Malmquist bias
Encyclopedia
The Malmquist bias refers to an effect in observational astronomy
which leads to the preferential detection of intrinsically bright objects. It was first popularized in 1922 by Swedish astronomer Gunnar Malmquist
(1893–1982), who then greatly elaborated upon this work in 1925. In statistics, this bias is referred to as a selection bias
and affects the survey results in a brightness
(or equivalently, apparent magnitude
) limited survey (referred to as being magnitude-limited
), where stars below a certain apparent brightness are not included. Since observed objects (stars, galaxies, etc.) appear dimmer when farther away, the brightness that is measured will fall off quickly with distance until their brightness falls below the observational threshold. Objects which are more luminous
, or intrinsically brighter, can be observed at a greater distance, creating a false trend of increasing average luminosity
(intrinsic brightness), and other related quantities, with distance. This effect has led to many spurious claims in the field of astronomy. Properly correcting for these effects has become an area of great focus.
Starlight also follows the inverse square law. Light rays leave the star
in equal amounts in all directions. The light rays create a sphere of light surrounding the star. As time progresses, the sphere grows as the light rays travel through space away from the star. While the sphere of light grows, the number of light rays stays the same. So, the amount of light per unit of surface area of the sphere (called flux
in astronomy) decreases with time. When observing a star, only the light rays that are in the given area being viewed can be detected. This is why a star appears dimmer the farther away it is.
If there are two stars with the same intrinsic brightness (called luminosity
in astronomy), each at a different distance, the closer star will appear brighter while the further will appear dimmer. In astronomy, the apparent brightness of a star, or any other luminous object, is called the apparent magnitude
. The apparent magnitude depends on the intrinsic brightness (also called absolute magnitude
) of the object and its distance.
If all stars had the same luminosity, the distance from Earth to a particular star could be easily determined. However, stars have a wide range in luminosities. Therefore, it can be difficult to distinguish a very luminous star that is very far away from a less luminous star that is closer. This is why it is so hard to calculate the distance to astronomical objects.
can be seen. As discussed above, the very luminous stars that are farther away will be seen, as well as luminous and faint stars that are closer. There will appear to be more luminous objects within a certain distance from Earth than faint objects. However, there are many more faint stars, they simply cannot be seen because they are so dim. The bias towards luminous stars when observing a patch of sky affects calculations of the average absolute magnitude
and average distance to a group of stars. Because of the luminous stars that are at a further distance, it will appear as if our sample of stars is farther away than it actually is, and that each star is intrinsically brighter than it actually is. This effect is known as the Malmquist bias.
When studying a sample of luminous objects, whether they be stars or galaxies, it is important to correct for the bias towards the more luminous objects. There are many different methods that can be used to correct for the Malmquist bias as discussed below.
from entering a data survey. However, magnitude limited
surveys are the simplest to perform, and other methods are difficult to put together, with their own uncertainties involved, and may be impossible for first observations of objects. As such, many different methods exist to attempt to correct the data, removing the bias
and allowing the survey to be usable. The methods are presented in order of increasing difficulty, but also increasing accuracy and effectiveness.
selected, there may be a range of distances in the data set over which all objects of any possible absolute magnitude
could be seen. As such, this small subset of data should be free of the Malmquist bias. This is easily accomplished by cutting off the data at the edge of where the lowest absolute magnitude
objects would be hitting the limiting magnitude
. Unfortunately, this method would waste a great deal of good data, and would limit the analysis to nearby objects only, making it less than desirable. (Looking at the figure to the right, only the first fifth of the data in distance could be kept before a data point is lost to the bias.) Of course, this method assumes that distances are known with relatively good accuracy, which as mentioned before, is a difficult process in astronomy.
(
) of the sample back to the true average absolute magnitude
(M0). The correction would be
To calculate the bias
correction, Malmquist, and others following this method follow six main assumptions:
Obviously, this is a very ideal situation, with the final assumption being particularly troubling, but allows for an approximate correction of simple form. By integrating the luminosity function
over all distances and all magnitudes brighter than mlim,
where A(mlim) is the total number of stars brighter than mlim. If the spatial distribution of stars can be assumed to be homogeneous, this relation is simplified even further, to the generally accepted form of
and the measurements from which distance is determined are from the same band, or predefined range, of wavelengths (e.g. the H band
, a range of infrared
wavelengths from roughly about 1300 - 2000 nanometers), and this leads to the correction form of cσ2, where c is some constant. Unfortunately, this is rarely the case, as many samples of objects are selected from one wavelength band but the distance is calculated from another. For example, astronomers frequently select galaxies from B-band catalogs, which are the most complete, and use these B band magnitudes, but the distances for the galaxies are calculated using the Tully-Fisher relation
and the H band. When this happens, the square of the variance is replaced by the covariance
between the scatter in the distance measurements and in the galaxy selection property (e.g. magnitude).
to properly account for the relative contributions at each magnitude. Since the objects at different absolute magnitudes
can be seen out to different distances, each point's contribution to the average absolute magnitude
or to the luminosity function
can be weighted by 1/Vmax, where Vmax is the maximum volume over which the objects could have been seen. Objects with a greater absolute magnitude
will have a larger volume over which they could have been detected, before falling under the threshold, and thus will be given less weight through this method since these bright objects will be more fully sampled. The maximum volume can be approximated as a sphere with radius found from the distance modulus
, using the object’s absolute magnitude
and the limiting apparent magnitude
.
However, there are two major complications to calculating Vmax. First is the completeness
of the area covered in the sky, which is the percentage of the sky that the objects were taken from. A full sky survey would collect objects from the entire sphere, 4π steradians, of sky but this is usually impractical, both from time constraints and geographical limitations (ground based telescopes can only see a limited amount of sky due to the Earth being in the way). Instead, astronomers will generally look at a small patch or area of sky and then infer universal distributions by assuming that space is either isotropic
, that it is generally the same in every direction, or is following a known distribution, such as that one will see more stars by looking toward the center of a galaxy than by looking directly away. Generally, the volume can be simply scaled down by the percentage actually viewed, giving the correct number of objects to volume relation. This effect could potentially be ignored in a single sample, all from the same survey, as the objects will basically all be altered by the same numerical factor, but it is incredibly important to account for in order to be able to compare between different surveys with different sky coverage.
