ASTR 551 (Majewski) Lecture Notes

• Eddington bias: Describes the effects of observational uncertainties on some measured trend (e.g., star/galaxy/radio source counts).

• Malmquist bias: Describes the effects of an imposed apparent magnitude survey limit on the mean absolute magnitudes of contributing objects in the survey.

• Lutz-Kelker bias: Describes the effects of parallax errors on the calibration of absolute magnitudes.

• Star/galaxy/radio source counts by apparent flux.

• Number of stars having different values of proper motion.

• Number of stars have different values of [Fe/H].

• Photometric redshifts of galaxies.

• The probable errors are at least approximately known.

• The errors are in general small (e.g., compared to the actual values measured), otherwise we would not have much trust in the data to begin with.

• Imagine the measured quantity as a differential distribution in successive bins of the measured quantity.

• We hope that the observed distribution resembles the true distribution, but due to observational errors some stars will find themselves tabulated in the incorrect bin.

• Clearly the smaller the random errors (e.g., relative to the bin width), the less "bin-mixing" there will be.

• But when there is bin mixing, it is clear that bins with a higher count are more likely to scatter members into bins with lower counts, than vice versa.

• The correction from the observed to the true distribution is most sensitive to the curvature (second derivative) of the distribution function at any given point.

• The correction is such that peaks get flattened out and minima get "filled in".

• The size of the correction scales by the spread in the errors.

(Top) Some observed distribution function A_o (in the starcount example we have been doing, this is A_o(m) ). (Middle) The first derivative. (Bottom) The second derivative, which is used as a multiplicative factor in the correction of the observed function in the top panel, to give the true distribution function, A_t .

• However, it is rare that we find ourselves in this situation.

• Typically we are magnitude-limited to some limiting flux/magnitude, m_lim .

• Because from the distance modulus formula
  m = M + 5 log r - 5 + A(r)
  it is clear that for a given m_lim and to a given r we can only see sources with
  M_max < m_lim - 5 log r + 5 - A(r)
  . Intrinsically fainter sources drop out of the apparent magnitude-limited survey.

• The key to the Malmquist bias is that we can see intrinsically brighter sources to larger distances.
  This means that in an apparent magnitude-limited survey, the volume probed by intrinsically brighter sources is larger than that probed by intrinsically fainter sources.
  Because different volumes are probed, an apparent magnitude-limited survey will have an over-representation of intrinsically brighter compared to fainter sources than in a volume-limited survey.

• Then, for any given apparent magnitude, objects from a set range of distance can contribute:
  From 5 log (r) = m_lim - M_bright + 5
  to 5 log (r) = m_lim - M_faint + 5.
  Note that, of course, the intrinsically brighter sources can be seen to larger distances.

• Now, impose an apparent magnitude limit to the survey:
  What is the mean M at each distance?
  • The horizontal line in the plot below shows that when
    5 log (r) > m_lim - M_faint + 5,
    the mean M (shown by the dotted line) starts to become brighter than (M_bright + M_faint) / 2 :
    
• Note that the mean distance at each apparent magnitude is not affected by the flux limitation.

• This effect plays an important role in almost every branch of astronomy.

• For example, calculation of a luminosity function from a volume limited sample will be seriously affected.

• Or imagine, for example, what is the mean [Fe/H] for a sample of RR Lyrae stars? Since [Fe/H] is correlated to the absolute magnitude of an RR Lyrae, one can imagine getting spurious results for a large sample (one can see more metal poor RR Lyraes to a greater distance in a magnitude limited sample).

• This is most easily figured out in the case that the luminosity function of what we are studying is Gaussian.
  
  The mathematical nicety of the exponential in the Gaussian, particularly in its derivative, delivers the correction,
  <M >_m - M_o, sought after.

• The derivation is not done here -- see Binney & Merrifield!.

• One finds:
  

• Now, dA / dM is almost always positive, so that objects in the magnitude-limited survey at magnitudes m will, on average, be of a smaller (brighter) absolute magnitude than the volume mean = M_o .

• Note, that to make this correction, one needs to know what σ is.
  The true σ can be obtained from the measured σ_m as (see Binney & Merrifield, Section 3.6.1):
  

• Now, d ln(A) / dM is often a constant (for example, in a homogeneous, Euclidean situation).
  Then σ ~ σ_m .

• As an example, for main sequence stars, σ_m ~ 0.5; thus, if we are in a homogeneous situation,
  

• At a dark site, we are limited to naked eye magnitudes near V = 6, whereas in a bright site (like a city) we are limited to something like being able to see V ~ 2-3 (or worse!).

• The table below from Binney & Merrifield gives the normalized (to 100 stars) luminosity function one would obtain if one were limited to V = 2.5 or V = 6 stellar samples, compared to a volume-limited sample.
  The differences are obvious.
  

• A way to correct luminosity functions for this severe Malmquist bias to obtain the true luminosity function (like that shown above) is to account for the effective volumes inhabited by stars of each luminosity contributing to your magnitude-limited survey.
  A simple way is to use the V / V_max method (see homework).
  The homework (Binney & Merrifield Problem 3.6) gives another exercise in calculating V_eff.

• Obviously, one can say little about the luminosity function for stars that are intrinsically faint and that have tiny effective volumes for a give apparent magnitude limit.

• As seen in the figure above, the faint end of the LF is not well established for this reason.

• Can infall velocities be exaggerated by Malmquist bias?

• Imagine a cluster of galaxies in an otherwise homogeneous distribution.
  
  • Imagine that we look at galaxies on the frontside of the cluster.
    Here, because the true density of galaxies may be rising quickly, d ln(A) / dM > 0.6, then the estimated mean absolute magnitude here will be brighter than they should be (e.g., in the homogeneous density case).
    In this case, we think galaxies are actually farther than they are, perhaps even to the point of attributing some of these galaxies to the backside.
    If these galaxies were generally following the Hubble flow, they will now seem to be moving too slow for their projected distance.

  • Imagine now that we look at the galaxies on the backside of the cluster.
    Here, because the true density of galaxies may be dropping quickly, the galaxy counts may even appear to level off, with d ln(A) / dM < 0.6.
    By the Malmquist bias, the estimated mean absolute magnitude here will be fainter than they should be (e.g., in the homogeneous density case).
    In this case, we may think the galaxies are closer than they really are, perhaps even to the point of attributing some of these galaxies to the frontside.
    If these galaxies were generally following the Hubble flow, they will now seem to be moving too fast for their projected distance.

  • The net effect may give us something that resembles peculiar galaxy motions of the kind used to infer a large infall velocity into the "attractor".
    

• a Gaussian distribution of width σ / π'

• Z ^-4 = ( π / π') ^-4

This means that (see Lutz-Kelker Fig. 1 below):

• When σ / π' is small (i.e. good parallaxes with small errors) the distribution is mainly Gaussian, but gets pushed slightly to π / π' < 1 (Table 1 and Fig.3 below).

• As σ / π' grows, the shift gets larger.

• When σ / π' gets large (σ / π' >~ 0.2), the Z ^-4 term dominates and we can't learn anything about the true π .