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Sunday, September 21, 2014

Redshift

From Wikipedia, the free encyclopedia

Absorption lines in the optical spectrum of a supercluster of distant galaxies (right), as compared to absorption lines in the optical spectrum of the Sun (left). Arrows indicate redshift. Wavelength increases up towards the red and beyond (frequency decreases).

In physics, redshift happens when light or other electromagnetic radiation from an object is increased in wavelength, or shifted to the red end of the spectrum. In general, whether or not the radiation is within the visible spectrum, "redder" means an increase in wavelength – equivalent to a lower frequency and a lower photon energy, in accordance with, respectively, the wave and quantum theories of light.

Some redshifts are an example of the Doppler effect, familiar in the change in the apparent pitches of sirens and frequency of the sound waves emitted by speeding vehicles. A redshift occurs whenever a light source moves away from an observer. Another kind of redshift is сosmological redshift, which is due to the expansion of the universe, and sufficiently distant light sources (generally more than a few million light years away) show redshift corresponding to the rate of increase in their distance from Earth. Finally, gravitational redshift is a relativistic effect observed in electromagnetic radiation moving out of gravitational fields. Conversely, a decrease in wavelength is called blueshift and is generally seen when a light-emitting object moves toward an observer or when electromagnetic radiation moves into a gravitational field.

Although knowledge of redshifts and blueshifts has been applied to develop several terrestrial technologies (such as Doppler radar and radar guns),^[1] redshifts are most famously seen in the spectroscopic observations of astronomical objects.^[2]

A special relativistic redshift formula (and its classical approximation) can be used to calculate the redshift of a nearby object when spacetime is flat. However, in many contexts, such as black holes and Big Bang cosmology, redshifts must be calculated using general relativity.^[3] Special relativistic, gravitational, and cosmological redshifts can be understood under the umbrella of frame transformation laws. There exist other physical processes that can lead to a shift in the frequency of electromagnetic radiation, including scattering and optical effects; however, the resulting changes are distinguishable from true redshift and are not generally referred to as such (see section on physical optics and radiative transfer).

History

The history of the subject began with the development in the 19th century of wave mechanics and the exploration of phenomena associated with the Doppler effect. The effect is named after Christian Doppler, who offered the first known physical explanation for the phenomenon in 1842.^[4] The hypothesis was tested and confirmed for sound waves by the Dutch scientist Christophorus Buys Ballot in 1845.^[5] Doppler correctly predicted that the phenomenon should apply to all waves, and in particular suggested that the varying colors of stars could be attributed to their motion with respect to the Earth.^[6] Before this was verified, however, it was found that stellar colors were primarily due to a star's temperature, not motion. Only later was Doppler vindicated by verified redshift observations.
The first Doppler redshift was described by French physicist Hippolyte Fizeau in 1848, who pointed to the shift in spectral lines seen in stars as being due to the Doppler effect. The effect is sometimes called the "Doppler–Fizeau effect". In 1868, British astronomer William Huggins was the first to determine the velocity of a star moving away from the Earth by this method.^[7] In 1871, optical redshift was confirmed when the phenomenon was observed in Fraunhofer lines using solar rotation, about 0.1 Å in the red.^[8] In 1887, Vogel and Scheiner discovered the annual Doppler effect, the yearly change in the Doppler shift of stars located near the ecliptic due to the orbital velocity of the Earth.^[9] In 1901, Aristarkh Belopolsky verified optical redshift in the laboratory using a system of rotating mirrors.^[10]

The earliest occurrence of the term "red-shift" in print (in this hyphenated form), appears to be by American astronomer Walter S. Adams in 1908, where he mentions "Two methods of investigating that nature of the nebular red-shift".^[11] The word does not appear unhyphenated until about 1934 by Willem de Sitter, perhaps indicating that up to that point its German equivalent, Rotverschiebung, was more commonly used.^[12]

Beginning with observations in 1912, Vesto Slipher discovered that most spiral galaxies, then mostly thought to be spiral nebulae, had considerable redshifts. Slipher first reports on his measurement in the inaugural volume of the Lowell Observatory Bulletin.^[13] Three years later, he wrote a review in the journal Popular Astronomy.^[14] In it he states, "[...] the early discovery that the great Andromeda spiral had the quite exceptional velocity of –300 km(/s) showed the means then available, capable of investigating not only the spectra of the spirals but their velocities as well."^[15] Slipher reported the velocities for 15 spiral nebulae spread across the entire celestial sphere, all but three having observable "positive" (that is recessional) velocities. Subsequently, Edwin Hubble discovered an approximate relationship between the redshifts of such "nebulae" and the distances to them with the formulation of his eponymous Hubble's law.^[16] These observations corroborated Alexander Friedmann's 1922 work, in which he derived the famous Friedmann equations.^[17] They are today considered strong evidence for an expanding universe and the Big Bang theory.^[18]

Measurement, characterization, and interpretation

High-redshift galaxy candidates in the Hubble Ultra Deep Field 2012.^[19]