The second complication is cosmological
concerns of redshift and the expanding universe, which must be considered when looking at distant objects. In these cases, the quantity of interest is the comoving distance
, which is a constant distance between two objects assuming that they are moving away from each other solely with the expansion of the universe, known as the Hubble flow. In effect, this comoving distance
is the object's separation if the universe's expansion were neglected, and it can be easily related to the actual distance by accounting for how it would have expanded. The comoving distance
can be used to calculate the respective comoving volume as usual, or a relation between the actual and comoving volumes can also be easily established. If z is the objects redshift, relating to how far emitted light is shifted toward longer wavelengths as a result of the object moving away from us with the universal expansion, DA and VA are the actual distance and volume (or what would be measured today) and DC and VC are the comoving distance
and volumes of interest, then
A large downside of the volume weighting method is its sensitivity to large-scale structures
, or parts of the universe with more or less objects than average, such as a star cluster
or a void
. Having very overdense or underdense regions of objects will cause an inferred change in our average absolute magnitude
and luminosity function
, according with the structure. This is a particular issue with the faint objects in calculating a luminosity function, as their smaller maximum volume means that a large-scale structure therein will have a large impact. Brighter objects with large maximum volumes will tend to average out and approach the correct value in spite of some large-scale structures.
of objects (such as stars or galaxies), which is a relation of how many objects are expected with certain intrinsic brightnesses
, distances, or other fundamental values. Each of these values have their own distribution function
which can be combined with a random number generator to create a theoretical sample of stars. This method takes the distribution function
of distances as a known, definite quantity, and then allows the distribution function
of absolute magnitudes
to change. In this way, it can check different distribution functions
of the absolute magnitudes
against the actual distribution of detected objects, and find the relation that provides the maximum probability of recreating the same set of objects. By starting with the detected, biased distribution of objects and the appropriate limits to detection, this method recreates the true distribution function
. However, this method requires heavy calculations and generally relies on computer programs.
found a very interesting relation between the logarithm of a spectral line's
line width and its apparent magnitude
, when working with galaxies. In an perfect, stationary case, spectral lines should be incredibly narrow bumps, looking like lines, but motions of the object such as rotation or motion in our line of sight will cause shifts and broadening of these lines. The relation is found by starting with the Tully-Fisher relation
, wherein the distance to a galaxy
is related to its apparent magnitude
and its velocity width, or the 'maximum' speed of its rotation curve
. From macroscopic Doppler broadening
, the logarithm of the line width of an observed spectral line can be related to the width of the velocity distribution. If the distances are assumed to be known very well, then the absolute magnitude
and the line width are closely related. For example, working with the commonly used 21cm line
, an important line relating to neutral hydrogen, the relation is generally calibrated with a linear regression
and given the form
where P is log(line width) and α and β are constants.
The reason that this estimator is useful is that the inverse regression line is actually unaffected by the Malmquist Bias, so long as the selection effects are only based on magnitude. As such, the expected value of P given M will be unbiased and will give an unbiased log distance estimator. This estimator has many properties and ramifications which can make it a very useful tool.
which relates the correction to the variance of the sample and the apparent magnitude
, absolute magnitude
, and the height above the galactic disk
. This gave a much more exact and accurate result, but also required an assumption about the spatial distribution of stars in the desired galaxy
. While useful individually, and there are many examples published, these have very limited scope and are not generally as broadly applicable as the other methods mentioned above.
, calibrate the Tully-Fisher relation
, or obtain the value of the Hubble constant, the Malmquist bias can strongly change the results.
The luminosity function gives the number of stars or galaxies per luminosity or absolute magnitude bin. When using a magnitude-limited sample, the number of faint objects is underrepresented as discussed above. This shifts the peak of the luminosity function from the faint end to a brighter luminosity and changes the shape of the luminosity function. Typically, the volume weighting method is used to correct the Malmquist bias so that the survey is equivalent to a distance-limited survey rather than a magnitude-limited survey. The figure to the right shows two luminosity functions for an example population of stars that is magnitude-limited. The dashed luminosity function shows the effect of the Malmquist bias, while the solid line shows the corrected luminosity function. Malmquist bias drastically changes the shape of the luminosity function.
Another application that is affected by the Malmquist bias is the Tully-Fisher relation
, which relates the luminosity of spiral galaxies to their respective velocity width. If a nearby cluster of galaxies is used to calibrate the Tully-Fisher relation, and then that relation is applied to a distant cluster, the distance to the farther cluster will be systematically underestimated. By underestimating the distance to clusters, anything found using those clusters will be incorrect; for example, when finding the value of the Hubble constant.
These are just a few examples where the Malmquist bias can strongly affect results. As mentioned above, anytime a magnitude-limited sample is used, the Malmquist bias needs to be corrected for. A correction is not limited to just the examples above.
but instead observe all objects within this volume. Clearly, in this case, the Malmquist bias is not an issue as the volume will be fully populated and any distribution
or luminosity function
will be appropriately sampled. Unfortunately, this method is not always practical. Finding distances to astronomical objects is very difficult, and even with the aid of objects with easily determined distances, called standard candles
, and similar things, there are great uncertainties. Further, distances are not generally known for objects until after they have already been observed and analyzed, and so a distance limited survey is usually only an option for a second round of observations, and not initially available. Finally, distance limited surveys
are generally only possible over small volumes where the distances are reliably known, and thus it is not practical for large surveys
.
again, but through very different means. Rather than trying to fix the absolute magnitudes
, this method takes the distances to the objects as being the random variables and attempts to rescale those. In effect, rather than giving the stars in the sample the correct distribution of absolute magnitudes
(and average absolute magnitude
), it attempts to 'move' the stars such that they would have a correct distribution of distances. Ideally, this should have the same end result as the magnitude correction methods and should result in a correctly represented sample. In either the homogeneous or inhomogeneous case, the bias is defined in terms of a prior distribution of distances, the distance estimator, and the likelihood function
of these two being the same distribution. The homogeneous case is much simpler and rescales the raw distance estimates by a constant factor. Unfortunately, this will be very insensitive to large scale structures
such as clustering as well as observational selection effects, and will not give a very accurate result. The inhomogeneous case attempts to correct this by creating a more complicated prior distribution of objects by taking into account structures seen in the observed distribution. In both cases though, it is assumed that the probability density function
is Gaussian with constant variance and a mean of the true average log distance, which is far from accurate. However, this method is debated and may not be accurate in any implementation due to uncertainties in calculating the raw, observed distance estimates causing the assumptions to use this method to be invalid.
sample, but in this case the low absolute magnitude
objects are overrepresented. In a sample with a magnitude limit
, there will be a margin of error near that boundary where objects that should be bright enough to make the cut are excluded and objects that are slightly below the limit are instead included. Since low absolute magnitude
objects are more common than brighter ones, and since these dimmer galaxies are more likely to be below the cutoff line and scattered up, while the brighter ones are more likely to be above the line and scattered down, an over-representation of the lower luminosity
objects result. However, in modern day literature and consensus, the Malmquist bias refers to the effect outlined above.
