430:
701:, that is the determination of exactly what the absolute magnitude of the candle is. This includes defining the class well enough that members can be recognized, and finding enough members of that class with well-known distances to allow their true absolute magnitude to be determined with enough accuracy. The second problem lies in recognizing members of the class, and not mistakenly using a standard candle calibration on an object which does not belong to the class. At extreme distances, which is where one most wishes to use a distance indicator, this recognition problem can be quite serious.
2872:
85:
410:
286:
2694:
2485:
1529:. By measuring these properties from a star's spectrum, the position of a main sequence star on the HâR diagram can be determined, and thereby the star's absolute magnitude estimated. A comparison of this value with the apparent magnitude allows the approximate distance to be determined, after correcting for interstellar extinction of the luminosity because of gas and dust.
162:
278:
43:
3055:
2022:
713:. If indeed the properties of Type Ia supernovae are different at large distances, i.e. if the extrapolation of their calibration to arbitrary distances is not valid, ignoring this variation can dangerously bias the reconstruction of the cosmological parameters, in particular the reconstruction of the matter
2762:(PNLF) was first proposed in the late 1970s by Holland Cole and David Jenner. They suggested that all planetary nebulae might all have similar maximum intrinsic brightness, now calculated to be M = â4.53. This would therefore make them potential standard candles for determining extragalactic distances.
145:) to Earth. The techniques for determining distances to more distant objects are all based on various measured correlations between methods that work at close distances and methods that work at larger distances. Several methods rely on a standard candle, which is an astronomical object that has a known
3002:
usually have long ceased to have large-scale star formation, they will not have
Cepheids. Instead, distance indicators whose origins are in an older stellar population (like novae and RR Lyrae variables) must be used. RR Lyrae variables are less luminous than Cepheids, and novae are unpredictable and
2012:
As detected thus far, NGC 3370, a spiral galaxy in the constellation Leo, contains the farthest
Cepheids yet found at a distance of 29 Mpc. Cepheid variable stars are in no way perfect distance markers: at nearby galaxies they have an error of about 7% and up to a 15% error for the most distant.
1347:
With few exceptions, distances based on direct measurements are available only out to about a thousand parsecs, which is a modest portion of our own Galaxy. For distances beyond that, measures depend upon physical assumptions, that is, the assertion that one recognizes the object in question, and the
449:
per year, while for halo stars the baseline is 40 AU per year. After several decades, the baseline can be orders of magnitude greater than the EarthâSun baseline used for traditional parallax. However, secular parallax introduces a higher level of uncertainty because the relative velocity of observed
3038:
observed that fainter galaxies are more redshifted. Finding the value of the Hubble constant was the result of decades of work by many astronomers, both in amassing the measurements of galaxy redshifts and in calibrating the steps of the distance ladder. Hubble's Law is the primary means we have for
2556:
US astronomer
William Alvin Baum first attempted to use globular clusters to measure distant elliptical galaxies. He compared the brightest globular clusters in Virgo A galaxy with those in Andromeda, assuming the luminosities of the clusters were the same in both. Knowing the distance to Andromeda,
2344:
Using Type Ia supernovae is one of the most accurate methods, particularly since supernova explosions can be visible at great distances (their luminosities rival that of the galaxy in which they are situated), much farther than
Cepheid Variables (500 times farther). Much time has been devoted to the
708:
that are of known distance have the same brightness, corrected by the shape of the light curve. The basis for this closeness in brightness is discussed below; however, the possibility exists that the distant Type Ia supernovae have different properties than nearby Type Ia supernovae. The use of Type
3101:
If the signal were to depend on the individual masses separately, there would not be enough observable information in the signal at the lowest order to infer its intrinsic loudness. This degeneracy between the masses therefore is crucial for the loudness measurement, but it is no accident: It has a
2255:
Once reached, the star becomes unstable and undergoes a runaway nuclear fusion reaction. Because all Type Ia supernovae explode at about the same mass, their absolute magnitudes are all the same. This makes them very useful as standard candles. All Type Ia supernovae have a standard blue and visual
2001:
These unresolved matters have resulted in cited values for the Hubble constant ranging between 60 km/s/Mpc and 80 km/s/Mpc. Resolving this discrepancy is one of the foremost problems in astronomy since some cosmological parameters of the
Universe may be constrained significantly better by
1322:
The total distance that these sound waves can travel before recombination determines a fixed scale, which simply expands with the universe after recombination. BAO therefore provide a standard ruler that can be measured in galaxy surveys from the effect of baryons on the clustering of galaxies. The
289:
