* Light green boxes: Technique
applicable to star-forming galaxies. * Light blue boxes: Technique applicable
to Population II galaxies. * Light Purple boxes: Geometric distance technique.
* Light Red box: The planetary nebula luminosity function technique is
applicable to all populations of the Virgo Supercluster. * Solid black lines:
Well calibrated ladder step. * Dashed black lines: Uncertain calibration ladder
step.
The cosmic distance ladder
(also known as the Extragalactic Distance Scale) is the succession of methods
by which astronomers determine the distances to celestial objects. A real
direct distance measurement of an astronomical object is possible only for
those objects that are "close enough" (within about a thousand
parsecs) to Earth. The techniques for determining distances to more distant
objects are all based on various measured correlations between methods that
work at close distances with methods that work at larger distances. Several
methods rely on a standard candle, which is an astronomical object that has a
known luminosity.
The ladder analogy arises
because no one technique can measure distances at all ranges encountered in
astronomy. Instead, one method can be used to measure nearby distances , a
second can be used to measure nearby to intermediate distances, and so on. Each
rung of the ladder provides information that can be used to determine the
distances at the next higher rung.
Direct measurement
Statue of an astronomer and the concept of the cosmic
distance ladder by the parallax method, made from the azimuth ring and other
parts of the Yale-Columbia Refractor (telescope) (c 1925) wrecked by the 2003
Canberra bushfires which burned out the Mount Stromlo Observatory; at Questacon,
Canberra, Australian Capital Territory.
At the base of the ladder are fundamental distance
measurements, in which distances are determined directly, with no physical
assumptions about the nature of the object in question. The precise measurement
of stellar positions is part of the discipline of astrometry.
Astronomical unit
Direct distance measurements are based upon precise
determination of the distance between the Earth and the Sun, which is called
the Astronomical Unit (AU). Historically, observations of transits of Venus
were crucial in determining the AU; in the first half of the 20th Century,
observations of asteroids were also important. Presently the AU is determined
with high precision using radar measurements of Venus and other nearby planets
and asteroids,[1] and by tracking interplanetary spacecraft in their orbits
around the Sun through the Solar System. Kepler's Laws provide precise ratios
of the sizes of the orbits of objects revolving around the Sun, but not a real
measure of the orbits themselves. Radar provides a value in kilometers for the
difference in two orbits' sizes, and from that and the ratio of the two orbit
sizes, the size of Earth's orbit comes directly.
Parallax
The most important fundamental distance measurements come
from trigonometric parallax. As the Earth orbits around the Sun, the position
of nearby stars will appear to shift slightly against the more distant
background. These shifts are angles in a right triangle, with 2 AU making the
short leg of the triangle and the distance to the star being the long leg. The
amount of shift is quite small, measuring 1 arcseconds for an object at a
distance of 1 parsec (3.26 light-years), thereafter decreasing in angular amount
as the reciprocal of the distance. Astronomers usually express distances in
units of parsecs; light-years are used in popular media, but almost invariably
values in light-years have been converted from numbers tabulated in parsecs in
the original source.
Because parallax becomes smaller for a greater stellar
distance, useful distances can be measured only for stars whose parallax is
larger than the precision of the measurement. Parallax measurements typically
have an accuracy measured in milliarcseconds.[2] In the 1990s, for example, the
Hipparcos mission obtained parallaxes for over a hundred thousand stars with a
precision of about a milliarcsecond,[3] providing useful distances for stars
out to a few hundred parsecs.