Observational astronomy
Observational astronomy is a division of the astronomical science that is concerned with getting data, in contrast with theoretical astrophysics which is mainly concerned with finding out the measurable implications of physical models...
which leads to the preferential detection of intrinsically bright objects. It was first popularized in 1922 by Swedish astronomer Gunnar Malmquist
Gunnar Malmquist
Karl Gunnar Malmquist was a Swedish astronomer.-Biography:Gunnar Malmquist was born in Ystad, where he completed his secondary school education before matriculating at the Lund University in 1911. He received his Ph.D. in 1921, was an amanuensis at the Lund Observatory 1915-1920 and a docent from...
(1893–1982), who then greatly elaborated upon this work in 1925. In statistics, this bias is referred to as a selection bias
Selection bias
Selection bias is a statistical bias in which there is an error in choosing the individuals or groups to take part in a scientific study. It is sometimes referred to as the selection effect. The term "selection bias" most often refers to the distortion of a statistical analysis, resulting from the...
and affects the survey results in a brightness
Brightness
Brightness is an attribute of visual perception in which a source appears to be radiating or reflecting light. In other words, brightness is the perception elicited by the luminance of a visual target...
(or equivalently, apparent magnitude
Apparent magnitude
The apparent magnitude of a celestial body is a measure of its brightness as seen by an observer on Earth, adjusted to the value it would have in the absence of the atmosphere...
) limited survey (referred to as being magnitude-limited
Limiting magnitude
In astronomy, limiting magnitude is the faintest apparent magnitude of a celestial body that is detectable or detected by a given instrument....
), where stars below a certain apparent brightness are not included. Since observed objects (stars, galaxies, etc.) appear dimmer when farther away, the brightness that is measured will fall off quickly with distance until their brightness falls below the observational threshold. Objects which are more luminous
Luminosity
Luminosity is a measurement of brightness.-In photometry and color imaging:In photometry, luminosity is sometimes incorrectly used to refer to luminance, which is the density of luminous intensity in a given direction. The SI unit for luminance is candela per square metre.The luminosity function...
, or intrinsically brighter, can be observed at a greater distance, creating a false trend of increasing average luminosity
Luminosity
Luminosity is a measurement of brightness.-In photometry and color imaging:In photometry, luminosity is sometimes incorrectly used to refer to luminance, which is the density of luminous intensity in a given direction. The SI unit for luminance is candela per square metre.The luminosity function...
(intrinsic brightness), and other related quantities, with distance. This effect has led to many spurious claims in the field of astronomy. Properly correcting for these effects has become an area of great focus.
Understanding the bias
For a derivation of the Malmquist bias, please see pages 111–15 in Galactic Astronomy by James Binney and Michael Merrifield.Magnitudes and brightness
In everyday life it is easy to see that light dims as it gets farther away. This can be seen with car headlights, candles, flashlights, and many other lit objects. This dimming follows the inverse square law, which states that the brightness of an object decreases as 1/d2, where d is the distance between the observer and the object.Starlight also follows the inverse square law. Light rays leave the star
Star
A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...
in equal amounts in all directions. The light rays create a sphere of light surrounding the star. As time progresses, the sphere grows as the light rays travel through space away from the star. While the sphere of light grows, the number of light rays stays the same. So, the amount of light per unit of surface area of the sphere (called flux
Flux
In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as flow per unit area, where flow is the movement of some quantity per time...
in astronomy) decreases with time. When observing a star, only the light rays that are in the given area being viewed can be detected. This is why a star appears dimmer the farther away it is.
If there are two stars with the same intrinsic brightness (called luminosity
Luminosity
Luminosity is a measurement of brightness.-In photometry and color imaging:In photometry, luminosity is sometimes incorrectly used to refer to luminance, which is the density of luminous intensity in a given direction. The SI unit for luminance is candela per square metre.The luminosity function...
in astronomy), each at a different distance, the closer star will appear brighter while the further will appear dimmer. In astronomy, the apparent brightness of a star, or any other luminous object, is called the apparent magnitude
Apparent magnitude
The apparent magnitude of a celestial body is a measure of its brightness as seen by an observer on Earth, adjusted to the value it would have in the absence of the atmosphere...
. The apparent magnitude depends on the intrinsic brightness (also called absolute magnitude
Absolute magnitude
Absolute magnitude is the measure of a celestial object's intrinsic brightness. it is also the apparent magnitude a star would have if it were 32.6 light years away from Earth...
) of the object and its distance.
If all stars had the same luminosity, the distance from Earth to a particular star could be easily determined. However, stars have a wide range in luminosities. Therefore, it can be difficult to distinguish a very luminous star that is very far away from a less luminous star that is closer. This is why it is so hard to calculate the distance to astronomical objects.
Source of the Malmquist bias
Typically, when looking at an area of sky filled with stars, only stars that are brighter than a limiting apparent magnitudeApparent magnitude
The apparent magnitude of a celestial body is a measure of its brightness as seen by an observer on Earth, adjusted to the value it would have in the absence of the atmosphere...
can be seen. As discussed above, the very luminous stars that are farther away will be seen, as well as luminous and faint stars that are closer. There will appear to be more luminous objects within a certain distance from Earth than faint objects. However, there are many more faint stars, they simply cannot be seen because they are so dim. The bias towards luminous stars when observing a patch of sky affects calculations of the average absolute magnitude
Absolute magnitude
Absolute magnitude is the measure of a celestial object's intrinsic brightness. it is also the apparent magnitude a star would have if it were 32.6 light years away from Earth...
and average distance to a group of stars. Because of the luminous stars that are at a further distance, it will appear as if our sample of stars is farther away than it actually is, and that each star is intrinsically brighter than it actually is. This effect is known as the Malmquist bias.
