increases with the square of the distance which is equivalent to the assumption that the distribution of stars in space is uniform.
is the true parallax), and therefore, there will be more stars in the volume shells at farther distance.
The existence (or otherwise) of this bias and the necessity of correcting for it has become relevant in astronomy with the precision parallax measurements made by the Hipparcos satellite and more recently with the high-precision data releases of the Gaia mission.
The correction method due to Lutz and Kelker placed a bound on the true parallax of stars.
Integrating over all true parallaxes (all space) assumes that stars are equally visible at all distances, and leads to divergent integrals yielding an invalid calculation.
In general, other corrections for systematic bias are required, depending on the selection criteria of the stars under consideration.
[4] The scope of effects of the bias are also discussed in the context of the current higher-precision measurements and the choice of stellar sample where the original stellar distribution assumptions are not valid.
These differences result in the original discussion of effects to be largely overestimated and highly dependent on the choice of stellar sample.
In the original treatment of the phenomenon by Lutz & Kelker, this probability, using Bayes theorem, is given as
is the probability density of finding a star with apparent magnitude m with a given distance
is the probability density function of the apparent magnitude independent of distance.
Assuming a uniform distribution of stars in space, the number density
Following the original description of the bias,[1] we can define a normalization by including the observed parallax as
represents the point where the measurement in parallax is equal to its true value, where the probability distribution should be centered.
Larger uncertainties in contrast would yield higher systematic deviations of the observed parallax from its true value.
Large errors in parallax measurement become apparent in luminosity calculations and are therefore easy to detect.
Consequently, the original treatment of the phenomenon considered the bias to be effective when the uncertainty in the observed parallax,
Several subsequent work on the phenomenon refuted this argument and it was shown that the scope is actually very sample based and may be dependent on other sources of bias.
Therefore, more recently it is argued that the scope for most stellar samples is not as drastic as first proposed.
Following the original statement, the scope of the effects of the bias, as well as its existence and relative methods of correction have been discussed in many works in recent literature, including subsequent work of Lutz himself.
[6] This suggests the bias is largely dependent on the specific choice of sample and measurement error distributions, although the term Lutz–Kelker bias is commonly used generically for the phenomenon on all stellar samples.
[9] Overall, such differences are discussed to result in effects of the bias to be largely overestimated in the original treatment.
More recently, the effects of the Lutz–Kelker bias became relevant in the context of the high-precision measurements of Gaia mission.
The scope of effects of Lutz–Kelker bias on certain samples is discussed in the recent Gaia data releases, including the original assumptions and the possibility of different distributions.
[10] It remains important to take bias effects with caution regarding sample selection as stellar distribution is expected to be non-uniform at large distance scales.
As a result, it is questioned whether correction methods, including the Lutz-Kelker correction proposed in the original work, are applicable for a given stellar sample, since effects are expected to depend on the stellar distribution.
Moreover, following the original description and the dependence of the bias on the measurement errors, the effects are expected to be lower due to the higher precision of current instruments such as Gaia.
The original description of the phenomenon was presented in a paper by Thomas E. Lutz and Douglas H. Kelker in the Publications of the Astronomical Society of the Pacific, Vol.
507, p. 573 article entitled "On the Use of Trigonometric Parallaxes for the Calibration of Luminosity Systems: Theory.
[11] The discussion on statistical bias on measurements in astronomy date back to as early as to Eddington in 1913.