 10 counts (from N.f / N = 1440 / 144)
•  then get the typical stellar magnitude from
mf – mpixel = -2.5 log [ f / 1440 ] = 5.4 mag
–  therefore mf = 25.4 (because mpixel=µ = 20 mag)
•  since m – M = 5 log d +25 with d in Mpc
d = 100.2(m – M – 25) = 0.76 Mpc
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Problems with SBF
•  need to assume that the typical stellar flux is the same
everywhere in the galaxy
–  works less well for spiral galaxies, because the arms
have bright young stars and the inter-arm regions
don’t
•  need to know what kind of stars are present
–  so stellar evolution theory can give us MV
•  fluctuations could have other causes, e.g.
–  differences between pixels of CCD
–  other things scattered around the galaxy, like globular
clusters
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Comparison of DIs
•  some examples for the Virgo cluster:
Method
Distance mod.
Data used
Tully-Fisher
31.18 ± 0.40
43 spirals
surface br.
31.03 ± 0.06
10 E/S0
glob. cluster
31.25 ± 0.2
4 E/S0
Cepheids
31.16 ± 0.2
in M100
•  so the scatter in the distance modulus (m – M) is about
0.22 mag, or a distance uncertainty of 10%
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