When studying a sample of luminous objects, whether they be stars or galaxies, it is important to correct for the bias towards the more luminous objects. There are many different methods that can be used to correct for the Malmquist bias as discussed below.
Correction Methods
The ideal situation is to somehow avoid this biasBias (statistics)
A statistic is biased if it is calculated in such a way that it is systematically different from the population parameter of interest. The following lists some types of, or aspects of, bias which should not be considered mutually exclusive:...
from entering a data survey. However, magnitude limited
Limiting magnitude
In astronomy, limiting magnitude is the faintest apparent magnitude of a celestial body that is detectable or detected by a given instrument....
surveys are the simplest to perform, and other methods are difficult to put together, with their own uncertainties involved, and may be impossible for first observations of objects. As such, many different methods exist to attempt to correct the data, removing the bias
Bias (statistics)
A statistic is biased if it is calculated in such a way that it is systematically different from the population parameter of interest. The following lists some types of, or aspects of, bias which should not be considered mutually exclusive:...
and allowing the survey to be usable. The methods are presented in order of increasing difficulty, but also increasing accuracy and effectiveness.
Limiting the Sample
The simplest method of correction is to only use the non-biased portions of the data set, if any, and throw away the rest of the data. Depending on the limiting magnitudeLimiting magnitude
In astronomy, limiting magnitude is the faintest apparent magnitude of a celestial body that is detectable or detected by a given instrument....
selected, there may be a range of distances in the data set over which all objects of any possible absolute magnitude
Absolute magnitude
Absolute magnitude is the measure of a celestial object's intrinsic brightness. it is also the apparent magnitude a star would have if it were 32.6 light years away from Earth...
could be seen. As such, this small subset of data should be free of the Malmquist bias. This is easily accomplished by cutting off the data at the edge of where the lowest absolute magnitude
Absolute magnitude
Absolute magnitude is the measure of a celestial object's intrinsic brightness. it is also the apparent magnitude a star would have if it were 32.6 light years away from Earth...
objects would be hitting the limiting magnitude
Limiting magnitude
In astronomy, limiting magnitude is the faintest apparent magnitude of a celestial body that is detectable or detected by a given instrument....
. Unfortunately, this method would waste a great deal of good data, and would limit the analysis to nearby objects only, making it less than desirable. (Looking at the figure to the right, only the first fifth of the data in distance could be kept before a data point is lost to the bias.) Of course, this method assumes that distances are known with relatively good accuracy, which as mentioned before, is a difficult process in astronomy.
Traditional Correction
The first solution, proposed by Malmquist in his 1922 work, was to correct the calculated average absolute magnitudeAbsolute magnitude
Absolute magnitude is the measure of a celestial object's intrinsic brightness. it is also the apparent magnitude a star would have if it were 32.6 light years away from Earth...
(
) of the sample back to the true average absolute magnitudeAbsolute magnitude
Absolute magnitude is the measure of a celestial object's intrinsic brightness. it is also the apparent magnitude a star would have if it were 32.6 light years away from Earth...
(M0). The correction would be
To calculate the bias
Bias (statistics)
A statistic is biased if it is calculated in such a way that it is systematically different from the population parameter of interest. The following lists some types of, or aspects of, bias which should not be considered mutually exclusive:...
correction, Malmquist, and others following this method follow six main assumptions:
- There exists no interstellar absorptionInterstellar mediumIn astronomy, the interstellar medium is the matter that exists in the space between the star systems in a galaxy. This matter includes gas in ionic, atomic, and molecular form, dust, and cosmic rays. It fills interstellar space and blends smoothly into the surrounding intergalactic space...
, or that the stuff in space between stars (like gas and dust) is not affecting the light and absorbing parts of it. This assumes that the brightnessBrightnessBrightness is an attribute of visual perception in which a source appears to be radiating or reflecting light. In other words, brightness is the perception elicited by the luminance of a visual target...
is simply following the inverse square law, mentioned above. - The luminosity functionLuminosity function (astronomy)In astronomy, the luminosity function gives the number of stars or galaxies per luminosity interval. Luminosity functions are used to study the properties of large groups or classes of objects, such as the stars in clusters or the galaxies in the Local Group....
(Φ) is independent of the distance (r). This basically just means that the universe is the same everywhere, and that stars will be similarly distributed somewhere else as they are here. - For a given area on the sky, or more specifically the celestial sphereCelestial sphereIn astronomy and navigation, the celestial sphere is an imaginary sphere of arbitrarily large radius, concentric with the Earth and rotating upon the same axis. All objects in the sky can be thought of as projected upon the celestial sphere. Projected upward from Earth's equator and poles are the...
, the spatial density of stars (ρ) depends only on distance. This assumes that there are the same number of stars in each direction, on average. - There is completenessCompletenessIn general, an object is complete if nothing needs to be added to it. This notion is made more specific in various fields.-Logical completeness:In logic, semantic completeness is the converse of soundness for formal systems...
, meaning the sample is complete and nothing is missed, to an apparent magnitude limitLimiting magnitudeIn astronomy, limiting magnitude is the faintest apparent magnitude of a celestial body that is detectable or detected by a given instrument....
(mlim). - The luminosity functionLuminosity functionThe luminosity function or luminous efficiency function describes the average visual sensitivity of the human eye to light of different wavelengths. It should not be considered perfectly accurate in every case, but it is a very good representation of visual sensitivity of the human eye and it is...
can be approximated as a Gaussian function, centered on an intrinsic mean absolute magnitudeAbsolute magnitudeAbsolute magnitude is the measure of a celestial object's intrinsic brightness. it is also the apparent magnitude a star would have if it were 32.6 light years away from Earth...
M0. - Stars are of the same spectral type, with intrinsic mean absolute magnitudeAbsolute magnitudeAbsolute magnitude is the measure of a celestial object's intrinsic brightness. it is also the apparent magnitude a star would have if it were 32.6 light years away from Earth...