The spectrum of light that comes from a single source (see idealized spectrum illustration top-right) can be measured. To determine the redshift, one searches for features in the spectrum such as absorption lines, emission lines, or other variations in light intensity. If found, these features can be compared with known features in the spectrum of various chemical compounds found in experiments where that compound is located on earth. A very common atomic element in space is hydrogen. The spectrum of originally featureless light shone through hydrogen will show a signature spectrum specific to hydrogen that has features at regular intervals. If restricted to absorption lines it would look similar to the illustration (top right). If the same pattern of intervals is seen in an observed spectrum from a distant source but occurring at shifted wavelengths, it can be identified as hydrogen too. If the same spectral line is identified in both spectra—but at different wavelengths—then the redshift can be calculated using the table below. Determining the redshift of an object in this way requires a frequency- or wavelength-range. In order to calculate the redshift one has to know the wavelength of the emitted light in the rest frame of the source, in other words, the wavelength that would be measured by an observer located adjacent to and comoving with the source. Since in astronomical applications this measurement cannot be done directly, because that would require travelling to the distant star of interest, the method using spectral lines described here is used instead. Redshifts cannot be calculated by looking at unidentified features whose rest-frame frequency is unknown, or with a spectrum that is featureless or white noise (random fluctuations in a spectrum).^[20]

Redshift (and blueshift) may be characterized by the relative difference between the observed and emitted wavelengths (or frequency) of an object. In astronomy, it is customary to refer to this change using a dimensionless quantity called z. If λ represents wavelength and f represents frequency (note, λf = c where c is the speed of light), then z is defined by the equations:^[21]

**Calculation of redshift, $z$**
Based on wavelength	Based on frequency
$z = \frac{\lambda_{\mathrm{obsv}} - \lambda_{\mathrm{emit}}}{\lambda_{\mathrm{emit}}}$	$z = \frac{f_{\mathrm{emit}} - f_{\mathrm{obsv}}}{f_{\mathrm{obsv}}}$
$1+z = \frac{\lambda_{\mathrm{obsv}}}{\lambda_{\mathrm{emit}}}$	$1+z = \frac{f_{\mathrm{emit}}}{f_{\mathrm{obsv}}}$

After z is measured, the distinction between redshift and blueshift is simply a matter of whether z is positive or negative. See the formula section below for some basic interpretations that follow when either a redshift or blueshift is observed. For example, Doppler effect blueshifts (z < 0) are associated with objects approaching (moving closer to) the observer with the light shifting to greater energies. Conversely, Doppler effect redshifts (z > 0) are associated with objects receding (moving away) from the observer with the light shifting to lower energies. Likewise, gravitational blueshifts are associated with light emitted from a source residing within a weaker gravitational field as observed from within a stronger gravitational field, while gravitational redshifting implies the opposite conditions.

Redshift formulae

In general relativity one can derive several important special-case formulae for redshift in certain special spacetime geometries, as summarized in the following table. In all cases the magnitude of the shift (the value of z) is independent of the wavelength.^[2]

**Redshift summary**
Redshift type	Geometry	Formula^[22]
Relativistic Doppler	Minkowski space (flat spacetime)	$1 + z = \gamma \left(1 + \frac{v_{\parallel}}{c}\right)$ $z \approx \frac{v_{\parallel}}{c}$ for small $v$ $1 + z = \sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}$ for motion completely in the radial direction. $1 + z=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$ for motion completely in the transverse direction.
Cosmological redshift	FLRW spacetime (expanding Big Bang universe)	$1 + z = \frac{a_{\mathrm{now}}}{a_{\mathrm{then}}}$
Gravitational redshift	any stationary spacetime (e.g. the Schwarzschild geometry)	$1 + z = \sqrt{\frac{g_{tt}(\text{receiver})}{g_{tt}(\text{source})}}$ (for the Schwarzschild geometry, $1 + z = \sqrt{\frac{1 - \frac{2GM}{ c^2 r_{\text{receiver}}}}{1 - \frac{2GM}{ c^2 r_{\text{source} }}}}$

Doppler effect

Doppler effect, yellow (~575 nm wavelength) ball appears greenish (blueshift to ~565 nm wavelength) approaching observer, turns orange (redshift to ~585 nm wavelength) as it passes, and returns to yellow when motion stops. To observe such a change in color, the object would have to be traveling at approximately 5200 km/s, or about 75 times faster than the speed record for the fastest manmade space probe.

If a source of the light is moving away from an observer, then redshift (z > 0) occurs; if the source moves towards the observer, then blueshift (z < 0) occurs. This is true for all electromagnetic waves and is explained by the Doppler effect. Consequently, this type of redshift is called the Doppler redshift. If the source moves away from the observer with velocity v, which is much less than the speed of light (

v \ll c

), the redshift is given by

z \approx \frac{v}{c}

(since

\gamma \approx 1

)

where c is the speed of light. In the classical Doppler effect, the frequency of the source is not modified, but the recessional motion causes the illusion of a lower frequency.

A more complete treatment of the Doppler redshift requires considering relativistic effects associated with motion of sources close to the speed of light. A complete derivation of the effect can be found in the article on the relativistic Doppler effect. In brief, objects moving close to the speed of light will experience deviations from the above formula due to the time dilation of special relativity which can be corrected for by introducing the Lorentz factor γ into the classical Doppler formula as follows:

1 + z = \left(1 + \frac{v}{c}\right) \gamma.