Stars can have a velocity relative to the Sun that causes
proper motion and radial velocity. The former is determined by plotting the
changing position of the stars over many years, while the latter comes from
measuring the Doppler shift in their spectrum caused by motion along the line
of sight. For a group of stars with the same spectral class and a similar
magnitude range, a mean parallax can be derived from statistical analysis of
the proper motions relative to their radial velocities. This statistical
parallax method is useful for measuring the distances of bright stars beyond 50
parsecs and giant variable stars, including Cepheids and the RR Lyrae
variables.[4]
The motion of the Sun through space provides a longer
baseline that will increase the accuracy of parallax measurements, known as
secular parallax. For stars in the Milky Way disk, this corresponds to a mean
baseline of 4 A.U. per year, while for halo stars the baseline is 40 A.U. per
year. After several decades, the baseline can be orders of magnitude greater
than the Earth-Sun baseline used for traditional parallax. However, secular
parallax introduces a higher level of uncertainty because the relative velocity
of other stars is an additional unknown. When applied to samples of multiple
stars, the uncertainty can be reduced; the precision is inversely proportion to
the square root of the sample size.[5]
Moving cluster parallax is a technique where the motions
of individual stars in a nearby star cluster can be used to find the distance
to the cluster. Only open clusters are near enough for this technique to be
useful. In particular the distance obtained for the Hyades has been an
important step in the distance ladder.
Other individual objects can have fundamental distance
estimates made for them under special circumstances. If the expansion of a gas
cloud, like a supernova remnant or planetary nebula, can be observed over time,
then an expansion parallax distance to that cloud can be estimated. Binary
stars which are both visual and spectroscopic binaries also can have their distance
estimated by similar means. The common characteristic to these is that a
measurement of angular motion is combined with a measurement of the absolute
velocity (usually obtained via the Doppler effect). The distance estimate comes
from computing how far away the object must be to make its observed absolute
velocity appear with the observed angular motion.
Expansion parallaxes in particular can give fundamental
distance estimates for objects that are very far away, because supernova ejecta
have large expansion velocities and large sizes (compared to stars). Further,
they can be observed with radio interferometers which can measure very small
angular motions. These combine to mean that some supernovae in other galaxies
have fundamental distance estimates.[6] Though valuable, such cases are quite
rare, so they serve as important consistency checks on the distance ladder
rather than workhorse steps by themselves.
Standard candles
Almost all of the physical distance indicators are
standard candles. These are objects that belong to some class that have a known
brightness. By comparing the known luminosity of the latter to its observed
brightness, the distance to the object can be computed using the inverse square
law. These objects of known brightness are termed standard candles.
In astronomy, the brightness of an object is given in
terms of its absolute magnitude. This quantity is derived from the logarithm of
its luminosity as seen from a distance of 10 parsecs. The apparent magnitude,
or the magnitude as seen by the observer, can be used to determine the distance
D to the object in kiloparsecs (where 1 kpc equals 103 parsecs) as follows:
where m the apparent magnitude and M the absolute
magnitude. For this to be accurate, both magnitudes must be in the same
frequency band and there can be no relative motion in the radial direction.
Some means of accounting for interstellar extinction,
which also makes objects appear fainter and more red, is also needed. The
difference between absolute and apparent magnitudes is called the distance
modulus, and astronomical distances, especially intergalactic ones, are
sometimes tabulated in this way.
Problems
Two problems exist for any class of standard candle. The
principal one is calibration, determining exactly what the absolute magnitude
of the candle is. This includes defining the class well enough that members can
be recognized, and finding enough members with well-known distances that their
true absolute magnitude can be determined with enough accuracy. The second lies
in recognizing members of the class, and not mistakenly using the standard
candle calibration upon an object which does not belong to the class. At
extreme distances, which is where one most wishes to use a distance indicator,
this recognition problem can be quite serious.
A significant issue with standard candles is the
recurring question of how standard they are. For example, all observations seem
to indicate that type Ia supernovae that are of known distance have the same
brightness (corrected by the shape of the light curve). The basis for this
closeness in brightness is discussed below; however, the possibility that the
distant type Ia supernovae have different properties than nearby type Ia
supernovae exists. The use of Supernovae type Ia is crucial in determining the
correct cosmological model. If indeed the properties of Supernovae type Ia are
different at large distances, i.e. if the extrapolation of their calibration to
arbitrary distances is not valid, ignoring this variation can dangerously bias
the reconstruction of the cosmological parameters, in particular the
reconstruction of the matter density parameter.[7]
That this is not merely a philosophical issue can be seen
from the history of distance measurements using Cepheid variables. In the
1950s, Walter Baade discovered that the nearby Cepheid variables used to
calibrate the standard candle were of a different type than the ones used to
measure distances to nearby galaxies. The nearby Cepheid variables were population
I stars with much higher metal content than the distant population II stars. As
a result, the population II stars were actually much brighter than believed,
and this had the effect of doubling the distances to the globular clusters, the
nearby galaxies, and the diameter of the Milky Way.
