Newtonian motivations for general relativity

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Newtonian_motivations_for_general_relativity

Some of the basic concepts of general relativity can be outlined outside the relativistic domain. In particular, the idea that mass–energy generates curvature in space and that curvature affects the motion of masses can be illustrated in a Newtonian setting. We use circular orbits as our prototype. This has the advantage that we know the kinetics of circular orbits. This allows us to calculate curvature of orbits in space directly and compare the results with dynamical forces.

The equivalence of gravitational and inertial mass

A unique feature of the gravitational force is that all massive objects accelerate in the same manner in a gravitational field. This is often expressed as "The gravitational mass is equal to the inertial mass." This allows us to think of gravity as a curvature of spacetime.

Test for flatness in spacetime

If initially parallel paths of two particles on nearby geodesics remain parallel within some accuracy, then spacetime is flat to within that accuracy.

Two nearby particles in a radial gravitational field

Newtonian mechanics for circular orbits

Circular orbits at the same radius.

The geodesic and field equations for circular orbits

Consider the situation in which there are two particles in nearby circular polar orbits of the Earth at radius ${\displaystyle r}$ and speed ${\displaystyle v}$. Since the orbits are circular, the gravitational force on the particles must equal the centripetal force,

${\displaystyle \frac{v^2}{r}=\frac{GM}{r^2}}$

where G is the gravitational constant and ${\displaystyle M}$ is the mass of the earth.

The particles execute simple harmonic motion about the earth and with respect to each other. They are at their maximum distance from each other as they cross the equator. Their trajectories intersect at the poles.

From Newton's Law of Gravitation the separation vector ${\displaystyle \mathbf{h}}$ can be shown to be given by the "geodesic equation"

${\displaystyle \frac{d^2\mathbf{h}}{d{\tau }^2}+R\mathbf{h}=0}$

where ${\displaystyle R=\frac{1}{r^2}\frac{v^2}{c^2}}$ is the curvature of the trajectory and ${\displaystyle \tau =ct}$ is the speed of light c times the time.

The curvature of the trajectory is generated by the mass of the earth ${\displaystyle M}$. This is represented by the "field equation"

${\displaystyle R=\frac{GM}{r^3}}$

In this example, the field equation is simply a statement of the Newtonian concept that centripetal force is equal to gravitational force for circular orbits. We refer to this expression as a field equation in order to highlight the similarities with the Einstein field equation. This equation is in a much different form than Gauss's law, which is the usual characterization of the field equation in Newtonian mechanics.

The position of the moving particle with respect to the particle at rest in the co-moving reference frame.

Relationship between curvature and mass density

Mass can be written in terms of the average mass density ${\displaystyle \rho (r)}$ inside a sphere of radius ${\displaystyle r}$ by the expression

${\displaystyle M=\frac{4\pi \rho (r)r^3}{3}}$.

The field equation becomes

${\displaystyle R=\frac{4\pi G}{3}\rho (r)}$.

The curvature of the particle trajectories is proportional to mass density.

Local measurements

A requirement of General Relativity is that all measurements must be made locally. Imagine that the particles are inside a windowless spacecraft co-orbiting the Earth with the center of mass of the spacecraft coincident with one of the particles. That particle would be at rest with respect to the spacecraft. An observer in the spacecraft would have no indication that the craft was orbiting Earth. The observer is only allowed to measure the behavior of the particles in the frame of the craft.

In this example, we can define a local coordinate system such that the ${\displaystyle z}$-direction is toward the ceiling of the craft and this is directed along ${\displaystyle \mathbf{r}}$. The ${\displaystyle x}$-direction is toward the front of the craft and is in the direction of ${\displaystyle \mathbf{v}}$. The ${\displaystyle y}$-direction is toward the left side of the craft.

In this frame, the vector ${\displaystyle \mathbf{h}}$ is the position vector for the second particle. An observer in the craft would think the second particle was oscillating in a potential well generated by a gravitational field. This is an example of a coordinate acceleration due to the choice of frames as opposed to a physical acceleration due to actual forces.

General motion in the earth's gravitational field

Elliptic and hyberbolic trajectories

Co-planar elliptic orbits. The particle in the outer orbit travels slower than the particle in the inner orbit. They will separate with time.

More generally, particles move in elliptic or hyberbolic trajectories in a plane that contains the earth center. The orbits need not be circular. One can obtain intuitive geodesic and field equations in those situations as well. Unlike circular orbits, however, the speed of the particles in elliptic or hyperbolic trajectories is not constant. We therefore do not have a constant speed with which to scale the curvature. Therefore, in anticipation of the transition to relativistic mechanics, the trajectories and curvatures are scaled with the speed of light ${\displaystyle c}$.

From Newton's Law of Gravitation

${\displaystyle \frac{d^2\mathbf{r}}{d{t}^2}=-\frac{GM}{r^3}\mathbf{r}}$

one can obtain the geodesic equation for the separation of two particles in nearby trajectories

${\displaystyle \frac{d^2\mathbf{h}}{d{\tau }^2}+R\mathbf{h}=0}$

and the field equation

${\displaystyle R=R_{\perp }=\frac{GM}{c^2{r}^3}=\frac{4\pi G}{3c^2}\rho (r)}$

if the particle separation is perpendicular to ${\displaystyle \mathbf{r}}$ and

${\displaystyle R=R_{\Vert }=-\frac{2GM}{c^2{r}^3}=-\frac{8\pi G}{3c^2}\rho (r)}$

if the separation is parallel to ${\displaystyle \mathbf{r}}$. In the calculation of ${\displaystyle R_{\Vert }}$ the radius was expanded in terms of ${\displaystyle \mathbf{h}}$. Only the linear term was retained.

