Proper motion is the astronomical measure of the observed changes in the apparent places of stars or other celestial objects in the sky, as seen from the center of mass of the Solar System, compared to the abstract background of the more distant stars. The components for proper motion in the equatorial coordinate system are given in the direction of right ascension and of declination, their combined value is computed as the total proper motion. It has dimensions of angle per time arcseconds per year or milliarcseconds per year. Knowledge of the proper motion and radial velocity allows calculations of true stellar motion or velocity in space in respect to the Sun, by coordinate transformation, the motion in respect to the Milky Way. Proper motion is not "proper", because it includes a component due to the motion of the Solar System itself. Over the course of centuries, stars appear to maintain nearly fixed positions with respect to each other, so that they form the same constellations over historical time.
Ursa Major or Crux, for example, looks nearly the same now. However, precise long-term observations show that the constellations change shape, albeit slowly, that each star has an independent motion; this motion is caused by the movement of the stars relative to the Solar System. The Sun travels in a nearly circular orbit about the center of the Milky Way at a speed of about 220 km/s at a radius of 8 kPc from the center, which can be taken as the rate of rotation of the Milky Way itself at this radius; the proper motion is a two-dimensional vector and is thus defined by two quantities: its position angle and its magnitude. The first quantity indicates the direction of the proper motion on the celestial sphere, the second quantity is the motion's magnitude expressed in arcseconds per year or milliarcsecond per year. Proper motion may alternatively be defined by the angular changes per year in the star's right ascension and declination, using a constant epoch in defining these; the components of proper motion by convention are arrived at.
Suppose an object moves from coordinates to coordinates in a time Δt. The proper motions are given by: μ α = α 2 − α 1 Δ t, μ δ = δ 2 − δ 1 Δ t; the magnitude of the proper motion μ is given by the Pythagorean theorem: μ 2 = μ δ 2 + μ α 2 ⋅ cos 2 δ, μ 2 = μ δ 2 + μ α ∗ 2, where δ is the declination. The factor in cos2δ accounts for the fact that the radius from the axis of the sphere to its surface varies as cosδ, for example, zero at the pole. Thus, the component of velocity parallel to the equator corresponding to a given angular change in α is smaller the further north the object's location; the change μα, which must be multiplied by cosδ to become a component of the proper motion, is sometimes called the "proper motion in right ascension", μδ the "proper motion in declination". If the proper motion in right ascension has been converted by cosδ, the result is designated μα*. For example, the proper motion results in right ascension in the Hipparcos Catalogue have been converted. Hence, the individual proper motions in right ascension and declination are made equivalent for straightforward calculations of various other stellar motions.
The position angle θ is related to these components by: μ sin θ = μ α cos δ = μ α ∗, μ cos θ = μ δ. Motions in equatorial coordinates can be converted to motions in galactic coordinates. For the majority of stars seen in the sky, the observed proper motions are small and unremarkable; such stars are either faint or are distant, have changes of below 10 milliarcseconds per year, do not appear to move appreciably over many millennia. A few do have significant motions, are called high-proper motion stars. Motions can be in seemingly random directions. Two or more stars, double stars or open star clusters, which are moving in similar directions, exhibit so-called shared or common proper motion, suggesting they may be gravitationally attached or share similar motion in space. Barnard's Star has the largest proper motion of all stars, moving at 10.3 seconds of arc per year. L
The celestial equator is the great circle of the imaginary celestial sphere on the same plane as the equator of Earth. This plane of reference bases the equatorial coordinate system. In other words, the celestial equator is an abstract projection of the terrestrial equator into outer space. Due to Earth's axial tilt, the celestial equator is inclined by about 23.44° with respect to the ecliptic. The inclination has varied from about 22.0° to 24.5° over the past 5 million years. An observer standing on Earth's equator visualizes the celestial equator as a semicircle passing through the zenith, the point directly overhead; as the observer moves north, the celestial equator tilts towards the opposite horizon. The celestial equator is defined to be infinitely distant. At the poles, the celestial equator coincides with the astronomical horizon. At all latitudes, the celestial equator is a uniform arc or circle because the observer is only finitely far from the plane of the celestial equator, but infinitely far from the celestial equator itself.