This phenomenon was first observed in a 1938 experiment performed by Herbert E. Ives and G.R. Stilwell, called the Ives–Stilwell experiment.^[23]

Since the Lorentz factor is dependent only on the magnitude of the velocity, this causes the redshift associated with the relativistic correction to be independent of the orientation of the source movement. In contrast, the classical part of the formula is dependent on the projection of the movement of the source into the line-of-sight which yields different results for different orientations. If θ is the angle between the direction of relative motion and the direction of emission in the observer's frame^[24] (zero angle is directly away from the observer), the full form for the relativistic Doppler effect becomes:

1+ z = \frac{1 + v \cos (\theta)/c}{\sqrt{1-v^2/c^2}}

and for motion solely in the line of sight (θ = 0°), this equation reduces to:

1 + z = \sqrt{\frac{1 + \frac{v}{c}}{1 - \frac{v}{c}}}

For the special case that the light is approaching at right angles (θ = 90°) to the direction of relative motion in the observer's frame,^[25] the relativistic redshift is known as the transverse redshift, and a redshift:

1 + z = \frac{1}{\sqrt{1-v^2/c^2}}

is measured, even though the object is not moving away from the observer. Even when the source is moving towards the observer, if there is a transverse component to the motion then there is some speed at which the dilation just cancels the expected blueshift and at higher speed the approaching source will be redshifted.^[26]

Expansion of space

In the early part of the twentieth century, Slipher, Hubble and others made the first measurements of the redshifts and blueshifts of galaxies beyond the Milky Way. They initially interpreted these redshifts and blueshifts as due solely to the Doppler effect, but later Hubble discovered a rough correlation between the increasing redshifts and the increasing distance of galaxies. Theorists almost immediately realized that these observations could be explained by a different mechanism for producing redshifts. Hubble's law of the correlation between redshifts and distances is required by models of cosmology derived from general relativity that have a metric expansion of space.^[18] As a result, photons propagating through the expanding space are stretched, creating the cosmological redshift.
There is a distinction between a redshift in cosmological context as compared to that witnessed when nearby objects exhibit a local Doppler-effect redshift. Rather than cosmological redshifts being a consequence of relative velocities, the photons instead increase in wavelength and redshift because of a feature of the spacetime through which they are traveling that causes space to expand.^[27] Due to the expansion increasing as distances increase, the distance between two remote galaxies can increase at more than 3×10⁸ m/s, but this does not imply that the galaxies move faster than the speed of light at their present location (which is forbidden by Lorentz covariance).

Mathematical derivation

The observational consequences of this effect can be derived using the equations from general relativity that describe a homogeneous and isotropic universe.

To derive the redshift effect, use the geodesic equation for a light wave, which is

ds^2=0=-c^2dt^2+\frac{a^2 dr^2}{1-kr^2}

where

$ds^2$ is the spacetime interval
$dt^2$ is the time interval
$dr^2$ is the spatial interval
$c$ is the speed of light
$a$ is the time-dependent cosmic scale factor
$k$ is the curvature per unit area.

For an observer observing the crest of a light wave at a position

r=0

and time

t=t_\mathrm{now}

, the crest of the light wave was emitted at a time

t=t_\mathrm{then}

in the past and a distant position

r=R

.
Integrating over the path in both space and time that the light wave travels yields:

c \int_{t_\mathrm{then}}^{t_\mathrm{now}} \frac{dt}{a}\; = \int_{R}^{0} \frac{dr}{\sqrt{1-kr^2}}\,.

In general, the wavelength of light is not the same for the two positions and times considered due to the changing properties of the metric. When the wave was emitted, it had a wavelength

\lambda_\mathrm{then}

. The next crest of the light wave was emitted at a time

t=t_\mathrm{then}+\lambda_\mathrm{then}/c\,.

The observer sees the next crest of the observed light wave with a wavelength

\lambda_\mathrm{now}

to arrive at a time

t=t_\mathrm{now}+\lambda_\mathrm{now}/c\,.

Since the subsequent crest is again emitted from

r=R

and is observed at

r=0

, the following equation can be written:

c \int_{t_\mathrm{then}+\lambda_\mathrm{then}/c}^{t_\mathrm{now}+\lambda_\mathrm{now}/c} \frac{dt}{a}\; = \int_{R}^{0} \frac{dr}{\sqrt{1-kr^2}}\,.

The right-hand side of the two integral equations above are identical which means

c \int_{t_\mathrm{then}+\lambda_\mathrm{then}/c}^{t_\mathrm{now}+\lambda_\mathrm{now}/c} \frac{dt}{a}\; = c \int_{t_\mathrm{then}}^{t_\mathrm{now}} \frac{dt}{a}\,

or, alternatively,

\int_{t_\mathrm{now}}^{t_\mathrm{now}+\lambda_\mathrm{now}/c} \frac{dt}{a}\; = \int_{t_\mathrm{then}}^{t_\mathrm{then}+\lambda_\mathrm{then}/c} \frac{dt}{a}\,.

For very small variations in time (over the period of one cycle of a light wave) the scale factor is essentially a constant (

a=a_\mathrm{now}

today and

a=a_\mathrm{then}

previously). This yields

\frac{t_\mathrm{now}+\lambda_\mathrm{now}/c}{a_\mathrm{now}}-\frac{t_\mathrm{now}}{a_\mathrm{now}}\; = \frac{t_\mathrm{then}+\lambda_\mathrm{then}/c}{a_\mathrm{then}}-\frac{t_\mathrm{then}}{a_\mathrm{then}}

which can be rewritten as

\frac{\lambda_\mathrm{now}}{\lambda_\mathrm{then}}=\frac{a_\mathrm{now}}{a_\mathrm{then}}\,.

Using the definition of redshift provided above, the equation

1+z = \frac{a_\mathrm{now}}{a_\mathrm{then}}

is obtained. In an expanding universe such as the one we inhabit, the scale factor is monotonically increasing as time passes, thus, z is positive and distant galaxies appear redshifted.