In the case that the separation of the particle is radial, the curvature is negative. This will cause the particles to separate rather than to be drawn toward each other as in the case in which they have the same radius. This is easy to understand. Outer orbits travel slower than inner orbits. This leads to particle separation.

Local coordinate system

Local "diagonal" coordinate system for an elliptic orbit.

A local coordinate system for a space craft co-moving with one of the particles can again be defined. The ${\displaystyle z}$-direction, toward the ceiling, is in the direction of ${\displaystyle \mathbf{r}}$. The ${\displaystyle x}$-direction, toward the front of the craft, is perpendicular to ${\displaystyle \mathbf{r}}$ but still in the plane of the trajectory. Unlike in a circular orbit, this craft no longer necessarily points in the direction of the velocity. The ${\displaystyle y}$-direction is toward the left side of the craft.

Tensor description

Simple diagonal frame

The geodesic equation in a radial gravitational field can be described succinctly in tensor notation in the co-moving frame in which the ceiling of the space craft is in the ${\displaystyle \hat{\mathbf{r}}}$ direction

${\displaystyle \frac{d^2{h}^i}{d{s}^2}+R_j^i{h}^j=0}$

where the Latin indices are over the spatial directions in the co-moving system, and we have used the Einstein summation convention in which repeated indices are summed. The curvature tensor ${\displaystyle R_j^i}$ is given by

${\displaystyle \Vert \begin{array}{ccc}{R}_1^1& R_1^2& R_1^3\\ R_2^1& R_2^2& R_2^3\\ R_3^1& R_3^2& R_3^3\end{array}\Vert =\Vert \begin{array}{ccc}{R}_{\perp }& 0& 0\\ 0& R_{\perp }& 0\\ 0& 0& R_{\Vert }\end{array}\Vert }$

and the separation vector is given by

${\displaystyle \Vert \begin{array}{ccc}{h}^1& h^2& h^3\end{array}\Vert =\Vert \begin{array}{ccc}\mathbf{h}\cdot \hat{\mathbf{x}}& \mathbf{h}\cdot \hat{\mathbf{y}}& \mathbf{h}\cdot \hat{\mathbf{z}}\end{array}\Vert }$

where ${\displaystyle \mathbf{h}\cdot \hat{\mathbf{x}}}$ is the component of ${\displaystyle \mathbf{h}}$ in the ${\displaystyle \hat{\mathbf{x}}}$ direction, ${\displaystyle \mathbf{h}\cdot \hat{\mathbf{y}}}$ is the component in the ${\displaystyle \hat{\mathbf{y}}}$ direction, and ${\displaystyle \mathbf{h}\cdot \hat{\mathbf{z}}}$ is the component in the ${\displaystyle \hat{\mathbf{z}}}$ direction.

In this co-moving coordinate system the curvature tensor is diagonal. This is not true in general.

Arbitrary orientation of the local frame

The co-moving spacecraft has no windows. An observer is not able to tell which direction is the ${\displaystyle \hat{\mathbf{r}}}$ direction, nor can he/she know which direction is the velocity with respect to earth. The orientation of the spacecraft may be quite different from the simple coordinate system in which the ceiling is in the ${\displaystyle \hat{\mathbf{r}}}$ direction and the front of the craft is in a direction coplanar with the radius and the velocity. We can transform our simple coordinates to an arbitrarily oriented coordinate system through rotations. This, however, destroys the diagonal nature of the curvature matrix.

Rotations are performed with a rotation matrix ${\displaystyle \mathcal{C}}$ such that the separation vector ${\displaystyle \overline{\mathbf{h}}}$ is related to the separation vector before rotation ${\displaystyle \mathbf{h}}$ by the relation

${\displaystyle {\overline{h}}^i={\mathcal{M}}_j^i{h}^j}$.

The inverse ${\displaystyle \overline{\mathcal{M}}}$ of ${\displaystyle \mathcal{M}}$ is defined by

${\displaystyle {\overline{\mathcal{M}}}_j^i{\mathcal{M}}_k^j={\delta }_k^i}$,

which yields

${\displaystyle h^i={\overline{\mathcal{M}}}_j^i{\overline{h}}^j}$.

Here ${\displaystyle {\delta }_k^i}$ is the Kronecker delta.

A simple rotation matrix that rotates the coordinate axis through an angle ${\displaystyle \theta }$ about the ${\displaystyle x}$-axis is

${\displaystyle \Vert \begin{array}{ccc}{\mathcal{M}}_1^1& {\mathcal{M}}_1^2& {\mathcal{M}}_1^3\\ {\mathcal{M}}_2^1& {\mathcal{M}}_2^2& {\mathcal{M}}_2^3\\ {\mathcal{M}}_3^1& {\mathcal{M}}_3^2& {\mathcal{M}}_3^3\end{array}\Vert =\Vert \begin{array}{ccc}1& 0& 0\\ 0& \cos (\theta )& \sin (\theta )\\ 0& -\sin (\theta )& \cos (\theta )\end{array}\Vert }$.

This is a rotation in the y-z plane. The inverse is obtained by switching the sign of ${\displaystyle \theta }$.

If the rotation matrix does not depend on time then the geodesic equation becomes, upon rotation

${\displaystyle \frac{d^2{\overline{h}}^i}{d{s}^2}+{\overline{R}}_j^i{\overline{h}}^j=0}$

where

${\displaystyle {\overline{R}}_j^i={\mathcal{M}}_k^i{R}_l^k{\overline{\mathcal{M}}}_j^l}$.

The curvature in the new coordinate system is non-diagonal. The inverse problem of transforming an arbitrary coordinate system into a diagonal system can be performed mathematically with the process of diagonalization.