M0 and dispersion σ.
Obviously, this is a very ideal situation, with the final assumption being particularly troubling, but allows for an approximate correction of simple form. By integrating the luminosity function
Luminosity function
The luminosity function or luminous efficiency function describes the average visual sensitivity of the human eye to light of different wavelengths. It should not be considered perfectly accurate in every case, but it is a very good representation of visual sensitivity of the human eye and it is...
over all distances and all magnitudes brighter than mlim,
where A(mlim) is the total number of stars brighter than mlim. If the spatial distribution of stars can be assumed to be homogeneous, this relation is simplified even further, to the generally accepted form of
Multiple-Band Observation Corrections
The traditional method assumes that the measurements of apparent magnitudeApparent magnitude
The apparent magnitude of a celestial body is a measure of its brightness as seen by an observer on Earth, adjusted to the value it would have in the absence of the atmosphere...
and the measurements from which distance is determined are from the same band, or predefined range, of wavelengths (e.g. the H band
H band
H band can refer to two different regions of the electromagnetic spectrum, in the radio and near-infrared.-Radio:The H band is the range of radio frequencies from 6 GHz to 8 GHz in the electromagnetic spectrum. This is equal to wave lengths between 5 cm and 3.75 cm...
, a range of infrared
Infrared astronomy
Infrared astronomy is the branch of astronomy and astrophysics that studies astronomical objects visible in infrared radiation. The wavelength of infrared light ranges from 0.75 to 300 micrometers...
wavelengths from roughly about 1300 - 2000 nanometers), and this leads to the correction form of cσ2, where c is some constant. Unfortunately, this is rarely the case, as many samples of objects are selected from one wavelength band but the distance is calculated from another. For example, astronomers frequently select galaxies from B-band catalogs, which are the most complete, and use these B band magnitudes, but the distances for the galaxies are calculated using the Tully-Fisher relation
Tully-Fisher relation
In astronomy, the Tully–Fisher relation, published by astronomers R. Brent Tully and J. Richard Fisher in 1977, is an empirical relationship between the intrinsic luminosity of a spiral galaxy and its velocity width...
and the H band. When this happens, the square of the variance is replaced by the covariance
Covariance
In probability theory and statistics, covariance is a measure of how much two variables change together. Variance is a special case of the covariance when the two variables are identical.- Definition :...
between the scatter in the distance measurements and in the galaxy selection property (e.g. magnitude).
Volume Weighting
Another fairly straightforward correction method is to use a weighted meanWeighted mean
The weighted mean is similar to an arithmetic mean , where instead of each of the data points contributing equally to the final average, some data points contribute more than others...
to properly account for the relative contributions at each magnitude. Since the objects at different absolute magnitudes
Absolute magnitude
Absolute magnitude is the measure of a celestial object's intrinsic brightness. it is also the apparent magnitude a star would have if it were 32.6 light years away from Earth...
can be seen out to different distances, each point's contribution to the average absolute magnitude
Absolute magnitude
Absolute magnitude is the measure of a celestial object's intrinsic brightness. it is also the apparent magnitude a star would have if it were 32.6 light years away from Earth...
or to the luminosity function
Luminosity function
The luminosity function or luminous efficiency function describes the average visual sensitivity of the human eye to light of different wavelengths. It should not be considered perfectly accurate in every case, but it is a very good representation of visual sensitivity of the human eye and it is...
can be weighted by 1/Vmax, where Vmax is the maximum volume over which the objects could have been seen. Objects with a greater absolute magnitude
Absolute magnitude
Absolute magnitude is the measure of a celestial object's intrinsic brightness. it is also the apparent magnitude a star would have if it were 32.6 light years away from Earth...
will have a larger volume over which they could have been detected, before falling under the threshold, and thus will be given less weight through this method since these bright objects will be more fully sampled. The maximum volume can be approximated as a sphere with radius found from the distance modulus
Distance modulus
-Definition:The distance modulus \mu=m-M is the difference between the apparent magnitude m and the absolute magnitude M of an astronomical object...
, using the object’s absolute magnitude
Absolute magnitude
Absolute magnitude is the measure of a celestial object's intrinsic brightness. it is also the apparent magnitude a star would have if it were 32.6 light years away from Earth...
and the limiting apparent magnitude
Limiting magnitude
In astronomy, limiting magnitude is the faintest apparent magnitude of a celestial body that is detectable or detected by a given instrument....
.
However, there are two major complications to calculating Vmax. First is the completeness
Completeness
In general, an object is complete if nothing needs to be added to it. This notion is made more specific in various fields.-Logical completeness:In logic, semantic completeness is the converse of soundness for formal systems...
of the area covered in the sky, which is the percentage of the sky that the objects were taken from. A full sky survey would collect objects from the entire sphere, 4π steradians, of sky but this is usually impractical, both from time constraints and geographical limitations (ground based telescopes can only see a limited amount of sky due to the Earth being in the way). Instead, astronomers will generally look at a small patch or area of sky and then infer universal distributions by assuming that space is either isotropic
Isotropy
Isotropy is uniformity in all orientations; it is derived from the Greek iso and tropos . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix an, hence anisotropy. Anisotropy is also used to describe situations where properties vary...
, that it is generally the same in every direction, or is following a known distribution, such as that one will see more stars by looking toward the center of a galaxy than by looking directly away. Generally, the volume can be simply scaled down by the percentage actually viewed, giving the correct number of objects to volume relation. This effect could potentially be ignored in a single sample, all from the same survey, as the objects will basically all be altered by the same numerical factor, but it is incredibly important to account for in order to be able to compare between different surveys with different sky coverage.
The second complication is cosmological
Cosmology
Cosmology is the discipline that deals with the nature of the Universe as a whole. Cosmologists seek to understand the origin, evolution, structure, and ultimate fate of the Universe at large, as well as the natural laws that keep it in order...
concerns of redshift and the expanding universe, which must be considered when looking at distant objects. In these cases, the quantity of interest is the comoving distance
Comoving distance
In standard cosmology, comoving distance and proper distance are two closely related distance measures used by cosmologists to define distances between objects...