Using a model of the expansion of the universe, redshift can be related to the age of an observed object, the so-called cosmic time–redshift relation. Denote a density ratio as Ω₀:

\Omega_0 = \frac {\rho}{ \rho_{crit}} \ ,

with ρ_crit the critical density demarcating a universe that eventually crunches from one that simply expands. This density is about three hydrogen atoms per thousand liters of space.^[28] At large redshifts one finds:

t(z) = \frac {2}{3 H_0 {\Omega_0}^{1/2} (1+ z )^{3/2}} \ ,

where H₀ is the present-day Hubble constant, and z is the redshift.^[29]^[30]^[31]

Distinguishing between cosmological and local effects

For cosmological redshifts of z < 0.01 additional Doppler redshifts and blueshifts due to the peculiar motions of the galaxies relative to one another cause a wide scatter from the standard Hubble Law.^[32] The resulting situation can be illustrated by the Expanding Rubber Sheet Universe, a common cosmological analogy used to describe the expansion of space. If two objects are represented by ball bearings and spacetime by a stretching rubber sheet, the Doppler effect is caused by rolling the balls across the sheet to create peculiar motion. The cosmological redshift occurs when the ball bearings are stuck to the sheet and the sheet is stretched.^[33]^[34]^[35]

The redshifts of galaxies include both a component related to recessional velocity from expansion of the universe, and a component related to peculiar motion (Doppler shift).^[36] The redshift due to expansion of the universe depends upon the recessional velocity in a fashion determined by the cosmological model chosen to describe the expansion of the universe, which is very different from how Doppler redshift depends upon local velocity.^[37] Describing the cosmological expansion origin of redshift, cosmologist Edward Robert Harrison said, "Light leaves a galaxy, which is stationary in its local region of space, and is eventually received by observers who are stationary in their own local region of space. Between the galaxy and the observer, light travels through vast regions of expanding space. As a result, all wavelengths of the light are stretched by the expansion of space. It is as simple as that....^[38] Steven Weinberg clarified, "The increase of wavelength from emission to absorption of light does not depend on the rate of change of a(t) [here a(t) is the Robertson-Walker scale factor] at the times of emission or absorption, but on the increase of a(t) in the whole period from emission to absorption."^[39]

Popular literature often uses the expression "Doppler redshift" instead of "cosmological redshift" to describe the redshift of galaxies dominated by the expansion of spacetime, but the cosmological redshift is not found using the relativistic Doppler equation^[40] which is instead characterized by special relativity; thus v > c is impossible while, in contrast, v > c is possible for cosmological redshifts because the space which separates the objects (for example, a quasar from the Earth) can expand faster than the speed of light.^[41] More mathematically, the viewpoint that "distant galaxies are receding" and the viewpoint that "the space between galaxies is expanding" are related by changing coordinate systems. Expressing this precisely requires working with the mathematics of the Friedmann-Robertson-Walker metric.^[42]

If the universe were contracting instead of expanding, we would see distant galaxies blueshifted by an amount proportional to their distance instead of redshifted.^[43]

Gravitational redshift

In the theory of general relativity, there is time dilation within a gravitational well. This is known as the gravitational redshift or Einstein Shift.^[44] The theoretical derivation of this effect follows from the Schwarzschild solution of the Einstein equations which yields the following formula for redshift associated with a photon traveling in the gravitational field of an uncharged, nonrotating, spherically symmetric mass:

1+z=\frac{1}{\sqrt{1-\frac{2GM}{rc^2}}},

where

$G\,$ is the gravitational constant,
$M\,$ is the mass of the object creating the gravitational field,
$r\,$ is the radial coordinate of the source (which is analogous to the classical distance from the center of the object, but is actually a Schwarzschild coordinate), and
$c\,$ is the speed of light.

This gravitational redshift result can be derived from the assumptions of special relativity and the equivalence principle; the full theory of general relativity is not required.^[45]

The effect is very small but measurable on Earth using the Mössbauer effect and was first observed in the Pound–Rebka experiment.^[46] However, it is significant near a black hole, and as an object approaches the event horizon the red shift becomes infinite. It is also the dominant cause of large angular-scale temperature fluctuations in the cosmic microwave background radiation (see Sachs-Wolfe effect).^[47]^[48]

Observations in astronomy

The redshift observed in astronomy can be measured because the emission and absorption spectra for atoms are distinctive and well known, calibrated from spectroscopic experiments in laboratories on Earth. When the redshift of various absorption and emission lines from a single astronomical object is measured, z is found to be remarkably constant. Although distant objects may be slightly blurred and lines broadened, it is by no more than can be explained by thermal or mechanical motion of the source. For these reasons and others, the consensus among astronomers is that the redshifts they observe are due to some combination of the three established forms of Doppler-like redshifts.
Alternative hypotheses and explanations for redshift such as tired light are not generally considered plausible.^[49]

Spectroscopy, as a measurement, is considerably more difficult than simple photometry, which measures the brightness of astronomical objects through certain filters.^[50] When photometric data is all that is available (for example, the Hubble Deep Field and the Hubble Ultra Deep Field), astronomers rely on a technique for measuring photometric redshifts.^[51] Due to the broad wavelength ranges in photometric filters and the necessary assumptions about the nature of the spectrum at the light-source, errors for these sorts of measurements can range up to δz = 0.5, and are much less reliable than spectroscopic determinations.^[52] However, photometry does at least allow a qualitative characterization of a redshift. For example, if a sun-like spectrum had a redshift of z = 1, it would be brightest in the infrared rather than at the yellow-green color associated with the peak of its blackbody spectrum, and the light intensity will be reduced in the filter by a factor of four,

{(1+z)^2}

. Both the photon count rate and the photon energy are redshifted. (See K correction for more details on the photometric consequences of redshift.)^[53]

Local observations

A picture of the solar corona taken with the LASCO C1 coronagraph. The picture is a color-coded image of the doppler shift of the FeXIV 5308 Å line, caused by the coronal plasma velocity towards or away from the satellite.