Diagram 1. Changing views of spacetime along the world line of a rapidly accelerating observer. In this animation, the dashed line is the spacetime trajectory ("world line") of a particle. The balls are placed at regular intervals of proper time along the world line. The solid diagonal lines are the light cones for the observer's current event, and intersect at that event. The small dots are other arbitrary events in the spacetime. For the observer's current instantaneous inertial frame of reference, the vertical direction indicates the time and the horizontal direction indicates distance. The slope of the world line (deviation from being vertical) is the velocity of the particle on that section of the world line. So at a bend in the world line the particle is being accelerated. Note how the view of spacetime changes when the observer accelerates, changing the instantaneous inertial frame of reference. These changes are governed by the Lorentz transformations. Also note that:
• the balls on the world line before/after future/past accelerations are more spaced out due to time dilation.
• events which were simultaneous before an acceleration are at different times afterwards (due to the relativity of simultaneity),
• events pass through the light cone lines due to the progression of proper time, but not due to the change of views caused by the accelerations, and
• the world line always remains within the future and past light cones of the current event.

Time dependent rotation of the local frame: Christoffel symbols

The space craft may tumble about its center of mass. In that case the rotation matrix is time dependent. If the rotation matrix is time dependent, then it does not commute with the time derivative.

In that case, the rotation of the separation velocity can be written

${\displaystyle {\mathcal{M}}_i^k\frac{d{h}^i}{ds}={\mathcal{M}}_i^k\frac{d{\overline{\mathcal{M}}}_j^i{\overline{h}}^j}{ds}}$

which becomes

${\displaystyle \frac{d{\overline{h}}^k}{ds}+{\mathrm{\Gamma }}_j^k{\overline{h}}^j\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{D{\overline{h}}^k}{Ds}}$

where

${\displaystyle {\mathrm{\Gamma }}_j^k\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{\mathcal{M}}_i^k\frac{d{\overline{\mathcal{M}}}_j^i}{ds}}$

is known as a Christoffel symbol.

The geodesic equation becomes

${\displaystyle \frac{D^2{\overline{h}}^i}{D{s}^2}+{\overline{R}}_j^i{\overline{h}}^j=0}$,

which is the same as before with the exception that the derivatives have been generalized.

Arbitrariness in the curvature

The velocity in the frame of the spacecraft can be written

${\displaystyle {\overline{u}}^i\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{D{\overline{h}}^i}{Ds}}$.

The geodesic equation becomes

${\displaystyle \frac{d{\overline{u}}^i}{ds}+{\mathrm{\Gamma }}_j^i{\overline{u}}^j+{\overline{R}}_j^i{\overline{h}}^j=0}$.

${\displaystyle \frac{d^2{\overline{h}}^i}{d{s}^2}+2{\mathrm{\Gamma }}_j^i\frac{d{\overline{h}}^i}{ds}+\frac{d{\mathrm{\Gamma }}_j^i}{ds}{\overline{h}}^j+{\mathrm{\Gamma }}_j^i{\mathrm{\Gamma }}_k^j{\overline{h}}^k+{\overline{R}}_j^i{\overline{h}}^j=0}$.

In an arbitrarily rotating spacecraft, the curvature of space is due to two terms, one due to the mass density and one due to the arbitrary rotation of the spacecraft. The arbitrary rotation is non-physical and must be eliminated in any real physical theory of gravitation. In General Relativity this is done with a process called Fermi–Walker transport. In a Euclidean sense, Fermi–Walker transport is simply a statement that the spacecraft is not allowed to tumble

${\displaystyle {\mathrm{\Gamma }}_j^i=0}$

for all i and j. The only time-dependent rotations allowed are those generated by the mass density.

General geodesic and field equations in a Newtonian setting

Geodesic equation

${\displaystyle \frac{D^2{\overline{h}}^i}{D{s}^2}+{\overline{R}}_j^i{\overline{h}}^j=0}$

where

${\displaystyle \frac{D}{Ds}\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{d}{ds}+\mathrm{\Gamma }}$

and ${\displaystyle \mathrm{\Gamma }}$ is a Christoffel symbol.

Field equation

${\displaystyle {\overline{R}}_j^i={\mathcal{M}}_k^i{R}_l^k{\overline{\mathcal{M}}}_j^l}$

where ${\displaystyle {\mathcal{M}}_j^i}$ is a rotation matrix and the curvature tensor is

${\displaystyle \Vert \begin{array}{ccc}{R}_1^1& R_1^2& R_1^3\\ R_2^1& R_2^2& R_2^3\\ R_3^1& R_3^2& R_3^3\end{array}\Vert =\Vert \begin{array}{ccc}{R}_{\perp }& 0& 0\\ 0& R_{\perp }& 0\\ 0& 0& R_{\Vert }\end{array}\Vert }$.

The curvature is proportional to the mass density

${\displaystyle R_{\perp }=\frac{4\pi G}{3c^2}\rho (r)}$

${\displaystyle R_{\Vert }=-\frac{8\pi G}{3c^2}\rho (r)}$.

Overview of the Newtonian picture

The geodesic and field equations simply are a restatement of Newton's Law of Gravitation as seen from a local frame of reference co-moving with the mass within the local frame. This picture contains many of the elements of General Relativity, including the concept that particles travel along geodesics in a curved space (spacetime in the relativistic case) and that the curvature is due to the presence of mass density (mass/energy density in the relativistic case). This picture also contains some of the mathematical machinery of General Relativity such as tensors, Christoffel symbols, and Fermi–Walker transport.

Relativistic generalization

Main article: Theoretical motivation for general relativity

World line of a circular orbit about the Earth depicted in two spatial dimensions X and Y (the plane of the orbit) and a time dimension, usually put as the vertical axis. Note that the orbit about the Earth is (almost) a circle in space, but its worldline is a helix in spacetime.