(Another class of physical distance indicator is the
standard ruler. In 2008, galaxy diameters have been proposed as a possible
standard ruler for cosmological parameter determination.[8])
Galactic distance indicators
With few exceptions, distances based on direct
measurements are available only out to about a thousand parsecs, which is a
modest portion of our own Galaxy. For distances beyond that, measures depend
upon physical assumptions, that is, the assertion that one recognizes the
object in question, and the class of objects is homogeneous enough that its
members can be used for meaningful estimation of distance.
Physical distance indicators, used on progressively
larger distance scales, include:
• Dynamical
parallax, using orbital parameters of visual binaries to measure the mass of
the system and the mass-luminosity relation to determine the luminosity
o Eclipsing
binaries — In the last decade, measurement of eclipsing binaries' fundamental
parameters has become possible with 8 meter class telescopes. This makes it
feasible to use them as indicators of distance. Recently, they have been used
to give direct distance estimates to the LMC, SMC, Andromeda Galaxy and Triangulum
Galaxy. Eclipsing binaries offer a direct method to gauge the distance to
galaxies to a new improved 5% level of accuracy which is feasible with current
technology up to a distance of around 3 Mpc.[9]
• RR Lyrae
variables — red giants typically used for measuring distances within the galaxy
and in nearby globular clusters.
• The
following four indicators all use stars in the old stellar populations
(Population II):[10]
o Tip of
the red giant branch (TRGB) distance indicator.
o Planetary
nebula luminosity function (PNLF)
o Globular
cluster luminosity function (GCLF)
o Surface
brightness fluctuation (SBF)
• In
galactic astronomy, X-ray bursts (thermonuclear flashes on the surface of a
neutron star) are used as standard candles. Observations of X-ray burst
sometimes show X-ray spectra indicating radius expansion. Therefore, the X-ray
flux at the peak of the burst should correspond to Eddington luminosity, which
can be calculated once the mass of the neutron star is known (1.5 solar masses
is a commonly used assumption). This method allows distance determination of
some low-mass X-ray binaries. Low-mass X-ray binaries are very faint in the
optical, making measuring their distances extremely difficult.
• Cepheids
and novae
• Individual
galaxies in clusters of galaxies
• The
Tully-Fisher relation
• The
Faber-Jackson relation
• Type Ia
supernovae that have a very well-determined maximum absolute magnitude as a
function of the shape of their light curve and are useful in determining extragalactic
distances up to a few hundred Mpc.[11] A notable exception is SN 2003fg, the
"Champagne Supernova," a type Ia supernova of unusual nature.
• Redshifts
and Hubble's Law
Main sequence fitting
When the absolute magnitude for a group of stars is plotted
against the spectral classification of the star, in a Hertzsprung-Russell
diagram, evolutionary patterns are found that relate to the mass, age and
composition of the star. In particular, during their hydrogen burning period,
stars lie along a curve in the diagram called the main sequence. By measuring
these properties from a star's spectrum, the position of a main sequence star
on the H-R diagram can be determined, and thereby the star's absolute magnitude
estimated. A comparison of this value with the apparent magnitude allows the
approximate distance to be determined, after correcting for interstellar
extinction of the luminosity because of gas and dust.
In a gravitationally-bound star cluster such as the
Hyades, the stars formed at approximately the same age and lie at the same
distance. This allows relatively accurate main sequence fitting, providing both
age and distance determination.