Parallax is an angle subtended by a line on a point. In the upper diagram, the Earth in its orbit sweeps the parallax angle subtended on the Sun. The lower diagram shows an equal angle swept by the Sun in a geostatic model. A similar diagram can be drawn for a star except that the angle of parallax
232:
is used to measure the distance between the orbits of the Earth and of a second body. From that measurement and the ratio of the two orbit sizes, the size of Earth's orbit is calculated. The Earth's orbit is known with an absolute precision of a few meters and a relative precision of a few parts in
4447:
We conclude that most of the knots are in fact likely to lie in front of the plane of the sky, thus invalidating the
Kervella et al. result. Although the Kervella et al. distance result is invalidated, we show that high-resolution polarimetric imaging has the potential to yield a valid geometric
2981:
This method has the potential to become one of the strongest methods of galactic distance calculators, perhaps exceeding the range of even the TullyâFisher method. As of today, however, elliptical galaxies are not bright enough to provide a calibration for this method through the use of techniques
152:
The ladder analogy arises because no single technique can measure distances at all ranges encountered in astronomy. Instead, one method can be used to measure nearby distances, a second can be used to measure nearby to intermediate distances, and so on. Each rung of the ladder provides information
3026:
of stars and galaxies, and is thus also subject to uncertainties in those aspects. For the most luminous of distance indicators, the Type Ia supernovae, this homogeneity is known to be poor. However, no other class of object is bright enough to be detected at such large distances, so the class is
1997:
Several problems complicate the use of
Cepheids as standard candles and are actively debated, chief among them are: the nature and linearity of the period-luminosity relation in various passbands and the impact of metallicity on both the zero-point and slope of those relations, and the effects of
1226:
greater than 1. It is difficult for detector networks to measure the polarization of a signal accurately if the binary system is observed nearly face-on. Such signals suffer significantly larger errors in the distance measurement. Unfortunately, binaries radiate most strongly perpendicular to the
1190:
Just as with standard candles, given the emitted and received amplitudes, the inverse-square law determines the distance to the source. There are some differences with standard candles, however. Gravitational waves are not emitted isotropically, but measuring the polarisation of the wave provides
2891:
cameras on telescopes. Because of spatial fluctuations in a galaxy's surface brightness, some pixels on these cameras will pick up more stars than others. As distance increases, the picture will become increasingly smoother. Analysis of this describes a magnitude of the pixel-to-pixel variation,
2990:
A succession of distance indicators, which is the distance ladder, is needed for determining distances to other galaxies. The reason is that objects bright enough to be recognized and measured at such distances are so rare that few or none are present nearby, so there are too few examples close
2916:
level of 20.75 B-mag arcsec. This surface brightness is independent of the galaxy's actual distance from us. Instead, D is inversely proportional to the galaxy's distance, represented as d. Thus, this relation does not employ standard candles. Rather, D provides a standard ruler. This relation
2329:), the shape of the light curve (taken at any reasonable time after the initial explosion) is compared to a family of parameterized curves that will determine the absolute magnitude at the maximum brightness. This method also takes into effect interstellar extinction/dimming from dust and gas.
2205:
This method works only if the supernova is close enough to be able to measure accurately the photosphere. Similarly, the expanding shell of gas is in fact not perfectly spherical nor a perfect blackbody. Also interstellar extinction can hinder the accurate measurements of the photosphere. This
1194:
Generally, if a wave is detected by a network of three detectors at different locations, the network will measure enough information to make these corrections and obtain the distance. Also unlike standard candles, gravitational waves need no calibration against other distance measures. The
749:
have been proposed as another type of standard candle. "Since kilonovae explosions are spherical, astronomers could compare the apparent size of a supernova explosion with its actual size as seen by the gas motion, and thus measure the rate of cosmic expansion at different distances."
511:
which can measure very small angular motions. These combine to provide fundamental distance estimates to supernovae in other galaxies. Though valuable, such cases are quite rare, so they serve as important consistency checks on the distance ladder rather than workhorse steps by
1319:) the baryons and photons scatter off each other, and form a tightly coupled fluid that can support sound waves. The waves are sourced by primordial density perturbations, and travel at speed that can be predicted from the baryon density and other cosmological parameters.
1675:
The extragalactic distance scale is a series of techniques used today by astronomers to determine the distance of cosmological bodies beyond our own galaxy, which are not easily obtained with traditional methods. Some procedures use properties of these objects, such as
2995:, cannot yet be satisfactorily calibrated by parallax alone, though the Gaia space mission can now weigh in on that specific problem. The situation is further complicated by the fact that different stellar populations generally do not have all types of stars in them.