General relativity generalizes the geodesic equation and the field equation to the relativistic realm in which trajectories in space are replaced with world lines in spacetime. The equations are also generalized to more complicated curvatures.

Astrophotography

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Astrophotography 

An image of Orion's Belt composited from digitized black-and-white photographic plates recorded through red and blue astronomical filters, with a computer synthesized green channel. The plates were taken using the Samuel Oschin Telescope between 1987 and 1991.

Astrophotography, also known as astronomical imaging, is the photography or imaging of astronomical objects, celestial events, or areas of the night sky. The first photograph of an astronomical object (the Moon) was taken in 1840, but it was not until the late 19th century that advances in technology allowed for detailed stellar photography. Besides being able to record the details of extended objects such as the Moon, Sun, and planets, modern astrophotography has the ability to image objects outside of the visible spectrum of the human eye such as dim stars, nebulae, and galaxies. This is accomplished through long time exposure as both film and digital cameras can accumulate and sum photons over long periods of time or using specialized optical filters which limit the photons to a certain wavelength.

Photography using extended exposure-times revolutionized the field of professional astronomical research, recording hundreds of thousands of new stars, and nebulae invisible to the human eye. Specialized and ever-larger optical telescopes were constructed as essentially big cameras to record images on photographic plates. Astrophotography had an early role in sky surveys and star classification but over time it has given way to more sophisticated equipment and techniques designed for specific fields of scientific research, with image sensors becoming just one of many forms of sensor.

Today, astrophotography is mostly a subdiscipline in amateur astronomy, usually seeking aesthetically pleasing images rather than scientific data. Amateurs use a wide range of special equipment and techniques.

Overview

The large 48" Oschin Schmidt Camera at Palomar Observatory

With a few exceptions, astronomical photography employs long exposures since both film and digital imaging devices can accumulate light photons over long periods of time. The amount of light hitting the film or detector is also increased by increasing the diameter of the primary optics (the objective) being used. Urban areas produce light pollution so equipment and observatories doing astronomical imaging are often located in remote locations to allow long exposures without the film or detectors being swamped with stray light.

Since the Earth is constantly rotating, telescopes and equipment are rotated in the opposite direction to follow the apparent motion of the stars overhead (called diurnal motion). This is accomplished by using either equatorial or computer-controlled altazimuth telescope mounts to keep celestial objects centered while the earth rotates. All telescope mount systems suffer from induced tracking errors due to imperfect motor drives, the mechanical sag of the telescope, and atmospheric refraction. Tracking errors are corrected by keeping a selected aiming point, usually a guide star, centered during the entire exposure. Sometimes (as in the case of comets) the object to be imaged is moving, so the telescope has to be kept constantly centered on that object. This guiding is done through a second co-mounted telescope called a "guide scope" or via some type of "off-axis guider", a device with a prism or optical beam splitter that allows the observer to view the same image in the telescope that is taking the picture. Guiding was formerly done manually throughout the exposure with an observer standing at (or riding inside) the telescope making corrections to keep a cross hair on the guide star. Since the advent of computer-controlled systems, this is accomplished by an automated system in professional and even amateur equipment.

Astronomical photography was one of the earliest types of scientific photography and almost from its inception it diversified into subdisciplines that each have a specific goal including star cartography, astrometry, stellar classification, photometry, spectroscopy, polarimetry, and the discovery of astronomical objects such as asteroids, meteors, comets, variable stars, novae, and even unknown planets. These often require specialized equipment such as telescopes designed for precise imaging, for wide field of view (such as Schmidt cameras), or for work at specific wavelengths of light. Astronomical CCD cameras may cool the sensor to reduce thermal noise and to allow the detector to record images in other spectra such as in infrared astronomy. Specialized filters are also used to record images in specific wavelengths.

History

Henry Draper with a refractor telescope set up for photography (photo probably taken in the 1860s or early 1870).

The development of astrophotography as a scientific tool was pioneered in the mid-19th century for the most part by experimenters and amateur astronomers, or so-called "gentleman scientists" (although, as in other scientific fields, these were not always men). Because of the very long exposures needed to capture relatively faint astronomical objects, many technological problems had to be overcome. These included making telescopes rigid enough so they would not sag out of focus during the exposure, building clock drives that could rotate the telescope mount at a constant rate, and developing ways to accurately keep a telescope aimed at a fixed point over a long period of time. Early photographic processes also had limitations. The daguerreotype process was far too slow to record anything but the brightest objects, and the wet plate collodion process limited exposures to the time the plate could stay wet.

The earliest surviving dagerrotype of the Moon by Draper (1840)

The first known attempt at astronomical photography was by Louis Jacques Mandé Daguerre, inventor of the daguerreotype process which bears his name, who attempted in 1839 to photograph the Moon. Tracking errors in guiding the telescope during the long exposure meant the photograph came out as an indistinct fuzzy spot. John William Draper, New York University Professor of Chemistry, physician and scientific experimenter managed to make the first successful photograph of the moon a year later on March 23, 1840, taking a 20-minute-long daguerreotype image using a 5-inch (13 cm) reflecting telescope.

The Sun may have been first photographed in an 1845 daguerreotype by the French physicists Léon Foucault and Hippolyte Fizeau. A failed attempt to obtain a photograph of a Total Eclipse of the Sun was made by the Italian physicist, Gian Alessandro Majocchi during an eclipse of the Sun that took place in his home city of Milan, on July 8, 1842. He later gave an account of his attempt and the Daguerreotype photographs he obtained, in which he wrote:

A few minutes before and after totality an iodized plate was exposed in a camera to the light of the thin crescent, and a distinct image was obtained, but another plate exposed to the light of the corona for two minutes during totality did not show the slightest trace of photographic action. No photographic alteration was caused by the light of the corona condensed by a lens for two minutes, during totality, on a sheet of paper prepared with bromide of silver.