In nearby objects (within our Milky Way galaxy) observed redshifts are almost always related to the line-of-sight velocities associated with the objects being observed. Observations of such redshifts and blueshifts have enabled astronomers to measure velocities and parametrize the masses of the orbiting stars in spectroscopic binaries, a method first employed in 1868 by British astronomer William Huggins.^[7] Similarly, small redshifts and blueshifts detected in the spectroscopic measurements of individual stars are one way astronomers have been able to diagnose and measure the presence and characteristics of planetary systems around other stars and have even made very detailed differential measurements of redshifts during planetary transits to determine precise orbital parameters.^[54] Finely detailed measurements of redshifts are used in helioseismology to determine the precise movements of the photosphere of the Sun.^[55] Redshifts have also been used to make the first measurements of the rotation rates of planets,^[56] velocities of interstellar clouds,^[57] the rotation of galaxies,^[2] and the dynamics of accretion onto neutron stars and black holes which exhibit both Doppler and gravitational redshifts.^[58] Additionally, the temperatures of various emitting and absorbing objects can be obtained by measuring Doppler broadening – effectively redshifts and blueshifts over a single emission or absorption line.^[59] By measuring the broadening and shifts of the 21-centimeter hydrogen line in different directions, astronomers have been able to measure the recessional velocities of interstellar gas, which in turn reveals the rotation curve of our Milky Way.^[2] Similar measurements have been performed on other galaxies, such as Andromeda.^[2] As a diagnostic tool, redshift measurements are one of the most important spectroscopic measurements made in astronomy.

Extragalactic observations

The most distant objects exhibit larger redshifts corresponding to the Hubble flow of the universe. The largest observed redshift, corresponding to the greatest distance and furthest back in time, is that of the cosmic microwave background radiation; the numerical value of its redshift is about z = 1089 (z = 0 corresponds to present time), and it shows the state of the Universe about 13.8 billion years ago,^[60] and 379,000 years after the initial moments of the Big Bang.^[61]

The luminous point-like cores of quasars were the first "high-redshift" (z > 0.1) objects discovered before the improvement of telescopes allowed for the discovery of other high-redshift galaxies.

For galaxies more distant than the Local Group and the nearby Virgo Cluster, but within a thousand megaparsecs or so, the redshift is approximately proportional to the galaxy's distance. This correlation was first observed by Edwin Hubble and has come to be known as Hubble's law. Vesto Slipher was the first to discover galactic redshifts, in about the year 1912, while Hubble correlated Slipher's measurements with distances he measured by other means to formulate his Law. In the widely accepted cosmological model based on general relativity, redshift is mainly a result of the expansion of space: this means that the farther away a galaxy is from us, the more the space has expanded in the time since the light left that galaxy, so the more the light has been stretched, the more redshifted the light is, and so the faster it appears to be moving away from us. Hubble's law follows in part from the Copernican principle.^[62] Because it is usually not known how luminous objects are, measuring the redshift is easier than more direct distance measurements, so redshift is sometimes in practice converted to a crude distance measurement using Hubble's law.

Gravitational interactions of galaxies with each other and clusters cause a significant scatter in the normal plot of the Hubble diagram. The peculiar velocities associated with galaxies superimpose a rough trace of the mass of virialized objects in the universe. This effect leads to such phenomena as nearby galaxies (such as the Andromeda Galaxy) exhibiting blueshifts as we fall towards a common barycenter, and redshift maps of clusters showing a Fingers of God effect due to the scatter of peculiar velocities in a roughly spherical distribution.^[62] This added component gives cosmologists a chance to measure the masses of objects independent of the mass to light ratio (the ratio of a galaxy's mass in solar masses to its brightness in solar luminosities), an important tool for measuring dark matter.^[63]

The Hubble law's linear relationship between distance and redshift assumes that the rate of expansion of the universe is constant. However, when the universe was much younger, the expansion rate, and thus the Hubble "constant", was larger than it is today. For more distant galaxies, then, whose light has been travelling to us for much longer times, the approximation of constant expansion rate fails, and the Hubble law becomes a non-linear integral relationship and dependent on the history of the expansion rate since the emission of the light from the galaxy in question. Observations of the redshift-distance relationship can be used, then, to determine the expansion history of the universe and thus the matter and energy content.

While it was long believed that the expansion rate has been continuously decreasing since the Big Bang, recent observations of the redshift-distance relationship using Type Ia supernovae have suggested that in comparatively recent times the expansion rate of the universe has begun to accelerate.

Highest redshifts

Plot of distance (in giga light-years) vs. redshift according to the Lambda-CDM model.

d_H

(in solid black) is the comoving distance from Earth to the location with the Hubble redshift z while

ct_{LB}

(in dotted red) is the speed of light multiplied by the lookback time to Hubble redshift z. The comoving distance is the physical space-like distance between here and the distant location, asymptoting to the size of the observable universe at some 47 billion light years. The lookback time is the distance a photon traveled from the time it was emitted to now divided by the speed of light, with a maximum distance of 13.8 billion light years corresponding to the age of the universe.