Astronomical objects near the celestial equator appear above the horizon from most places on earth, but they culminate highest near the equator. The celestial equator passes through these constellations: These, by definition, are the most globally visible constellations. Celestial bodies other than Earth have defined celestial equators. Celestial pole Rotation around a fixed axis Celestial sphere Declination Equatorial coordinate system
Parallax is a displacement or difference in the apparent position of an object viewed along two different lines of sight, is measured by the angle or semi-angle of inclination between those two lines. Due to foreshortening, nearby objects show a larger parallax than farther objects when observed from different positions, so parallax can be used to determine distances. To measure large distances, such as the distance of a planet or a star from Earth, astronomers use the principle of parallax. Here, the term parallax is the semi-angle of inclination between two sight-lines to the star, as observed when Earth is on opposite sides of the Sun in its orbit; these distances form the lowest rung of what is called "the cosmic distance ladder", the first in a succession of methods by which astronomers determine the distances to celestial objects, serving as a basis for other distance measurements in astronomy forming the higher rungs of the ladder. Parallax affects optical instruments such as rifle scopes, binoculars and twin-lens reflex cameras that view objects from different angles.
Many animals, including humans, have two eyes with overlapping visual fields that use parallax to gain depth perception. In computer vision the effect is used for computer stereo vision, there is a device called a parallax rangefinder that uses it to find range, in some variations altitude to a target. A simple everyday example of parallax can be seen in the dashboard of motor vehicles that use a needle-style speedometer gauge; when viewed from directly in front, the speed may show 60. As the eyes of humans and other animals are in different positions on the head, they present different views simultaneously; this is the basis of stereopsis, the process by which the brain exploits the parallax due to the different views from the eye to gain depth perception and estimate distances to objects. Animals use motion parallax, in which the animals move to gain different viewpoints. For example, pigeons down to see depth; the motion parallax is exploited in wiggle stereoscopy, computer graphics which provide depth cues through viewpoint-shifting animation rather than through binocular vision.
Parallax arises due to change in viewpoint occurring due to motion of the observer, of the observed, or of both. What is essential is relative motion. By observing parallax, measuring angles, using geometry, one can determine distance. Astronomers use the word "parallax" as a synonym for "distance measurement" by other methods: see parallax #Astronomy. Stellar parallax created by the relative motion between the Earth and a star can be seen, in the Copernican model, as arising from the orbit of the Earth around the Sun: the star only appears to move relative to more distant objects in the sky. In a geostatic model, the movement of the star would have to be taken as real with the star oscillating across the sky with respect to the background stars. Stellar parallax is most measured using annual parallax, defined as the difference in position of a star as seen from the Earth and Sun, i. e. the angle subtended at a star by the mean radius of the Earth's orbit around the Sun. The parsec is defined as the distance.
Annual parallax is measured by observing the position of a star at different times of the year as the Earth moves through its orbit. Measurement of annual parallax was the first reliable way to determine the distances to the closest stars; the first successful measurements of stellar parallax were made by Friedrich Bessel in 1838 for the star 61 Cygni using a heliometer. Stellar parallax remains the standard for calibrating other measurement methods. Accurate calculations of distance based on stellar parallax require a measurement of the distance from the Earth to the Sun, now based on radar reflection off the surfaces of planets; the angles involved in these calculations are small and thus difficult to measure. The nearest star to the Sun, Proxima Centauri, has a parallax of 0.7687 ± 0.0003 arcsec. This angle is that subtended by an object 2 centimeters in diameter located 5.3 kilometers away. The fact that stellar parallax was so small that it was unobservable at the time was used as the main scientific argument against heliocentrism during the early modern age.