The first solar eclipse photograph was taken on July 28, 1851, by a daguerrotypist named Berkowski.

The Sun's solar corona was first successfully imaged during the Solar eclipse of July 28, 1851. Dr. August Ludwig Busch, the Director of the Königsberg Observatory gave instructions for a local daguerreotypist named Johann Julius Friedrich Berkowski to image the eclipse. Busch himself was not present at Königsberg (now Kaliningrad, Russia), but preferred to observe the eclipse from nearby Rixhoft. The telescope used by Berkowski was attached to 6+1⁄2-inch (17 cm) Königsberg heliometer and had an aperture of only 2.4 in (6.1 cm), and a focal length of 32 in (81 cm). Commencing immediately after the beginning of totality, Berkowski exposed a daguerreotype plate for 84 seconds in the focus of the telescope, and on developing an image of the corona was obtained. He also exposed a second plate for about 40 to 45 seconds but was spoiled when the sun broke out from behind the moon. More detailed photographic studies of the Sun were made by the British astronomer Warren De la Rue starting in 1861.

The first photograph of a star was a daguerreotype of the star Vega by astronomer William Cranch Bond and daguerreotype photographer and experimenter John Adams Whipple, on July 16 and 17, 1850 with Harvard College Observatory's 15 inch Great refractor. In 1863 the English chemist William Allen Miller and English amateur astronomer Sir William Huggins used the wet collodion plate process to obtain the first ever photographic spectrogram of a star, Sirius and Capella. In 1872 American physician Henry Draper, the son of John William Draper, recorded the first spectrogram of a star (Vega) to show absorption lines.

Henry Draper's 1880 photograph of the Orion Nebula, the first ever taken.

One of Andrew Ainslie Common's 1883 photographs of the same nebula, the first to show that a long exposure could record stars and nebulae invisible to the human eye.

Astronomical photography did not become a serious research tool until the late 19th century, with the introduction of dry plate photography. It was first used by Sir William Huggins and his wife Margaret Lindsay Huggins, in 1876, in their work to record the spectra of astronomical objects. In 1880, Henry Draper used the new dry plate process with photographically corrected 11 in (28 cm) refracting telescope made by Alvan Clark to make a 51-minute exposure of the Orion Nebula, the first photograph of a nebula ever made. A breakthrough in astronomical photography came in 1883, when amateur astronomer Andrew Ainslie Common used the dry plate process to record several images of the same nebula in exposures up to 60 minutes with a 36 in (91 cm) reflecting telescope that he constructed in the backyard of his home in Ealing, outside London. These images for the first time showed stars too faint to be seen by the human eye.

The first all-sky photographic astrometry project, Astrographic Catalogue and Carte du Ciel, was started in 1887. It was conducted by 20 observatories all using special photographic telescopes with a uniform design called normal astrographs, all with an aperture of around 13 in (330 mm) and a focal length of 11 ft (3.4 m), designed to create images with a uniform scale on the photographic plate of approximately 60 arcsecs/mm while covering a 2° × 2° field of view. The attempt was to accurately map the sky down to the 14th magnitude but it was never completed.

The beginning of the 20th century saw the worldwide construction of refracting telescopes and sophisticated large reflecting telescopes specifically designed for photographic imaging. Towards the middle of the century, giant telescopes such as the 200 in (5.1 m) Hale Telescope and the 48 in (120 cm) Samuel Oschin telescope at Palomar Observatory were pushing the limits of film photography.

Some progress was made in the field of photographic emulsions and in the techniques of forming gas hypersensitization, cryogenic cooling, and light amplification, but starting in the 1970s after the invention of the CCD, photographic plates were gradually replaced by electronic imaging in professional and amateur observatories. CCD's are far more light sensitive, do not drop off in sensitivity over long exposures the way film does ("reciprocity failure"), have the ability to record in a much wider spectral range, and simplify storage of information. Telescopes now use many configurations of CCD sensors including linear arrays and large mosaics of CCD elements equivalent to 100 million pixels, designed to cover the focal plane of telescopes that formerly used 10–14-inch (25–36 cm) photographic plates.

The Hubble Space Telescope shortly after the STS-125 maintenance mission in 2009.

The late 20th century saw advances in astronomical imaging take place in the form of new hardware, with the construction of giant multi-mirror and segmented mirror telescopes. It would also see the introduction of space-based telescopes, such as the Hubble Space Telescope. Operating outside the atmosphere's turbulence, scattered ambient light and the vagaries of weather allows the Hubble Space Telescope, with a mirror diameter of 2.4 metres (94 in), to record stars down to the 30th magnitude, some 100 times dimmer than what the 5-meter Mount Palomar Hale telescope could record in 1949.

Amateur astrophotography

For broader coverage of this topic, see Amateur astronomy.

2 minute time exposure of the comet Hale-Bopp imaged using a camera on a fixed tripod. The tree in the foreground was illuminated using a small flashlight.

Astrophotography is a popular hobby among photographers and amateur astronomers. Techniques ranges from basic film and digital cameras on tripods up to methods and equipment geared toward advanced imaging. Amateur astronomers and amateur telescope makers also use homemade equipment and modified devices.

Media

Images are recorded on many types of media and imaging devices including single-lens reflex cameras, 35 mm film, 120 film, digital single-lens reflex cameras, simple amateur-level, and professional-level commercially manufactured astronomical CCD and CMOS cameras, video cameras, and even off-the-shelf webcams used for Lucky imaging.