Extragalactic distance scale
Extragalactic distance indicators[12]
Method Uncertainty
for Single Galaxy (mag) Distance to Virgo
Cluster (Mpc)
Range (Mpc)
Classical Cepheids 0.16 15 - 25 29
Novae 0.4 21.1 ± 3.9 20
Planetary Nebula Luminosity Function 0.3 15.4
± 1.1 50
Globular Cluster Luminosity Function 0.4 18.8
± 3.8 50
Surface Brightness Fluctuations 0.3 15.9 ± 0.9 50
D - σ relation 0.5 16.8 ± 2.4 > 100
Type Ia Supernovae 0.10 19.4 ± 5.0 >
1000
The extragalactic distance scale is a series of
techniques used today by astronomers to determine the distance of cosmological
bodies beyond our own galaxy, which are not easily obtained with traditional
methods. Some procedures utilize properties of these objects, such as stars,
globular clusters, nebulae, and galaxies as a whole. Other methods are based
more on the statistics and probabilities of things such as entire galaxy clusters.
Wilson-Bappu Effect
Discovered in 1956 by Olin Wilson and M.K. Vainu Bappu,
The Wilson-Bappu Effect utilizes the effect known as spectroscopic parallax.
Certain stars have features in their emission/absorption spectra allowing
relatively easy absolute magnitude calculation. Certain spectral lines are
directly related to an object's magnitude, such as the K absorption line of
calcium. Distance to the star can be calculated from magnitude by the distance
modulus:
Though in theory this method has the ability to provide
reliable distance calculations to stars roughly 7 megaparsecs (Mpc) away, it is
generally only used for stars hundreds of kiloparsecs (kpc) away.
This method is only valid for stars over 15 magnitudes.
Classical Cepheids
Beyond the reach of the Wilson-Bappu effect, the next
method relies on the period-luminosity relation of classical Cepheid variable
stars, first discovered by Henrietta Leavitt. The following relation can be
used to calculate the distance to Galactic and extragalactic classical
Cepheids:
Several problems complicate the use of Cepheids as
standard candles and are actively debated, chief among them are: the nature and
linearity of the period-luminosity relation in various passbands and the impact
of metallicity on both the zero-point and slope of those relations, and the
effects of photometric contamination (blending) and a changing (typically
unknown) extinction law on Cepheid
distances.
These unresolved matters have resulted in cited values
for the Hubble Constant ranging between 60 km/s/Mpc and 80 km/s/Mpc. Resolving
this discrepancy is one of the foremost problems in astronomy since the
cosmological parameters of the Universe may be constrained by supplying a
precise value of the Hubble constant.
Cepheid variable stars were the key instrument in Edwin
Hubble’s 1923 conclusion that M31 (Andromeda) was an external galaxy, as
opposed to a smaller nebula within the Milky Way. He was able to calculate the
distance of M31 to 285 Kpc, today’s value being 770 Kpc.
As detected thus far, NGC 3370, a spiral galaxy in the
constellation Leo, contains the farthest Cepheids yet found at a distance of 29
Mpc. Cepheid variable stars are in no way perfect distance markers: at nearby
galaxies they have an error of about 7% and up to a 15% error for the most
distant.
Supernovae
SN 1994D in
the NGC 4526 galaxy (bright spot on the lower left). Image by NASA, ESA, The
Hubble Key Project Team, and The High-Z Supernova Search Team
There are
several different methods for which supernovae can be used to measure
extragalactic distances, here we cover the most used.
Measuring a
supernova's photosphere
We can
assume that a supernova expands spherically symmetric. If the supernova is
close enough such that we can measure the angular extent, θ(t), of its photosphere,
we can use the equation
.
Where ω is
angular velocity, θ is angular extent. In order to get an accurate measurement,
it is necessary to make two observations separated by time Δt. Subsequently, we
can use
.
Where d is
the distance to the SN, Vej is the supernova's ejecta's radial velocity (it can
be assumed that Vej equals Vθ if spherically symmetric).
This method
works only if the supernova is close enough to be able to measure accurately
the photosphere. Similarly, the expanding shell of gas is in fact not perfectly
spherical nor a perfect blackbody. Also interstellar extinction can hinder the
accurate measurements of the photosphere. This problem is further exacerbated
by core-collapse supernova. All of these factors contribute to the distance
error of up to 25%.