, which is a constant distance between two objects assuming that they are moving away from each other solely with the expansion of the universe, known as the Hubble flow. In effect, this comoving distance
Comoving distance
In standard cosmology, comoving distance and proper distance are two closely related distance measures used by cosmologists to define distances between objects...
is the object's separation if the universe's expansion were neglected, and it can be easily related to the actual distance by accounting for how it would have expanded. The comoving distance
Comoving distance
In standard cosmology, comoving distance and proper distance are two closely related distance measures used by cosmologists to define distances between objects...
can be used to calculate the respective comoving volume as usual, or a relation between the actual and comoving volumes can also be easily established. If z is the objects redshift, relating to how far emitted light is shifted toward longer wavelengths as a result of the object moving away from us with the universal expansion, DA and VA are the actual distance and volume (or what would be measured today) and DC and VC are the comoving distance
Comoving distance
In standard cosmology, comoving distance and proper distance are two closely related distance measures used by cosmologists to define distances between objects...
and volumes of interest, then
A large downside of the volume weighting method is its sensitivity to large-scale structures
Observable universe
In Big Bang cosmology, the observable universe consists of the galaxies and other matter that we can in principle observe from Earth in the present day, because light from those objects has had time to reach us since the beginning of the cosmological expansion...
, or parts of the universe with more or less objects than average, such as a star cluster
Star cluster
Star clusters or star clouds are groups of stars. Two types of star clusters can be distinguished: globular clusters are tight groups of hundreds of thousands of very old stars which are gravitationally bound, while open clusters, more loosely clustered groups of stars, generally contain less than...
or a void
Void (astronomy)
In astronomy, voids are the empty spaces between filaments, the largest-scale structures in the Universe, that contain very few, or no, galaxies. They were first discovered in 1978 during a pioneering study by Stephen Gregory and Laird A. Thompson at the Kitt Peak National Observatory...
. Having very overdense or underdense regions of objects will cause an inferred change in our average absolute magnitude
Absolute magnitude
Absolute magnitude is the measure of a celestial object's intrinsic brightness. it is also the apparent magnitude a star would have if it were 32.6 light years away from Earth...
and luminosity function
Luminosity function
The luminosity function or luminous efficiency function describes the average visual sensitivity of the human eye to light of different wavelengths. It should not be considered perfectly accurate in every case, but it is a very good representation of visual sensitivity of the human eye and it is...
, according with the structure. This is a particular issue with the faint objects in calculating a luminosity function, as their smaller maximum volume means that a large-scale structure therein will have a large impact. Brighter objects with large maximum volumes will tend to average out and approach the correct value in spite of some large-scale structures.
Advanced Methods
Many more methods exist which become increasingly complicated and powerful in application. A few of the most common are summarized here, with more specific information found in the references.Stepwise Maximum Likelihood Method
This method is based on the distribution functionsDistribution function
In molecular kinetic theory in physics, a particle's distribution function is a function of seven variables, f, which gives the number of particles per unit volume in phase space. It is the number of particles per unit volume having approximately the velocity near the place and time...
of objects (such as stars or galaxies), which is a relation of how many objects are expected with certain intrinsic brightnesses
Brightness
Brightness is an attribute of visual perception in which a source appears to be radiating or reflecting light. In other words, brightness is the perception elicited by the luminance of a visual target...
, distances, or other fundamental values. Each of these values have their own distribution function
Distribution function
In molecular kinetic theory in physics, a particle's distribution function is a function of seven variables, f, which gives the number of particles per unit volume in phase space. It is the number of particles per unit volume having approximately the velocity near the place and time...
which can be combined with a random number generator to create a theoretical sample of stars. This method takes the distribution function
Distribution function
In molecular kinetic theory in physics, a particle's distribution function is a function of seven variables, f, which gives the number of particles per unit volume in phase space. It is the number of particles per unit volume having approximately the velocity near the place and time...
of distances as a known, definite quantity, and then allows the distribution function
Distribution function
In molecular kinetic theory in physics, a particle's distribution function is a function of seven variables, f, which gives the number of particles per unit volume in phase space. It is the number of particles per unit volume having approximately the velocity near the place and time...
of absolute magnitudes
Absolute magnitude
Absolute magnitude is the measure of a celestial object's intrinsic brightness. it is also the apparent magnitude a star would have if it were 32.6 light years away from Earth...
to change. In this way, it can check different distribution functions
Distribution function
In molecular kinetic theory in physics, a particle's distribution function is a function of seven variables, f, which gives the number of particles per unit volume in phase space. It is the number of particles per unit volume having approximately the velocity near the place and time...
of the absolute magnitudes
Absolute magnitude
Absolute magnitude is the measure of a celestial object's intrinsic brightness. it is also the apparent magnitude a star would have if it were 32.6 light years away from Earth...
against the actual distribution of detected objects, and find the relation that provides the maximum probability of recreating the same set of objects. By starting with the detected, biased distribution of objects and the appropriate limits to detection, this method recreates the true distribution function
Distribution function
In molecular kinetic theory in physics, a particle's distribution function is a function of seven variables, f, which gives the number of particles per unit volume in phase space. It is the number of particles per unit volume having approximately the velocity near the place and time...
. However, this method requires heavy calculations and generally relies on computer programs.
Schechter Estimators
Paul SchechterPaul L. Schechter
Paul L. Schechter is an astrophysicist and observational cosmologist. He is the William A. M. Burden Professor of Astrophysics at MIT.Schechter received his bachelor's degree from Cornell in 1968, and his Ph.D. degree from Caltech in 1975...
found a very interesting relation between the logarithm of a spectral line's
Spectral line
A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from a deficiency or excess of photons in a narrow frequency range, compared with the nearby frequencies.- Types of line spectra :...
line width and its apparent magnitude
Apparent magnitude
The apparent magnitude of a celestial body is a measure of its brightness as seen by an observer on Earth, adjusted to the value it would have in the absence of the atmosphere...
, when working with galaxies. In an perfect, stationary case, spectral lines should be incredibly narrow bumps, looking like lines, but motions of the object such as rotation or motion in our line of sight will cause shifts and broadening of these lines. The relation is found by starting with the Tully-Fisher relation
Tully-Fisher relation
In astronomy, the Tully–Fisher relation, published by astronomers R. Brent Tully and J. Richard Fisher in 1977, is an empirical relationship between the intrinsic luminosity of a spiral galaxy and its velocity width...