Currently, the objects with the highest known redshifts are galaxies and the objects producing gamma ray bursts. The most reliable redshifts are from spectroscopic data, and the highest confirmed spectroscopic redshift of a galaxy is that of UDFy-38135539 ^[64] at a redshift of

z=8.6

, corresponding to just 600 million years after the Big Bang. The previous record was held by IOK-1,^[65] at a redshift

z = 6.96

, corresponding to just 750 million years after the Big Bang. Slightly less reliable are Lyman-break redshifts, the highest of which is the lensed galaxy A1689-zD1 at a redshift

z = 7.6

^[66] and the next highest being

z=7.0

.^[67] The most distant observed gamma ray burst was GRB 090423, which had a redshift of

z = 8.2

.^[68] The most distant known quasar, ULAS J1120+0641, is at

z = 7.1

.^[69]^[70] The highest known redshift radio galaxy (TN J0924-2201) is at a redshift

z = 5.2

^[71] and the highest known redshift molecular material is the detection of emission from the CO molecule from the quasar SDSS J1148+5251 at

z = 6.42

^[72]

Extremely red objects (EROs) are astronomical sources of radiation that radiate energy in the red and near infrared part of the electromagnetic spectrum. These may be starburst galaxies that have a high redshift accompanied by reddening from intervening dust, or they could be highly redshifted elliptical galaxies with an older (and therefore redder) stellar population.^[73] Objects that are even redder than EROs are termed hyper extremely red objects (HEROs).^[74]

The cosmic microwave background has a redshift of

z=1089

, corresponding to an age of approximately 379,000 years after the Big Bang and a comoving distance of more than 46 billion light years.^[75] The yet-to-be-observed first light from the oldest Population III stars, not long after atoms first formed and the CMB ceased to be absorbed almost completely, may have redshifts in the range of

20<z<100

.^[76] Other high-redshift events predicted by physics but not presently observable are the cosmic neutrino background from about two seconds after the Big Bang (and a redshift in excess of

z>10^{10}

)^[77] and the cosmic gravitational wave background emitted directly from inflation at a redshift in excess of

z>10^{25}

.^[78]

Redshift surveys

Rendering of the 2dFGRS data

With advent of automated telescopes and improvements in spectroscopes, a number of collaborations have been made to map the universe in redshift space. By combining redshift with angular position data, a redshift survey maps the 3D distribution of matter within a field of the sky. These observations are used to measure properties of the large-scale structure of the universe. The Great Wall, a vast supercluster of galaxies over 500 million light-years wide, provides a dramatic example of a large-scale structure that redshift surveys can detect.^[79]

The first redshift survey was the CfA Redshift Survey, started in 1977 with the initial data collection completed in 1982.^[80] More recently, the 2dF Galaxy Redshift Survey determined the large-scale structure of one section of the Universe, measuring redshifts for over 220,000 galaxies; data collection was completed in 2002, and the final data set was released 30 June 2003.^[81] The Sloan Digital Sky Survey (SDSS), is ongoing as of 2013 and aims to measure the redshifts of around 3 million objects.^[82] SDSS has recorded redshifts for galaxies as high as 0.8, and has been involved in the detection of quasars beyond z = 6. The DEEP2 Redshift Survey uses the Keck telescopes with the new "DEIMOS" spectrograph; a follow-up to the pilot program DEEP1, DEEP2 is designed to measure faint galaxies with redshifts 0.7 and above, and it is therefore planned to provide a high redshift complement to SDSS and 2dF.^[83]

Effects due to physical optics or radiative transfer

The interactions and phenomena summarized in the subjects of radiative transfer and physical optics can result in shifts in the wavelength and frequency of electromagnetic radiation. In such cases the shifts correspond to a physical energy transfer to matter or other photons rather than being due to a transformation between reference frames. These shifts can be due to such physical phenomena as coherence effects or the scattering of electromagnetic radiation whether from charged elementary particles, from particulates, or from fluctuations of the index of refraction in a dielectric medium as occurs in the radio phenomenon of radio whistlers.^[2] While such phenomena are sometimes referred to as "redshifts" and "blueshifts", in astrophysics light-matter interactions that result in energy shifts in the radiation field are generally referred to as "reddening" rather than "redshifting" which, as a term, is normally reserved for the effects discussed above.^[2]

In many circumstances scattering causes radiation to redden because entropy results in the predominance of many low-energy photons over few high-energy ones (while conserving total energy).^[2] Except possibly under carefully controlled conditions, scattering does not produce the same relative change in wavelength across the whole spectrum; that is, any calculated z is generally a function of wavelength. Furthermore, scattering from random media generally occurs at many angles, and z is a function of the scattering angle. If multiple scattering occurs, or the scattering particles have relative motion, then there is generally distortion of spectral lines as well.^[2]

In interstellar astronomy, visible spectra can appear redder due to scattering processes in a phenomenon referred to as interstellar reddening^[2] – similarly Rayleigh scattering causes the atmospheric reddening of the Sun seen in the sunrise or sunset and causes the rest of the sky to have a blue color. This phenomenon is distinct from redshifting because the spectroscopic lines are not shifted to other wavelengths in reddened objects and there is an additional dimming and distortion associated with the phenomenon due to photons being scattered in and out of the line-of-sight.

Stellar structure

From Wikipedia, the free encyclopedia

This diagram shows a cross-section of the sun

Stars of different mass and age have varying internal structures. Stellar structure models describe the internal structure of a star in detail and make detailed predictions about the luminosity, the color and the future evolution of the star.