It is clear from Euclid's geometry that the effect would be undetectable if the stars were far enough away, but for various reasons such gigantic distances involved seemed implausible: it was one of Tycho's principal objections to Copernican heliocentrism that in order for it to be compatible with the lack of observable stellar parallax, there would have to be an enormous and unlikely void between the orbit of Saturn and the eighth sphere. In 1989, the satellite Hipparcos was launched for obtaining improved parallaxes and proper motions for over 100,000 nearby stars, increasing the reach of the method tenfold. So, Hipparcos is only able to measure parallax angles for stars up to about 1,600 light-years away, a little more than one percent of the diameter of the Milky Way Galaxy; the European Space Agency's Gaia mission, launched in December 2013, will be able to measure parallax angles to an accuracy of 10 microarcseconds, thus mapping nearby stars up to a distance of tens of thousands of ligh
Stellar evolution is the process by which a star changes over the course of time. Depending on the mass of the star, its lifetime can range from a few million years for the most massive to trillions of years for the least massive, longer than the age of the universe; the table shows the lifetimes of stars as a function of their masses. All stars are born from collapsing clouds of gas and dust called nebulae or molecular clouds. Over the course of millions of years, these protostars settle down into a state of equilibrium, becoming what is known as a main-sequence star. Nuclear fusion powers a star for most of its life; the energy is generated by the fusion of hydrogen atoms at the core of the main-sequence star. As the preponderance of atoms at the core becomes helium, stars like the Sun begin to fuse hydrogen along a spherical shell surrounding the core; this process causes the star to grow in size, passing through the subgiant stage until it reaches the red giant phase. Stars with at least half the mass of the Sun can begin to generate energy through the fusion of helium at their core, whereas more-massive stars can fuse heavier elements along a series of concentric shells.
Once a star like the Sun has exhausted its nuclear fuel, its core collapses into a dense white dwarf and the outer layers are expelled as a planetary nebula. Stars with around ten or more times the mass of the Sun can explode in a supernova as their inert iron cores collapse into an dense neutron star or black hole. Although the universe is not old enough for any of the smallest red dwarfs to have reached the end of their lives, stellar models suggest they will become brighter and hotter before running out of hydrogen fuel and becoming low-mass white dwarfs. Stellar evolution is not studied by observing the life of a single star, as most stellar changes occur too to be detected over many centuries. Instead, astrophysicists come to understand how stars evolve by observing numerous stars at various points in their lifetime, by simulating stellar structure using computer models. Stellar evolution starts with the gravitational collapse of a giant molecular cloud. Typical giant molecular clouds are 100 light-years across and contain up to 6,000,000 solar masses.
As it collapses, a giant molecular cloud breaks into smaller pieces. In each of these fragments, the collapsing gas releases gravitational potential energy as heat; as its temperature and pressure increase, a fragment condenses into a rotating sphere of superhot gas known as a protostar. A protostar continues to grow by accretion of gas and dust from the molecular cloud, becoming a pre-main-sequence star as it reaches its final mass. Further development is determined by its mass. Mass is compared to the mass of the Sun: 1.0 M☉ means 1 solar mass. Protostars are encompassed in dust, are thus more visible at infrared wavelengths. Observations from the Wide-field Infrared Survey Explorer have been important for unveiling numerous Galactic protostars and their parent star clusters. Protostars with masses less than 0.08 M☉ never reach temperatures high enough for nuclear fusion of hydrogen to begin. These are known as brown dwarfs; the International Astronomical Union defines brown dwarfs as stars massive enough to fuse deuterium at some point in their lives.
Objects smaller than 13 MJ are classified as sub-brown dwarfs. Both types, deuterium-burning and not, shine dimly and die away cooling over hundreds of millions of years. For a more-massive protostar, the core temperature will reach 10 million kelvin, initiating the proton–proton chain reaction and allowing hydrogen to fuse, first to deuterium and to helium. In stars of over 1 M☉, the carbon–nitrogen–oxygen fusion reaction contributes a large portion of the energy generation; the onset of nuclear fusion leads quickly to a hydrostatic equilibrium in which energy released by the core maintains a high gas pressure, balancing the weight of the star's matter and preventing further gravitational collapse. The star thus evolves to a stable state, beginning the main-sequence phase of its evolution. A new star will sit at a specific point on the main sequence of the Hertzsprung–Russell diagram, with the main-sequence spectral type depending upon the mass of the star. Small cold, low-mass red dwarfs fuse hydrogen and will remain on the main sequence for hundreds of billions of years or longer, whereas massive, hot O-type stars will leave the main sequence after just a few million years.