, wherein the distance to a galaxy
Galaxy
A galaxy is a massive, gravitationally bound system that consists of stars and stellar remnants, an interstellar medium of gas and dust, and an important but poorly understood component tentatively dubbed dark matter. The word galaxy is derived from the Greek galaxias , literally "milky", a...
is related to its apparent magnitude
Apparent magnitude
The apparent magnitude of a celestial body is a measure of its brightness as seen by an observer on Earth, adjusted to the value it would have in the absence of the atmosphere...
and its velocity width, or the 'maximum' speed of its rotation curve
Galaxy rotation curve
The rotation curve of a galaxy can be represented by a graph that plots the orbital velocity of the stars or gas in the galaxy on the y-axis against the distance from the center of the galaxy on the x-axis....
. From macroscopic Doppler broadening
Doppler broadening
In atomic physics, Doppler broadening is the broadening of spectral lines due to the Doppler effect caused by a distribution of velocities of atoms or molecules. Different velocities of the emitting particles result in different shifts, the cumulative effect of which is the line broadening.The...
, the logarithm of the line width of an observed spectral line can be related to the width of the velocity distribution. If the distances are assumed to be known very well, then the absolute magnitude
Absolute magnitude
Absolute magnitude is the measure of a celestial object's intrinsic brightness. it is also the apparent magnitude a star would have if it were 32.6 light years away from Earth...
and the line width are closely related. For example, working with the commonly used 21cm line
Hydrogen line
The hydrogen line, 21 centimeter line or HI line refers to the electromagnetic radiation spectral line that is created by a change in the energy state of neutral hydrogen atoms. This electromagnetic radiation is at the precise frequency of 1420.40575177 MHz, which is equivalent to the vacuum...
, an important line relating to neutral hydrogen, the relation is generally calibrated with a linear regression
Linear regression
In statistics, linear regression is an approach to modeling the relationship between a scalar variable y and one or more explanatory variables denoted X. The case of one explanatory variable is called simple regression...
and given the form
where P is log(line width) and α and β are constants.
The reason that this estimator is useful is that the inverse regression line is actually unaffected by the Malmquist Bias, so long as the selection effects are only based on magnitude. As such, the expected value of P given M will be unbiased and will give an unbiased log distance estimator. This estimator has many properties and ramifications which can make it a very useful tool.
Complex Mathematical Relations
Advanced versions of the traditional correction mentioned above can be found in the literature, limiting or changing the initial assumptions to suit the appropriate author's needs. Often, these other methods will provide very complicated mathematical expressions with very powerful but specific applications. For example, work by Luri et al. found a relation for the bias for stars in a galaxyGalaxy
A galaxy is a massive, gravitationally bound system that consists of stars and stellar remnants, an interstellar medium of gas and dust, and an important but poorly understood component tentatively dubbed dark matter. The word galaxy is derived from the Greek galaxias , literally "milky", a...
which relates the correction to the variance of the sample and the apparent magnitude
Apparent magnitude
The apparent magnitude of a celestial body is a measure of its brightness as seen by an observer on Earth, adjusted to the value it would have in the absence of the atmosphere...
, absolute magnitude
Absolute magnitude
Absolute magnitude is the measure of a celestial object's intrinsic brightness. it is also the apparent magnitude a star would have if it were 32.6 light years away from Earth...
, and the height above the galactic disk
Disc (galaxy)
A disc is a component of disc galaxies, such as spiral galaxies, or lenticular galaxies.The galactic disc is the plane in which the spirals, bars and discs of disc galaxies exist. Galaxy discs tend to have more gas and dust, and younger stars than galactic bulges, or galactic haloes.The galactic...
. This gave a much more exact and accurate result, but also required an assumption about the spatial distribution of stars in the desired galaxy
Galaxy
A galaxy is a massive, gravitationally bound system that consists of stars and stellar remnants, an interstellar medium of gas and dust, and an important but poorly understood component tentatively dubbed dark matter. The word galaxy is derived from the Greek galaxias , literally "milky", a...
. While useful individually, and there are many examples published, these have very limited scope and are not generally as broadly applicable as the other methods mentioned above.
Applications
Anytime a magnitude-limited sample is used, one of the methods described above should be used to correct for the Malmquist bias. For instance, when trying to obtain a luminosity functionLuminosity function
The luminosity function or luminous efficiency function describes the average visual sensitivity of the human eye to light of different wavelengths. It should not be considered perfectly accurate in every case, but it is a very good representation of visual sensitivity of the human eye and it is...
, calibrate the Tully-Fisher relation
Tully-Fisher relation
In astronomy, the Tully–Fisher relation, published by astronomers R. Brent Tully and J. Richard Fisher in 1977, is an empirical relationship between the intrinsic luminosity of a spiral galaxy and its velocity width...
, or obtain the value of the Hubble constant, the Malmquist bias can strongly change the results.
The luminosity function gives the number of stars or galaxies per luminosity or absolute magnitude bin. When using a magnitude-limited sample, the number of faint objects is underrepresented as discussed above. This shifts the peak of the luminosity function from the faint end to a brighter luminosity and changes the shape of the luminosity function. Typically, the volume weighting method is used to correct the Malmquist bias so that the survey is equivalent to a distance-limited survey rather than a magnitude-limited survey. The figure to the right shows two luminosity functions for an example population of stars that is magnitude-limited. The dashed luminosity function shows the effect of the Malmquist bias, while the solid line shows the corrected luminosity function. Malmquist bias drastically changes the shape of the luminosity function.
Another application that is affected by the Malmquist bias is the Tully-Fisher relation
Tully-Fisher relation
In astronomy, the Tully–Fisher relation, published by astronomers R. Brent Tully and J. Richard Fisher in 1977, is an empirical relationship between the intrinsic luminosity of a spiral galaxy and its velocity width...
, which relates the luminosity of spiral galaxies to their respective velocity width. If a nearby cluster of galaxies is used to calibrate the Tully-Fisher relation, and then that relation is applied to a distant cluster, the distance to the farther cluster will be systematically underestimated. By underestimating the distance to clusters, anything found using those clusters will be incorrect; for example, when finding the value of the Hubble constant.