The conventional over-the-counter film has long been used for astrophotography. Film exposures range from seconds to over an hour. Commercially available color film stock is subject to reciprocity failure over long exposures, in which sensitivity to light of different wavelengths appears to drop off at different rates as the exposure time increases, leading to a color shift in the image and reduced sensitivity over all as a function of time. This is compensated for, or at least reduced, by cooling the film (see Cold camera photography). This can also be compensated for by using the same technique used in professional astronomy of taking photographs at different wavelengths that are then combined to create a correct color image. Since the film is much slower than digital sensors, tiny errors in tracking can be corrected without much noticeable effect on the final image. Film astrophotography is becoming less popular due to the lower ongoing costs, greater sensitivity, and the convenience of digital photography.

Since the late 1990s amateurs have been following the professional observatories in the switch from film to digital CCDs for astronomical imaging. CCDs are more sensitive than film, allowing much shorter exposure times, and have a linear response to light. Images can be captured in many short exposures to create a synthetic long exposure. Digital cameras also have minimal or no moving parts and the ability to be operated remotely via an infrared remote or computer tethering, limiting vibration. Simple digital devices such as webcams can be modified to allow access to the focal plane and even (after the cutting of a few wires), for long exposure photography. Digital video cameras are also used. There are many techniques and pieces of commercially manufactured equipment for attaching digital single-lens reflex (DSLR) cameras and even basic point and shoot cameras to telescopes. Consumer-level digital cameras suffer from image noise over long exposures, so there are many techniques for cooling the camera, including cryogenic cooling. Astronomical equipment companies also now offer a wide range of purpose-built astronomical CCD cameras complete with hardware and processing software. Many commercially available DSLR cameras have the ability to take long time exposures combined with sequential (time-lapse) images allowing the photographer to create a motion picture of the night sky. CMOS cameras are increasingly replacing CCD cameras in the amateur sector.

Post-processing

The Pleiades Star Cluster photographed with a 6 megapixel DSLR connected to an 80mm refracting telescope piggybacked on a larger telescope. Made from seven 180 second images combined and processed in Photoshop with a noise reduction plugin.

Both digital camera images and scanned film images are usually adjusted in image processing software to improve the image in some way. Images can be brightened and manipulated in a computer to adjust color and increase the contrast. More sophisticated techniques involve capturing multiple images (sometimes thousands) to composite together in an additive process to sharpen images to overcome atmospheric seeing, negating tracking issues, bringing out faint objects with a poor signal-to-noise ratio, and filtering out light pollution.

Digital camera images may also need further processing to reduce the image noise from long exposures, including subtracting a “dark frame” and a processing called image stacking or "Shift-and-add". Commercial, freeware and free software packages are available specifically for astronomical photographic image manipulation.

"Lucky imaging" is a secondary technique that involves taking a video of an object rather than standard long exposure photos. Software can then select the highest quality images which can then be stacked. 

Hardware

Astrophotographic hardware among non-professional astronomers varies widely since the photographers themselves range from general photographers shooting some form of aesthetically pleasing images to very serious amateur astronomers collecting data for scientific research. As a hobby, astrophotography has many challenges that have to be overcome that differ from conventional photography and from what is normally encountered in professional astronomy.

NGC281, popularly the 'Pacman Nebula', imaged from a suburban location using a 130mm amateur telescope and a DSLR camera.

Since most people live in urban areas, equipment often needs to be portable so that it can be taken far away from the lights of major cities or towns to avoid urban light pollution. Urban astrophotographers may use special light-pollution or narrow-band filters and advanced computer processing techniques to reduce ambient urban light in the background of their images. They may also stick to imaging bright targets like the Sun, Moon and planets. Another method used by amateurs to avoid light pollution is to set up, or rent time, on a remotely operated telescope at a dark sky location. Other challenges include setup and alignment of portable telescopes for accurate tracking, working within the limitations of “off the shelf” equipment, the endurance of monitoring equipment, and sometimes manually tracking astronomical objects over long exposures in a wide range of weather conditions.

Some camera manufacturers modify their products to be used as astrophotography cameras, such as Canon's EOS 60Da, based on the EOS 60D but with a modified infrared filter and a low-noise sensor with heightened hydrogen-alpha sensitivity for improved capture of red hydrogen emission nebulae.

There are also cameras specifically designed for amateur astrophotography based on commercially available imaging sensors. They may also allow the sensor to be cooled to reduce thermal noise in long exposures, provide raw image readout, and to be controlled from a computer for automated imaging. Raw image readout allows later better image processing by retaining all the original image data which along with stacking can assist in imaging faint deep sky objects.

With very low light capability, a few specific models of webcams are popular for Solar, Lunar, and Planetary imaging. Mostly, these are manually focused cameras containing a CCD sensor instead of the more common CMOS. The lenses of these cameras are removed and then these are attached to telescopes to record images, videos, or both. In newer techniques, videos of very faint objects are taken and the sharpest frames of the video are 'stacked' together to obtain a still image of respectable contrast. The Philips PCVC 740K and SPC 900 are among the few webcams liked by astrophotographers. Any smartphone that allows long exposures can be used for this purpose, but some phones have a specific mode for astrophotography that will stitch together multiple exposures.

Equipment setups

An amateur astrophotography set up with an automated guide system connected to a laptop.