A mid-sized yellow dwarf star, like the Sun, will remain on the main sequence for about 10 billion years. The Sun is thought to be in the middle of its main sequence lifespan; the core exhausts its supply of hydrogen and the star begins to evolve off of the main sequence. Without the outward pressure generated by the fusion of hydrogen to counteract the force of gravity the core contracts until either electron degeneracy pressure becomes sufficient to oppose gravity or the core becomes hot enough for helium fusion to begin. Which of these happens first depends upon the star's mass. What happens after a low-mass star ceases to produce energy through fusion has not been directly observed. Recent astrophysical models suggest that red dwarfs of 0.1 M☉ may stay on the main sequence for some six to twelve tril
Margin of error
The margin of error is a statistic expressing the amount of random sampling error in a survey's results. The larger the margin of error, the less confidence one should have that the poll's reported results are close to the "true" figures. Margin of error is positive whenever a population is incompletely sampled and the outcome measure has positive variance; the term "margin of error" is used in non-survey contexts to indicate observational error in reporting measured quantities. Margin of error is defined as the "radius" of a confidence interval for a particular statistic from a survey. One example is the percent of people who prefer product A versus product B; when a single, global margin of error is reported for a survey, it refers to the maximum margin of error for all reported percentages using the full sample from the survey. If the statistic is a percentage, this maximum margin of error can be calculated as the radius of the confidence interval for a reported percentage of 50%; the margin of error has been described as an "absolute" quantity, equal to a confidence interval radius for the statistic.
For example, if the true value is 50 percentage points, the statistic has a confidence interval radius of 5 percentage points we say the margin of error is 5 percentage points. As another example, if the true value is 50 people, the statistic has a confidence interval radius of 5 people we might say the margin of error is 5 people. In some cases, the margin of error is not expressed as an "absolute" quantity. For example, suppose the true value is 50 people, the statistic has a confidence interval radius of 5 people. If we use the "absolute" definition, the margin of error would be 5 people. If we use the "relative" definition we express this absolute margin of error as a percent of the true value. Hence, in this case, the absolute margin of error is 5 people, but the "percent relative" margin of error is 10%. However, the distinction is not explicitly made, yet is apparent from context. Like confidence intervals, the margin of error can be defined for any desired confidence level, but a level of 90%, 95% or 99% is chosen.
This level is the confidence that a margin of error around the reported percentage would include the "true" percentage. Hence, for example, we can be confident, at the 95% level, that out of every 100 simple random samples taken from a given population, 95 of them will contain the true percentage or other statistic under investigation, within the margin of error associated with each. Along with the confidence level, the sample design for a survey, in particular its sample size, determines the magnitude of the margin of error. A larger sample size produces a smaller margin of all else remaining equal. If the exact confidence intervals are used the margin of error takes into account both sampling error and non-sampling error. If an approximate confidence interval is used the margin of error may only take random sampling error into account, it does not represent other potential sources of error or bias such as a non-representative sample-design, poorly phrased questions, people lying or refusing to respond, the exclusion of people who could not be contacted, or miscounts and miscalculations.
An example from the 2004 U. S. presidential campaign will be used to illustrate concepts throughout this article. According to an October 2, 2004 survey by Newsweek, 47% of registered voters would vote for John Kerry/John Edwards if the election were held on that day, 45% would vote for George W. Bush/Dick Cheney, 2% would vote for Ralph Nader/Peter Camejo; the size of the sample was 1,013. Unless otherwise stated, the remainder of this article uses a 95% level of confidence. Polls involve taking a sample from a certain population. In the case of the Newsweek poll, the population of interest is the population of people who will vote; because it is impractical to poll everyone who will vote, pollsters take smaller samples that are intended to be representative, that is, a random sample of the population. It is possible that pollsters sample 1,013 voters who happen to vote for Bush when in fact the population is evenly split between Bush and Kerry, but this is unlikely given that the sample is random.