These are just a few examples where the Malmquist bias can strongly affect results. As mentioned above, anytime a magnitude-limited sample is used, the Malmquist bias needs to be corrected for. A correction is not limited to just the examples above.
Alternatives
Some alternatives do exist to attempt to avoid the Malmquist bias, or to approach it in a different way, with a few of the more common ones summarized below.Distance Limited Sampling
One ideal method to avoid the Malmquist bias is to only select objects within a set distance, and have no limiting magnitudeLimiting magnitude
In astronomy, limiting magnitude is the faintest apparent magnitude of a celestial body that is detectable or detected by a given instrument....
but instead observe all objects within this volume. Clearly, in this case, the Malmquist bias is not an issue as the volume will be fully populated and any distribution
Distribution function
In molecular kinetic theory in physics, a particle's distribution function is a function of seven variables, f, which gives the number of particles per unit volume in phase space. It is the number of particles per unit volume having approximately the velocity near the place and time...
or luminosity function
Luminosity function
The luminosity function or luminous efficiency function describes the average visual sensitivity of the human eye to light of different wavelengths. It should not be considered perfectly accurate in every case, but it is a very good representation of visual sensitivity of the human eye and it is...
will be appropriately sampled. Unfortunately, this method is not always practical. Finding distances to astronomical objects is very difficult, and even with the aid of objects with easily determined distances, called standard candles
Standard Candles
Standard Candles is a compilation of short stories by American science fiction author Jack McDevitt. The sixteen stories in the anthology were originally published in various magazines from 1982 to 1996...
, and similar things, there are great uncertainties. Further, distances are not generally known for objects until after they have already been observed and analyzed, and so a distance limited survey is usually only an option for a second round of observations, and not initially available. Finally, distance limited surveys
Astronomical surveys
An astronomical survey is a general map or image of a region of the sky which lacks a specific observational target. Alternatively, an astronomical survey may comprise a set of many images of objects which share a common type or feature...
are generally only possible over small volumes where the distances are reliably known, and thus it is not practical for large surveys
Astronomical surveys
An astronomical survey is a general map or image of a region of the sky which lacks a specific observational target. Alternatively, an astronomical survey may comprise a set of many images of objects which share a common type or feature...
.
Homogeneous and Inhomogeneous Malmquist Correction
This method attempts to correct the biasBias (statistics)
A statistic is biased if it is calculated in such a way that it is systematically different from the population parameter of interest. The following lists some types of, or aspects of, bias which should not be considered mutually exclusive:...
again, but through very different means. Rather than trying to fix the absolute magnitudes
Absolute magnitude
Absolute magnitude is the measure of a celestial object's intrinsic brightness. it is also the apparent magnitude a star would have if it were 32.6 light years away from Earth...
, this method takes the distances to the objects as being the random variables and attempts to rescale those. In effect, rather than giving the stars in the sample the correct distribution of absolute magnitudes
Absolute magnitude
Absolute magnitude is the measure of a celestial object's intrinsic brightness. it is also the apparent magnitude a star would have if it were 32.6 light years away from Earth...
(and average absolute magnitude
Absolute magnitude
Absolute magnitude is the measure of a celestial object's intrinsic brightness. it is also the apparent magnitude a star would have if it were 32.6 light years away from Earth...
), it attempts to 'move' the stars such that they would have a correct distribution of distances. Ideally, this should have the same end result as the magnitude correction methods and should result in a correctly represented sample. In either the homogeneous or inhomogeneous case, the bias is defined in terms of a prior distribution of distances, the distance estimator, and the likelihood function
Likelihood function
In statistics, a likelihood function is a function of the parameters of a statistical model, defined as follows: the likelihood of a set of parameter values given some observed outcomes is equal to the probability of those observed outcomes given those parameter values...
of these two being the same distribution. The homogeneous case is much simpler and rescales the raw distance estimates by a constant factor. Unfortunately, this will be very insensitive to large scale structures
Observable universe
In Big Bang cosmology, the observable universe consists of the galaxies and other matter that we can in principle observe from Earth in the present day, because light from those objects has had time to reach us since the beginning of the cosmological expansion...
such as clustering as well as observational selection effects, and will not give a very accurate result. The inhomogeneous case attempts to correct this by creating a more complicated prior distribution of objects by taking into account structures seen in the observed distribution. In both cases though, it is assumed that the probability density function
Probability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...
is Gaussian with constant variance and a mean of the true average log distance, which is far from accurate. However, this method is debated and may not be accurate in any implementation due to uncertainties in calculating the raw, observed distance estimates causing the assumptions to use this method to be invalid.
Historical Alternatives
The term 'Malmquist bias' has not always been definitively used to refer to the bias outlined above. As recently as the year 2000, the Malmquist bias has appeared in the literature clearly referring to a different types of bias and statistical effect. The most common of these other uses is to refer to an effect that takes place with a magnitude limitedLimiting magnitude
In astronomy, limiting magnitude is the faintest apparent magnitude of a celestial body that is detectable or detected by a given instrument....
sample, but in this case the low absolute magnitude
Absolute magnitude
Absolute magnitude is the measure of a celestial object's intrinsic brightness. it is also the apparent magnitude a star would have if it were 32.6 light years away from Earth...
objects are overrepresented. In a sample with a magnitude limit
Limiting magnitude
In astronomy, limiting magnitude is the faintest apparent magnitude of a celestial body that is detectable or detected by a given instrument....
, there will be a margin of error near that boundary where objects that should be bright enough to make the cut are excluded and objects that are slightly below the limit are instead included. Since low absolute magnitude
Absolute magnitude
Absolute magnitude is the measure of a celestial object's intrinsic brightness. it is also the apparent magnitude a star would have if it were 32.6 light years away from Earth...
objects are more common than brighter ones, and since these dimmer galaxies are more likely to be below the cutoff line and scattered up, while the brighter ones are more likely to be above the line and scattered down, an over-representation of the lower luminosity
Luminosity
Luminosity is a measurement of brightness.-In photometry and color imaging:In photometry, luminosity is sometimes incorrectly used to refer to luminance, which is the density of luminous intensity in a given direction. The SI unit for luminance is candela per square metre.The luminosity function...
objects result. However, in modern day literature and consensus, the Malmquist bias refers to the effect outlined above.