Energy transport

The different transport mechanisms of low-mass, intermediate-mass, and high-mass stars.

Different layers of the stars transport heat up and outwards in different ways, primarily convection and radiative transfer, but thermal conduction is important in white dwarfs.

Convection is the dominant mode of energy transport when the temperature gradient is steep enough so that a given parcel of gas within the star will continue to rise if it rises slightly via an adiabatic process. In this case, the rising parcel is buoyant and continues to rise if it is warmer than the surrounding gas; if the rising particle is cooler than the surrounding gas, it will fall back to its original height.^[1] In regions with a low temperature gradient and a low enough opacity to allow energy transport via radiation, radiation is the dominant mode of energy transport.

The internal structure of a main sequence star depends upon the mass of the star.

In solar mass stars (0.3–1.5 solar masses), including the Sun, hydrogen-to-helium fusion occurs primarily via proton-proton chains, which do not establish a steep temperature gradient. Thus, radiation dominates in the inner portion of solar mass stars. The outer portion of solar mass stars is cool enough that hydrogen is neutral and thus opaque to ultraviolet photons, so convection dominates. Therefore, solar mass stars have radiative cores with convective envelopes in the outer portion of the star.

In massive stars (greater than about 1.5 solar masses), the core temperature is above about 1.8×10⁷ K, so hydrogen-to-helium fusion occurs primarily via the CNO cycle. In the CNO cycle, the energy generation rate scales as the temperature to the 17th power, whereas the rate scales as the temperature to the 4th power in the proton-proton chains.^[2] Due to the strong temperature sensitivity of the CNO cycle, the temperature gradient in the inner portion of the star is steep enough to make the core convective. In the outer portion of the star, the temperature gradient is shallower but the temperature is high enough that the hydrogen is nearly fully ionized, so the star remains transparent to ultraviolet radiation. Thus, massive stars have a radiative envelope.

The lowest mass main sequence stars have no radiation zone; the dominant energy transport mechanism throughout the star is convection. Giants are also fully convective.^[3]

Equations of stellar structure

The simplest commonly used model of stellar structure is the spherically symmetric quasi-static model, which assumes that a star is in a steady state and that it is spherically symmetric. It contains four basic first-order differential equations: two represent how matter and pressure vary with radius; two represent how temperature and luminosity vary with radius.^[4]

In forming the stellar structure equations (exploiting the assumed spherical symmetry), one considers the matter density

\rho(r)

, temperature

T(r)

, total pressure (matter plus radiation)

P(r)

, luminosity

l(r)

, and energy generation rate per unit mass

\epsilon(r)

in a spherical shell of a thickness

\mbox{d}r

at a distance

r

from the center of the star. The star is assumed to be in local thermodynamic equilibrium (LTE) so the temperature is identical for matter and photons. Although LTE does not strictly hold because the temperature a given shell "sees" below itself is always hotter than the temperature above, this approximation is normally excellent because the photon mean free path,

\lambda

, is much smaller than the length over which the temperature varies considerably, i. e.

\lambda \ll T/|\nabla T|

.

First is a statement of hydrostatic equilibrium: the outward force due to the pressure gradient within the star is exactly balanced by the inward force due to gravity.

{\mbox{d} P \over \mbox{d} r} = - { G m \rho \over r^2 }

where

m(r)

is the cumulative mass inside the shell at

r

and G is the gravitational constant. The cumulative mass increases with radius according to the mass continuity equation:

{\mbox {d} m \over \mbox{d} r} = 4 \pi r^2 \rho .

Integrating the mass continuity equation from the star center (

r=0

) to the radius of the star (

r=R

) yields the total mass of the star.

Considering the energy leaving the spherical shell yields the energy equation:

{\mbox{d} l \over \mbox{d} r} = 4 \pi r^2 \rho ( \epsilon - \epsilon_\nu )

where

\epsilon_\nu

is the luminosity produced in the form of neutrinos (which usually escape the star without interacting with ordinary matter) per unit mass. Outside the core of the star, where nuclear reactions occur, no energy is generated, so the luminosity is constant.

The energy transport equation takes differing forms depending upon the mode of energy transport.
For conductive luminosity transport (appropriate for a white dwarf), the energy equation is

{\mbox{d} T \over \mbox{d} r} = - {1 \over k} { l \over 4 \pi r^2 },

where k is the thermal conductivity.

In the case of radiative energy transport, appropriate for the inner portion of a solar mass main sequence star and the outer envelope of a massive main sequence star,

{\mbox{d} T \over \mbox{d} r} = - {3 \kappa \rho l \over 64 \pi r^2 \sigma T^3},

where

\kappa

is the opacity of the matter,

\sigma

is the Stefan-Boltzmann constant, and the Boltzmann constant is set to one.