1372:â In the last decade, measurement of eclipsing binaries' fundamental parameters has become possible with 8-meter class telescopes. This makes it feasible to use them as indicators of distance. Recently, they have been used to give direct distance estimates to the
2336:
fits the particular supernovae magnitude light curves to a template light curve. This template, as opposed to being several light curves at different wavelengths (MLCS) is just a single light curve that has been stretched (or compressed) in time. By using this
4604:
Ferrarese, L; et al. (2000). "A Database of
Cepheid Distance Moduli and Tip of the Red Giant Branch, Globular Cluster Luminosity Function, Planetary Nebula Luminosity Function, and Surface Brightness Fluctuation Data Useful for Distance Determinations".
932:
4289:
4822:; Wyrzykowski, L.; Pietrzynski, G.; Szewczyk, O.; Szymanski, M.; Kubiak, M.; Soszynski, I.; Zebrun, K. (2001). "The Optical Gravitational Lensing Experiment. Cepheids in the Galaxy IC1613: No Dependence of the Period-Luminosity Relation on Metallicity".
3017:
Another concern, especially for the very brightest standard candles, is their "standardness": how homogeneous the objects are in their true absolute magnitude. For some of these different standard candles, the homogeneity is based on theories about the
495:
binaries also can have their distance estimated by similar means, and do not suffer from the above geometric uncertainty. The common characteristic to these methods is that a measurement of angular motion is combined with a measurement of the absolute
320:(the distance between the extreme positions of Earth's orbit around the Sun) making the base leg of the triangle and the distance to the star being the long equal-length legs. The amount of shift is quite small, even for the nearest stars, measuring 1
741:
stars. As a result, the population II stars were actually much brighter than believed, and when corrected, this had the effect of doubling the estimates of distances to the globular clusters, the nearby galaxies, and the diameter of the Milky Way.
2470:
After novae fade, they are about as bright as the most luminous
Cepheid variable stars, therefore both these techniques have about the same max distance: ~ 20 Mpc. The error in this method produces an uncertainty in magnitude of about ±0.4
2320:
Therefore, when observing a Type Ia supernova, if it is possible to determine what its peak magnitude was, then its distance can be calculated. It is not intrinsically necessary to capture the supernova directly at its peak magnitude; using the
506:
Expansion parallaxes in particular can give fundamental distance estimates for objects that are very far, because supernova ejecta have large expansion velocities and large sizes (compared to stars). Further, they can be observed with radio
5379:
Tonry, John L.; Dressler, Alan; Blakeslee, John P.; Ajhar, Edward A.; Fletcher, Andre B.; Luppino, Gerard A.; Metzger, Mark R.; Moore, Christopher B. (2001), "The SBF Survey of Galaxy
Distances. IV. SBF Magnitudes, Colors, and Distances",
1191:
enough information to determine the angle of emission. Gravitational wave detectors also have anisotropic antenna patterns, so the position of the source on the sky relative to the detectors is needed to determine the angle of reception.
2356:
can be used in much the same way as supernovae to derive extragalactic distances. There is a direct relation between a nova's max magnitude and the time for its visible light to decline by two magnitudes. This relation is shown to be:
2879:
The following method deals with the overall inherent properties of galaxies. These methods, though with varying error percentages, have the ability to make distance estimates beyond 100 Mpc, though it is usually applied more locally.
4772:
Majaess, Daniel; Turner, David; Moni Bidin, Christian; Mauro, Francesco; Geisler, Douglas; Gieren, Wolfgang; Minniti, Dante; Chené, André-Nicolas; Lucas, Philip; Borissova, Jura; Kurtev, Radostn; Dékåny, Istvan; Saito, Roberto K.
1327:). The scale does depend on cosmological parameters like the baryon and matter densities, and the number of neutrinos, so distances based on BAO are more dependent on cosmological model than those based on local measurements.
1177:
4965:
Bono, G.; Caputo, F.; Fiorentino, G.; Marconi, M.; Musella, I. (2008). "Cepheids in External Galaxies. I. The MaserâHost Galaxy NGC 4258 and the Metallicity Dependence of PeriodâLuminosity and PeriodâWesenheit Relations".
4460:
Kervella, Pierre; Bond, Howard E.; Cracraft, Misty; Szabados, Låszló; Breitfelder, Joanne; Mérand2, Antoine; Sparks, William B.; Gallenne, Alexandre; Bersier, David; Fouqué, Pascal; Anderson, Richard I. (December 2014).
2217:
are some of the best ways to determine extragalactic distances. Ia's occur when a binary white dwarf star begins to accrete matter from its companion star. As the white dwarf gains matter, eventually it reaches its
1993:
1881:
1195:
measurement of distance does of course require the calibration of the gravitational wave detectors, but then the distance is fundamentally given as a multiple of the wavelength of the laser light being used in the
193:
distance measurements, in which distances are determined directly, with no physical assumptions about the nature of the object in question. The precise measurement of stellar positions is part of the discipline of
685:, which also makes objects appear fainter and more red, is needed, especially if the object lies within a dusty or gaseous region. The difference between an object's absolute and apparent magnitudes is called its
4247:
2436:
1388:. Eclipsing binaries offer a direct method to gauge the distance to galaxies to a new improved 5% level of accuracy which is feasible with current technology to a distance of around 3 Mpc (3 million parsecs).