Type Ia
light curves
Type Ia SN
are some of the best ways to determine extragalactic distances. Ia's occur when
a binary white dwarf star begins to accrete matter from its companion Red Dwarf
star. As the white dwarf gains matter, eventually it reaches its Chandrasekhar
Limit of . Once reached, the star
becomes unstable and undergoes a runaway nuclear fusion reaction. Because all
Type Ia SN explode at about the same mass, their absolute magnitudes are all
the same. This makes them very useful as standard candles. All type Ia SN have
a standard blue and visual magnitude of
Therefore,
when observing a type Ia SN, if it is possible to determine what its peak
magnitude was, then its distance can be calculated. It is not intrinsically
necessary to capture the SN directly at its peak magnitude; using the
multicolor light curve shape method (MLCS), the shape of the light curve (taken
at any reasonable time after the initial explosion) is compared to a family of
parameterized curves that will determine the absolute magnitude at the maximum
brightness. This method also takes into effect interstellar extinction/dimming
from dust and gas.
Similarly,
the stretch method fits the particular SN magnitude light curves to a template
light curve. This template, as opposed to being several light curves at
different wavelengths (MLCS) is just a single light curve that has been
stretched (or compressed) in time. By using this Stretch Factor, the peak
magnitude can be determined [citation].
Using Type
Ia SN is one of the most accurate methods, particularly since SN explosions can
be visible at great distances (their luminosities rival that of the galaxy in
which they are situated), much farther than Cepheid Variables (500 times
farther). Much time has been devoted to the refining of this method. The
current uncertainty approaches a mere 5%, corresponding to an uncertainty of
just 0.1 magnitudes.
Novae in
distance determinations
Novae can be
used in much the same way as supernovae to derive extragalactic distances.
There is a direct relation between a nova's max magnitude and the time for its
visible light to decline by two magnitudes. This relation is shown to be:
Where is the time derivative of the nova's mag,
describing the average rate of decline over the first 2 magnitudes.
After novae
fade, they are about as bright as the most luminous Cepheid Variable stars,
therefore both these techniques have about the same max distance: ~ 20 Mpc. The
error in this method produces an uncertainty in magnitude of about ± 0.4
Globular
cluster luminosity function
Based on the
method of comparing the luminosities of globular clusters (located in galactic
halos) from distant galaxies to that of the Virgo cluster, the globular cluster
luminosity function carries an uncertainty of distance of about 20% (or .4
magnitudes).
US
astronomer William Alvin Baum first attempted to use globular clusters to
measure distant elliptical galaxies. He compared the brightest globular
clusters in Virgo A galaxy with those in Andromeda, assuming the luminosities
of the clusters were the same in both. Knowing the distance to Andromeda, has
assumed a direct correlation and estimated Virgo A’s distance.
Baum used
just a single globular cluster, but individual formations are often poor
standard candles. Canadian astronomer Racine assumed the use of the globular
cluster luminosity function (GCLF) would lead to a better approximation. The
number of globular clusters as a function of magnitude given by:
Where m0 is
the turnover magnitude, and M0 the magnitude of the Virgo cluster, sigma the
dispersion ~ 1.4 mag.
It is
important to remember that it is assumed that globular clusters all have
roughly the same luminosities within the universe. There is no universal
globular cluster luminosity function that applies to all galaxies.
Planetary
nebula luminosity function
Like the
GCLF method, a similar numerical analysis can be used for planetary nebulae
(note the use of more than one!) within far off galaxies. The planetary nebula
luminosity function (PNLF) was first proposed in the late 1970s by Holland Cole
and David Jenner. They suggested that all planetary nebulae might all have
similar maximum intrinsic brightness, now calculated to be M = -4.53. This
would therefore make them potential standard candles for determining
extragalactic distances.
Astronomer
George Howard Jacoby and his fellow colleagues later proposed that the PNLF
function equaled:
Where N(M)
is number of planetary nebula, having absolute magnitude M. M* is equal to the
nebula with the brightest magnitude.