Fixed or tripod

The most basic types of astronomical photographs are made with standard cameras and photographic lenses mounted in a fixed position or on a tripod. Foreground objects or landscapes are sometimes composed in the shot. Objects imaged are constellations, interesting planetary configurations, meteors, and bright comets. Exposure times must be short (under a minute) to avoid having the stars point image become an elongated line due to the Earth's rotation. Camera lens focal lengths are usually short, as longer lenses will show image trailing in a matter of seconds. A rule of thumb called the 500 rule states that, to keep stars point-like,

Maximum exposure time in seconds = 500/Focal length in mm × Crop factor

regardless of aperture or ISO setting. For example, with a 35 mm lens on an APS-C sensor, the maximum time is 500/35 × 1.5 ≈ 9.5 s. A more accurate calculation takes into account pixel pitch and declination.

Allowing the stars to intentionally become elongated lines in exposures lasting several minutes or even hours, called “star trails”, is an artistic technique sometimes used.

Tracking mounts

Telescope mounts that compensate for the Earth's rotation are used for longer exposures without objects being blurred. They include commercial equatorial mounts and homemade equatorial devices such as barn door trackers and equatorial platforms. Mounts can suffer from inaccuracies due to backlash in the gears, wind, and imperfect balance, and so a technique called auto guiding is used as a closed feedback system to correct for these inaccuracies.

Tracking mounts can come in two forms; single axis and dual axis. Single axis mounts are often known as star trackers. Star trackers have a single motor which drives the right ascension axis. This allows the mount to compensate for the Earth's rotation. Star trackers rely on the user ensuring the mount is polar aligned with high accuracy, as it is unable correct in the secondary declination axis, limiting exposure times.

Dual axis mounts use two motors to drive both the right ascension and the declination axis together. This mount will compensate for the Earth's rotation by driving the right ascension axis, similar to a star tracker. However using an auto-guiding system, the secondary declination axis can also be driven, compensating for errors in polar alignment, allowing for significantly longer exposure times.

"Piggyback" photography

Piggyback astronomical photography is a method where a camera/lens is mounted on an equatorially mounted astronomical telescope. The telescope is used as a guide scope to keep the field of view centered during the exposure. This allows the camera to use a longer exposure and/or a longer focal length lens or even be attached to some form of photographic telescope co-axial with the main telescope.

Telescope focal plane photography

In this type of photography, the telescope itself is used as the "lens" collecting light for the film or CCD of the camera. Although this allows for the magnification and light-gathering power of the telescope to be used, it is one of the most difficult astrophotography methods. This is because of the difficulties in centering and focusing sometimes very dim objects in the narrow field of view, contending with magnified vibration and tracking errors, and the added expense of equipment (such as sufficiently sturdy telescope mounts, camera mounts, camera couplers, off-axis guiders, guide scopes, illuminated cross-hairs, or auto-guiders mounted on primary telescope or the guide-scope.) There are several different ways cameras (with removable lenses) are attached to amateur astronomical telescopes including:

• Prime focus – In this method the image produced by the telescope falls directly on the film or CCD with no intervening optics or telescope eyepiece.
• Positive projection – A method in which the telescope eyepiece (eyepiece projection) or a positive lens (placed after the focal plane of the telescope objective) is used to project a much more magnified image directly onto the film or CCD. Since the image is magnified with a narrow field of view this method is generally used for lunar and planetary photography.
• Negative projection – This method, like positive projection, produces a magnified image. A negative lens, usually a Barlow or a photographic teleconverter, is placed in the light cone before the focal plane of the telescope objective.
• Compression – Compression uses a positive lens (also called a focal reducer), placed in the converging cone of light before the focal plane of the telescope objective, to reduce overall image magnification. It is used on very long focal length telescopes, such as Maksutovs and Schmidt–Cassegrains, to obtain a wider field of view, or to reduce the focal ratio of the setup thereby increasing the speed of the system.

When the camera lens is not removed (or cannot be removed) a common method used is afocal photography, also called afocal projection. In this method, both the camera lens and the telescope eyepiece are attached. When both are focused at infinity the light path between them is parallel (afocal), allowing the camera to basically photograph anything the observer can see. This method works well for capturing images of the moon and brighter planets, as well as narrow field images of stars and nebulae. Afocal photography was common with early 20th-century consumer-level cameras since many models had non-removable lenses. It has grown in popularity with the introduction of point and shoot digital cameras since most models also have non-removable lenses.

Filters

Filters can be categorised into two classes; broadband and narrowband. Broadband filters allow a wide range of wavelengths to pass through, removing small amounts of light pollution. Narrowband filters only allow light from very specific wavelengths to pass through, blocking out the vast majority of the spectrum.

Astronomical filters usually come as sets and are manufactured to specific standards, in order to allow different observatories to make observations at the same standard. A common filter standard in the astronomy community is the Johnson Morgan UVB, designed to match a CCD’s color response to that of photographic film. However there are over 200 standards available.

Remote Telescope

Fast Internet access in the last part of the 20th century, and advances in computer-controlled telescope mounts and CCD cameras, allows use of 'Remote Telescopes' for amateur astronomers not aligned with major telescope facilities to partake in research and deep-sky imaging. This enables the imager to control a telescope far away in a dark location. The observers can image through the telescopes using CCD cameras.

Imaging can be done regardless of the location of the user or the telescopes they wish to use. The digital data collected by the telescope is then transmitted and displayed to the user by means of the Internet. An example of a digital remote telescope operation for public use via the Internet is The Bareket Observatory.

Speckle imaging

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Speckle_imaging 

Typical short-exposure image of a binary star (ζ Boötis) as seen through atmospheric turbulence. Each star should appear as a single point, but the atmosphere causes the images of the two stars to break up into two patterns of speckles. The speckles move around rapidly, so that each star appears as a single fuzzy blob in long exposure images.

Slow-motion speckle imaging movie, showing how a high-magnification (negative) image of a star breaks up into multiple blobs (speckles), entirely an atmospheric effect.