The case of convective luminosity transport (appropriate for non-radiative portions of main sequence stars and all of giants and low mass stars) does not have a known rigorous mathematical formulation, and involves turbulence in the gas. Convective energy transport is usually modeled using mixing length theory. This treats the gas in the star as containing discrete elements which roughly retain the temperature, density, and pressure of their surroundings but move through the star as far as a characteristic length, called the mixing length.^[5] For a monatomic ideal gas, when the convection is adiabatic, meaning that the convective gas bubbles don't exchange heat with their surroundings, mixing length theory yields

{\mbox{d} T \over \mbox{d} r} = \left(1 - {1 \over \gamma} \right) {T \over P } { \mbox{d} P \over \mbox{d} r},

where

\gamma = c_p / c_v

is the adiabatic index, the ratio of specific heats in the gas. (For a fully ionized ideal gas,

\gamma = 5/3

.) When the convection is not adiabatic, the true temperature gradient is not given by this equation. For example, in the Sun the convection at the base of the convection zone, near the core, is adiabatic but that near the surface is not. The mixing length theory contains two free parameters which must be set to make the model fit observations, so it is a phenomelogical theory rather than a rigorous mathematical formulation.^[6]

Also required are the equations of state, relating the pressure, opacity and energy generation rate to other local variables appropriate for the material, such as temperature, density, chemical composition, etc. Relevant equations of state for pressure may have to include the perfect gas law, radiation pressure, pressure due to degenerate electrons, etc. Opacity cannot be expressed exactly by a single formula. It is calculated for various compositions at specific densities and temperatures and presented in tabular form.^[7] Stellar structure codes (meaning computer programs calculating the model's variables) either interpolate in a density-temperature grid to obtain the opacity needed, or use a fitting function based on the tabulated values. A similar situation occurs for accurate calculations of the pressure equation of state. Finally, the nuclear energy generation rate is computed from particle physics experiments, using reaction networks to compute reaction rates for each individual reaction step and equilibrium abundances for each isotope in the gas.^[6]^[8]

Combined with a set of boundary conditions, a solution of these equations completely describes the behavior of the star. Typical boundary conditions set the values of the observable parameters appropriately at the surface (

r=R

) and center (

r=0

) of the star:

P(R) = 0

, meaning the pressure at the surface of the star is zero;

m(0) = 0

, there is no mass inside the center of the star, as required if the mass density remains finite;

m(R) = M

, the total mass of the star is the star's mass; and

T(R) = T_{eff}

, the temperature at the surface is the effective temperature of the star.

Although nowadays stellar evolution models describes the main features of color magnitude diagrams, important improvements have to be made in order to remove uncertainties which are linked to the limited knowledge of transport phenomena. The most difficult challenge remains the numerical treatment of turbulence. Some research teams are developing simplified modelling of turbulence in 3D calculations.

Rapid evolution

The above simplified model is not adequate without modification in situations when the composition changes are sufficiently rapid. The equation of hydrostatic equilibrium may need to be modified by adding a radial acceleration term if the radius of the star is changing very quickly, for example if the star is radially pulsating.^[9] Also, if the nuclear burning is not stable, or the star's core is rapidly collapsing, an entropy term must be added to the energy equation.^[10]

Team improves solar-cell efficiency

Sep 19, 2014 Original post: http://phys.org/news/2014-09-team-solar-cell-efficiency.html

This polymer solar cell consists of a new polymer, called PID2, which was developed in the laboratory of Luping Yu, professor in chemistry at the University of Chicago. The new polymer improves the efficiency of electrical power generation by …more

New light has been shed on solar power generation using devices made with polymers, thanks to a collaboration between scientists in the University of Chicago's chemistry department, the Institute for Molecular Engineering, and Argonne National Laboratory.

Researchers identified a new polymer—a type of large molecule that forms plastics and other familiar materials—which improved the efficiency of solar cells. The group also determined the method by which the polymer improved the cells' efficiency. The polymer allowed electrical charges to move more easily throughout the cell, boosting the production of electricity—a mechanism never before demonstrated in such devices.

"Polymer solar cells have great potential to provide low-cost, lightweight and flexible electronic devices to harvest solar energy," said Luyao Lu, graduate student in chemistry and lead author of a paper describing the result, published online last month in the journal Nature Photonics.

Solar cells made from polymers are a popular topic of research due to their appealing properties. But researchers are still struggling to efficiently generate electrical power with these materials.

"The field is rather immature—it's in the infancy stage," said Luping Yu, professor in chemistry, fellow in the Institute for Molecular Engineering, who led the UChicago group carrying out the research.

The active regions of such solar cells are composed of a mixture of polymers that give and receive electrons to generate electrical current when exposed to light. The new polymer developed by Yu's group, called PID2, improves the efficiency of electrical power generation by 15 percent when added to a standard polymer-fullerene mixture.

"Fullerene, a small carbon molecule, is one of the standard materials used in polymer solar cells," Lu said. "Basically, in polymer solar cells we have a polymer as electron donor and fullerene as electron acceptor to allow charge separation." In their work, the UChicago-Argonne researchers added another polymer into the device, resulting in solar cells with two polymers and one fullerene.

Luyao Lu, a graduate student in chemistry, works in the solar cell characterization facility of the University of Chicago's Gordon Center for Integrative Science. Lu is the lead author of a Nature Photonics article describing the development of a new type of polymer solar cell that displays enhanced power conversion efficiency. Credit: Andrew Nelles

Read more at: http://phys.org/news/2014-09-team-solar-cell-efficiency.html#jCp

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Sunday, September 21, 2014

Redshift

Redshift

History

Measurement, characterization, and interpretation

Redshift formulae

Doppler effect

Expansion of space

Mathematical derivation

Distinguishing between cosmological and local effects

Gravitational redshift

Observations in astronomy

Local observations

Extragalactic observations

Highest redshifts

Redshift surveys

Effects due to physical optics or radiative transfer

Stellar structure

Stellar structure

Energy transport

Equations of stellar structure

Rapid evolution

Team improves solar-cell efficiency

Team improves solar-cell efficiency

Sep 19, 2014 Original post: http://phys.org/news/2014-09-team-solar-cell-efficiency.html

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