Sampling theory provides methods for calculating the probability that the poll results differ from reality by more than a certain amount due to chance. This theory and some Bayesian assumptions suggest that the "true" percentage will be close to 47%; the more people that are sampled, the more confident pollsters can be that the "true" percentage is close to the observed percentage. The margin of error is a measure of how close the results are to be. However, the margin of error only accounts for random sampling error, so it is blind to systematic errors that may be introduced by non-response or by interactions between the survey and subjects' memory, motivation and knowledge; this section will discuss the standard error of a percentage, the corresponding confidence interval, connect these two concepts to the margin of error. For simplicity, the calculations here assume the poll was based on a simple random sample from a large population; the standard error of a reported proportion or percentage p measures its accuracy, is the estimated standard deviation of that perc
Stellar parallax is the apparent shift of position of any nearby star against the background of distant objects. Created by the different orbital positions of Earth, the small observed shift is largest at time intervals of about six months, when Earth arrives at opposite sides of the Sun in its orbit, giving a baseline distance of about two astronomical units between observations; the parallax itself is considered to be half of this maximum, about equivalent to the observational shift that would occur due to the different positions of Earth and the Sun, a baseline of one astronomical unit. Stellar parallax is so difficult to detect that its existence was the subject of much debate in astronomy for hundreds of years, it was first observed in 1806 by Giuseppe Calandrelli who reported parallax in α-Lyrae in his work "Osservazione e riflessione sulla parallasse annua dall’alfa della Lira". In 1838 Friedrich Bessel made the first successful parallax measurement, for the star 61 Cygni, using a Fraunhofer heliometer at Königsberg Observatory.
Once a star's parallax is known, its distance from Earth can be computed trigonometrically. But the more distant an object is, the smaller its parallax. With 21st-century techniques in astrometry, the limits of accurate measurement make distances farther away than about 100 parsecs too approximate to be useful when obtained by this technique; this limits the applicability of parallax as a measurement of distance to objects that are close on a galactic scale. Other techniques, such as spectral red-shift, are required to measure the distance of more remote objects. Stellar parallax measures are given in the tiny units of arcseconds, or in thousandths of arcseconds; the distance unit parsec is defined as the length of the leg of a right triangle adjacent to the angle of one arcsecond at one vertex, where the other leg is 1 AU long. Because stellar parallaxes and distances all involve such skinny right triangles, a convenient trigonometric approximation can be used to convert parallaxes to distance.
The approximate distance is the reciprocal of the parallax: d ≃ 1 / p. For example, Proxima Centauri, whose parallax is 0.7687, is 1 / 0.7687 parsecs = 1.3009 parsecs distant. Stellar parallax is so small that its apparent absence was used as a scientific argument against heliocentrism during the early modern age, it is clear from Euclid's geometry that the effect would be undetectable if the stars were far enough away, but for various reasons such gigantic distances involved seemed implausible: it was one of Tycho Brahe's principal objections to Copernican heliocentrism that in order for it to be compatible with the lack of observable stellar parallax, there would have to be an enormous and unlikely void between the orbit of Saturn and the eighth sphere. James Bradley first tried to measure stellar parallaxes in 1729; the stellar movement proved too insignificant for his telescope, but he instead discovered the aberration of light and the nutation of Earth's axis, catalogued 3222 stars. Stellar parallax is most measured using annual parallax, defined as the difference in position of a star as seen from Earth and Sun, i.e. the angle subtended at a star by the mean radius of Earth's orbit around the Sun.
The parsec is defined as the distance. Annual parallax is measured by observing the position of a star at different times of the year as Earth moves through its orbit. Measurement of annual parallax was the first reliable way to determine the distances to the closest stars; the first successful measurements of stellar parallax were made by Friedrich Bessel in 1838 for the star 61 Cygni using a heliometer. Being difficult to measure, only about 60 stellar parallaxes had been obtained by the end of the 19th century by use of the filar micrometer. Astrographs using astronomical photographic plates sped the process in the early 20th century. Automated plate-measuring machines and more sophisticated computer technology of the 1960s allowed more efficient compilation of star catalogues. In the 1980s, charge-coupled devices replaced photographic plates and reduced optical uncertainties to one milliarcsecond. Stellar parallax remains the standard for calibrating other measurement methods. Accurate calculations of distance based on stellar parallax require a measurement of the distance from Earth to the Sun, now known to exquisite accuracy based on radar reflection off the surfaces of planets.