Speckle imaging comprises a range of high-resolution astronomical imaging techniques based on the analysis of large numbers of short exposures that freeze the variation of atmospheric turbulence. They can be divided into the shift-and-add ("image stacking") method and the speckle interferometry methods. These techniques can dramatically increase the resolution of ground-based telescopes, but are limited to bright targets.

Explanation

The principle of all the techniques is to take very short exposure images of astronomical targets, and then process those so as to remove the effects of astronomical seeing. Use of these techniques led to a number of discoveries, including thousands of binary stars that would otherwise appear as a single star to a visual observer working with a similar-sized telescope, and the first images of sunspot-like phenomena on other stars. Many of the techniques remain in wide use today, notably when imaging relatively bright targets.

The resolution of a telescope is limited by the size of the main mirror, due to the effects of Fraunhofer diffraction. This results in images of distant objects being spread out to a small spot known as the Airy disk. A group of objects whose images are closer together than this limit appear as a single object. Thus larger telescopes can image not only dimmer objects (because they collect more light), but resolve objects that are closer together as well.

This improvement of resolution breaks down due to the practical limits imposed by the atmosphere, whose random nature disrupts the single spot of the Airy disk into a pattern of similarly-sized spots scattered over a much larger area (see the adjacent image of a binary). For typical seeing, the practical resolution limits are at mirror sizes much less than the mechanical limits for the size of mirrors, namely at a mirror diameter equal to the astronomical seeing parameter r₀ – about 20 cm in diameter for observations with visible light under good conditions. For many years, telescope performance was limited by this effect, until the introduction of speckle interferometry and adaptive optics provided a means of removing this limitation.

Speckle imaging recreates the original image through image processing techniques. The key to the technique, found by the American astronomer David L. Fried in 1966, was to take very fast images in which case the atmosphere is effectively "frozen" in place. At infrared wavelengths, coherence times τ₀ are on the order of 100 ms, but for the visible region they drop to as little as 10 ms. When exposure times are shorter than τ₀, the movement of the atmosphere is too sluggish to have an effect; the speckles recorded in the image are a snapshot of the atmospheric seeing at that instant. Coherence time τ₀ = r₀/v is a function of wavelength, because r₀ is a function of wavelength.

Of course there is a downside: taking images at this short an exposure is difficult, and if the object is too dim, not enough light will be captured to make analysis possible. Early uses of the technique in the early 1970s were made on a limited scale using photographic techniques, but since photographic film captures only about 7% of the incoming light, only the brightest of objects could be viewed in this way. The introduction of the CCD into astronomy, which captures more than 70% of the light, lowered the bar on practical applications by an order of magnitude, and today the technique is widely used on bright astronomical objects (e.g. stars and star systems).

Many of the simpler speckle imaging methods have multiple names, largely from amateur astronomers re-inventing existing speckle imaging techniques and giving them new names.

Another use of the technique is in industry. By shining a laser (whose smooth wavefront is an excellent simulation of the light from a distant star) on a surface, the resulting speckle pattern can be processed to give detailed images of flaws in the material.

Types

Shift-and-add method

This section is about an imaging method. For the multiplication algorithm, see multiplication algorithm § Shift and add.

Lucky imaging images of Jupiter at 5 µm, using stacks of individual Gemini Observatory frames each with a relatively long 309-msec exposure time, illustrate the principle that coherence time τ₀ increases with wavelength.

The shift-and-add method (more recently "image-stacking" method) is a form of speckle imaging commonly used for obtaining high quality images from a number of short exposures with varying image shifts. It has been used in astronomy for several decades, and is the basis for the image stabilisation feature on some cameras. The short exposure images are aligned by using the brightest speckle and averaged to give a single output image.

The method involves calculation of the differential shifts of the images. This is easily accomplished in astronomical images since they can be aligned with the stars. Once the images are aligned they are averaged together. It is a basic principle of statistics that variation in a sample can be reduced by averaging together the individual values. In fact, when using an average, the signal-to-noise ratio should be increased by a factor of the square root of the number of images. A number of software packages exist for performing this, including IRAF, RegiStax, Autostakkert, Keiths Image Stacker, Hugin, and Iris.

In the lucky imaging approach, only the best short exposures are selected for averaging. Early shift-and-add techniques aligned images according to the image centroid, giving a lower overall Strehl ratio.

Speckle interferometry

In 1970, the French astronomer Antoine Labeyrie showed that Fourier analysis (speckle interferometry) can obtain information about the high-resolution structure of the object from the statistical properties of the speckle patterns. This technique was first implemented in 1971 at Palomar Observatory (200-inch telescope) by Daniel Y.Gezari, Antoine Labeyrie and Robert V.Stachnick. Methods developed in the 1980s allowed simple images to be reconstructed from this power spectrum information.

One more recent type of speckle interferometry called speckle masking' involves calculation of the bispectrum or closure phases from each of the short exposures. The "average bispectrum" can then be calculated and then inverted to obtain an image. This works particularly well using aperture masks. In this arrangement the telescope aperture is blocked except for a few holes which allow light through, creating a small optical interferometer with better resolving power than the telescope would otherwise have. This aperture masking technique was pioneered by the Cavendish Astrophysics Group.

One limitation of the technique is that it requires extensive computer processing of the image, which was hard to come by when the technique was first developed. This limitation has faded away over the years as computing power has increased, and nowadays desktop computers have more than enough power to make such processing a trivial task.

Biology

Speckle imaging in biology refers to the underlabeling of periodic cellular components (such as filaments and fibers) so that instead of appearing as a continuous and uniform structure, it appears as a discrete set of speckles. This is due to statistical distribution of the labeled component within unlabeled components. The technique, also known as dynamic speckle enables real-time monitoring of dynamical systems and video image analysis to understand biological processes.