Surface
brightness fluctuation method
Galaxy cluster
The following method deals with the overall
inherent properties of galaxies. These methods, though with varying error
percentages, have the ability to make distance estimates beyond 100 Mpc, though
it is usually applied more locally.
The surface brightness fluctuation (SBF)
method takes advantage of the use of CCD cameras on telescopes. Because of
spatial fluctuations in a galaxy’s surface brightness, some pixels on these
cameras will pick up more stars than others. However, as distance increases the
picture will become increasingly smoother. Analysis of this describes a
magnitude of the pixel-to-pixel variation, which is directly related to a
galaxy’s distance.
D-σ Relation
The D- σ relation, used in elliptical
galaxies, relates the angular diameter (D) of the galaxy to its velocity
dispersion. It is important to describe exactly what D represents in order to
have a more fitting understanding of this method. It is, more precisely, the
galaxy’s angular diameter out to the surface brightness level of 20.75 B-mag
arcsec − 2. This surface brightness is independent of the galaxy’s actual
distance from us. Instead, D is inversely proportional to the galaxy’s
distance, represented as d. So instead of this relation imploring standard
candles, instead D provides a standard ruler. This relation between D and σ is
Where C is a constant which depends on the
distance to the galaxy clusters.
This method has the possibility of become
one of the strongest methods of galactic distance calculators, perhaps
exceeding the range of even the Tully-Fisher method. As of today, however,
elliptical galaxies aren’t bright enough to provide a calibration for this
method through the use of techniques such as Cepheids. So instead calibration
is done using more crude methods.
Overlap and scaling
A succession of distance indicators, which
is the distance ladder, is needed for determining distances to other galaxies.
The reason is that objects bright enough to be recognized and measured at such
distances are so rare that few or none are present nearby, so there are too few
examples close enough with reliable trigonometric parallax to calibrate the
indicator. For example, Cepheid variables, one of the best indicators for
nearby spiral galaxies, cannot be satisfactorily calibrated by parallax alone.
The situation is further complicated by the fact that different stellar
populations generally do not have all types of stars in them. Cepheids in
particular are massive stars, with short lifetimes, so they will only be found
in places where stars have very recently been formed. Consequently, because
elliptical galaxies usually have long ceased to have large-scale star
formation, they will not have Cepheids. Instead, distance indicators whose
origins are in an older stellar population (like novae and RR Lyrae variables)
must be used. However, RR Lyrae variables are less luminous than Cepheids (so
they cannot be seen as far away as Cepheids can), and novae are unpredictable
and an intensive monitoring program — and luck during that program — is needed
to gather enough novae in the target galaxy for a good distance estimate.
Because the more distant steps of the
cosmic distance ladder depend upon the nearer ones, the more distant steps
include the effects of errors in the nearer steps, both systematic and statistical
ones. The result of these propagating errors means that distances in astronomy
are rarely known to the same level of precision as measurements in the other
sciences, and that the precision necessarily is poorer for more distant types
of object.
Another concern, especially for the very
brightest standard candles, is their "standardness": how homogeneous
the objects are in their true absolute magnitude. For some of these different
standard candles, the homogeneity is based on theories about the formation and
evolution of stars and galaxies, and is thus also subject to uncertainties in
those aspects. For the most luminous of distance indicators, the Type Ia
supernovae, this homogeneity is known to be poor; however, no other class of
object is bright enough to be detected at such large distances, so the class is
useful simply because there is no real alternative.
The observational result of Hubble's Law,
the proportional relationship between distance and the speed with which a
galaxy is moving away from us (usually referred to as redshift) is a product of
the cosmic distance ladder. Hubble observed that fainter galaxies are more
redshifted. Finding the value of the Hubble constant was the result of decades
of work by many astronomers, both in amassing the measurements of galaxy
redshifts and in calibrating the steps of the distance ladder. Hubble's Law is
the primary means we have for estimating the distances of quasars and distant
galaxies in which individual distance indicators cannot be seen.
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