Extragalactic distance scale

The extragalactic distance scale is a series of techniques used today by astronomers to determine the distance of cosmological bodies (beyond our own galaxy) 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. It is important to note that the following methods are for, as the name implies, extragalactic objects, so the more traditional and well known method of trigonometric parallax will not be covered as it is only accurate to around a kiloparsec or so.

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 which makes it relatively easy to calculate their absolute magnitudes. Certain spectral lines are directly related to an objects magnitude, such as the K absorption line of calcium. From there one can use the distance modulus

:$M\; -\; m\; =\; -\; 2.5\; log\_\{10\}(F\_1/F\_2)\; ,.$

to calculate the star’s distance. 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. It is also important to note that this method is only valid for stars over 15 magnitudes.

Cepheid Scale Distance

Beyond the reach of the Wilson-Bappu effect, the next method relies on the period-luminosity relation of Cepheid variable stars, first discovered by Henrietta Leavitt. The following Cepheid relations can be used to calculate the distance to Galactic and extragalactic Cepheids:

: $5log\_\{10\}\{d\}=V+\; (3.43)\; log\_\{10\}\{P\}\; -\; (2.58)\; (V-I)\; +\; 7.50\; ,.$: $5log\_\{10\}\{d\}=V+\; (3.30)\; log\_\{10\}\{P\}\; -\; (1.48)\; (V-J)\; +\; 7.63\; ,.$Majaess D. J., Turner D. G., Lane D. J. (2008). [http://arxiv.org/abs/0808.2937 "Assessing potential cluster Cepheids from a new distance and reddening parameterization and 2MASS photometry"] , MNRAS]

The use of Cepheid variable stars is not without its problems however. The largest source of error with Cepheids as standard candles is the possibility that the period-luminosity relation is affected by metallicity. For Galactic use only, the following relation is also valid in addition to those highlighted above:

: $5log\_\{10\}\{d\}=V+\; (4.42)\; log\_\{10\}\{P\}\; -\; (3.43)\; (B-V)\; +\; 7.15\; ,.$

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 as distance indicators

There are several different methods for which supernovae can be used to measure extragalactic distances, here we cover the most used.

Measuring SN's photosphere

We can assume that a SN expands spherically symmetric. If the SN is close enough such that we can measure the angular extent, θ(t), of its photosphere, we can use the equation

:$\{omega\}\; =\; frac$Delta}{ hetaDelta}{t ,..

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

:$d\; =\; frac\{V\_\{ej\{omega\}\; ,.$.

Where d is the distance to the SN, V_ej is the SN’s ejecta’s radial velocity (it can be assumed that V_ej equals V_θ if spherically symmetric.

This method works only if the SN 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 dwarfs gains matter, eventually it reaches its Chandrasekhar Limit of $1.4\; M\_\{odot\}$. 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 great standard candles. All Type Ia SN all have a standard blue and visual magnitude of

$M\_B\; approx\; M\_V\; approx\; -19.3\; pm\; 0.03\; ,.$

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 method (MCLS), 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 mag 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 (MCLS) 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.

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:

$M^\{max\}\_\{V\}\; =\; -9.96\; -\; 2.31\; log\_\{10\}\; dot\{x\}\; ,.$

Where $dot\{x\}$ 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:

$Phi\; (m)\; =\; A\; e^\{(m-m\_0)^2/2\{sigma\}^2\}\; ,.$

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 1970’s 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:

$N\; (M)\; propto\; e^\{0.307\; M\}\; (1\; -\; e^\{3(M^\{*\}\; -\; M)\}\; )\; ,.$

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

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

:$log\_\{10\}(D)\; =\; 1.333\; log\; (sigma)\; +\; C\; ,.$

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 techniques such as Cepheids. So instead calibration is done using more crude methods.

Summary

All the methods mentioned above are used today by astronomers for measuring objects beyond our own galaxy. Like all methods, different techniques and calibrations are used by other astronomers. The following table shows at-a-glance information for most of the methods mentioned above. It lists each error (in magnitudes), distance to the Virgo Cluster as calculated by each technique, and overall range of how far out each method can be used effectively.

ee also

* Cosmic distance ladder
* Standard candle

References

1) "An Introduction to Modern Astrophysics", Carroll and Ostlie, copyright 2007

2) "Measuring the Universe The Cosmological Distance Ladder", Stephen Webb, copyright 2001

3) "The Cosmos", Pasachoff and Filippenko, copyright 2007

4) "The Astrophysical Journal", "The Globular Cluster Luminosity Function as a Distance Indicator: Dynamical Effects", Ostriker and Gnedin, May 5 1997

External links

* [http://heasarc.gsfc.nasa.gov/docs/cosmic/ NASA Cosmic Distance Scale]
* [http://www.noao.edu/jacoby/pnlf/pnlf.html PNLF information database]
* [http://www.journals.uchicago.edu/toc/apj/current The Astrophysical Journal]