The angles involved in these calculations are small and thus difficult to measure. The nearest star to the Sun, Proxima Centauri, has a parallax of 0.7687 ± 0.0003 arcsec. This angle is that subtended by an object 2 centimeters in diameter located 5.3 kilometers away. In 1989 the satellite Hipparcos was launched for obtaining parallaxes and proper motions of nearby stars, increasing the number of stellar parallaxes measured to milliarcsecond accuracy a thousandfold. So, Hipparcos is only able to measure parallax angles for stars up to about 1,600 light-years away, a little more than one percent of the diameter of the Milky Way Galaxy; the Hubble telescope WFC3 now has a precision of 20 to 40 microarcseconds, enabling reliable distance measurements u
In astronomy, metallicity is used to describe the abundance of elements present in an object that are heavier than hydrogen or helium. Most of the physical matter in the Universe is in the form of hydrogen and helium, so astronomers use the word "metals" as a convenient short term for "all elements except hydrogen and helium"; this usage is distinct from the usual physical definition of a solid metal. For example and nebulae with high abundances of carbon, nitrogen and neon are called "metal-rich" in astrophysical terms though those elements are non-metals in chemistry; the presence of heavier elements hails from stellar nucleosynthesis, the theory that the majority of elements heavier than hydrogen and helium in the Universe are formed in the cores of stars as they evolve. Over time, stellar winds and supernovae deposit the metals into the surrounding environment, enriching the interstellar medium and providing recycling materials for the birth of new stars, it follows that older generations of stars, which formed in the metal-poor early Universe have lower metallicities than those of younger generations, which formed in a more metal-rich Universe.
Observed changes in the chemical abundances of different types of stars, based on the spectral peculiarities that were attributed to metallicity, led astronomer Walter Baade in 1944 to propose the existence of two different populations of stars. These became known as Population I and Population II stars. A third stellar population was introduced in 1978, known as Population III stars; these metal-poor stars were theorised to have been the "first-born" stars created in the Universe. Astronomers use several different methods to describe and approximate metal abundances, depending on the available tools and the object of interest; some methods include determining the fraction of mass, attributed to gas versus metals, or measuring the ratios of the number of atoms of two different elements as compared to the ratios found in the Sun. Stellar composition is simply defined by the parameters X, Y and Z. Here X is the mass fraction of hydrogen, Y is the mass fraction of helium, Z is the mass fraction of all the remaining chemical elements.
Thus X + Y + Z = 1.00. In most stars, nebulae, H II regions, other astronomical sources and helium are the two dominant elements; the hydrogen mass fraction is expressed as X ≡ m H / M, where M is the total mass of the system, m H is the fractional mass of the hydrogen it contains. The helium mass fraction is denoted as Y ≡ m He / M; the remainder of the elements are collectively referred to as "metals", the metallicity—the mass fraction of elements heavier than helium—can be calculated as Z = ∑ i > He m i M = 1 − X − Y. For the surface of the Sun, these parameters are measured to have the following values: Due to the effects of stellar evolution, neither the initial composition nor the present day bulk composition of the Sun is the same as its present-day surface composition; the overall stellar metallicity is defined using the total iron content of the star, as iron is among the easiest to measure with spectral observations in the visible spectrum. The abundance ratio is defined as the logarithm of the ratio of a star's iron abundance compared to that of the Sun and is expressed thus: = log 10 star − log 10 sun, where N Fe and N H are the number of iron and hydrogen atoms per unit of volume respectively.
The unit used for metallicity is the dex, contraction of "decimal exponent". By this formulation, stars with a higher metallicity than the Sun have a positive logarithmic value, whereas those with a lower metallicity than the Sun have a negative value. For example, stars with a value of +1 have 10 times the metallicity of the Sun. Young Population I stars have higher iron-to-hydrogen ratios than older Population II stars. Primordial Population III stars are estimated to have a metallicity of less than −6.0, that is, less than a millionth of the abundance of iron in the Sun. The same notation is used to express variations in abundances between other the individual elements as compared to solar proportions. For example, the notati