2861:
2115:
2657:
1272:, which was used to make the first such measurement. Even if no electromagnetic counterpart can be identified for an ensemble of signals, it is possible to use a statistical method to infer the value of
680:
the absolute magnitude. For this to be accurate, both magnitudes must be in the same frequency band and there can be no relative motion in the radial direction. Some means of correcting for interstellar
2318:
728:
discovered that the nearby Cepheid variables used to calibrate the standard candle were of a different type than the ones used to measure distances to nearby galaxies. The nearby Cepheid variables were
808:
3014:
means that distances in astronomy are rarely known to the same level of precision as measurements in the other sciences, and that the precision necessarily is poorer for more distant types of object.
4798:
Stanek, K. Z.; Udalski, A. (1999). "The Optical Gravitational Lensing Experiment. Investigating the Influence of Blending on the Cepheid Distance Scale with Cepheids in the Large Magellanic Cloud".
1749:. Though in theory this method has the ability to provide reliable distance calculations to stars up to 7 megaparsecs (Mpc), it is generally only used for stars at hundreds of kiloparsecs (kpc).
1525:, evolutionary patterns are found that relate to the mass, age and composition of the star. In particular, during their hydrogen burning period, stars lie along a curve in the diagram called the
3275:
Riess, A. G.; Casertano, S.; Anderson, J.; MacKenty, J.; Filippenko, A. V. (2014). "Parallax Beyond a Kiloparsec from Spatially Scanning the Wide Field Camera 3 on the Hubble Space Telescope".
5693:
2179:
1444:) are used as standard candles. Observations of X-ray burst sometimes show X-ray spectra indicating radius expansion. Therefore, the X-ray flux at the peak of the burst should correspond to
3790:
Cutler, Curt; Flanagan, Ăanna E. (15 March 1994). "Gravitational waves from merging compact binaries: How accurately can one extract the binary's parameters from the inspiral waveform?".
2976:
3253:
4912:
Macri, L. M.; Stanek, K. Z.; Bersier, D.; Greenhill, L. J.; Reid, M. J. (2006). "A New Cepheid Distance to the MaserâHost Galaxy NGC 4258 and Its Implications for the Hubble Constant".
2009:(Andromeda) was an external galaxy, as opposed to a smaller nebula within the Milky Way. He was able to calculate the distance of M31 to 285 kpc, today's value being 770 kpc.
4704:
611:
386:
of the star's spectrum caused by motion along the line of sight. For a group of stars with the same spectral class and a similar magnitude range, a mean parallax can be derived from
337:
Because parallax becomes smaller for a greater stellar distance, useful distances can be measured only for stars which are near enough to have a parallax larger than a few times the
3592:
1540:, the stars formed at approximately the same age and lie at the same distance. This allows relatively accurate main sequence fitting, providing both age and distance determination.
1218:, then it might be received as multiple events, separated in time, the analogue of multiple images of a quasar, for example. Less easy to discern and control for is the effect of
5874:
1448:, which can be calculated once the mass of the neutron star is known (1.5 solar masses is a commonly used assumption). This method allows distance determination of some low-mass
2564:
assumed the use of the globular cluster luminosity function (GCLF) would lead to a better approximation. The number of globular clusters as a function of magnitude is given by:
198:. Early fundamental distancesâsuch as the radii of the earth, moon and sun, and the distances between themâwere well estimated with very low technology by the ancient Greeks.
4108:
Nissanke, Samaya; Holz, Daniel E.; Hughes, Scott A.; Dalal, Neal; Sievers, Jonathan L. (2010-12-10). "Exploring Short Gamma-Ray Bursts as Gravitational-Wave Standard Sirens".
1745:
There are major limitations to this method for finding stellar distances. The calibration of the spectral line strengths has limited accuracy and it requires a correction for
670:
1238:
explosion that may allow the position to be accurately identified by electromagnetic telescopes. In such cases, the redshift of the host galaxy allows a determination of the
1004:
2250:
1054:
1222:, where the signal's path through space is affected by many small magnification and demagnification events. This will be important for signals originating at cosmological
5115:
4698:
George H. Jacoby; David Branch; Robin Ciardullo; Roger L. Davies; William E. Harris; Michael J. Pierce; Christopher J. Pritchet; John L. Tonry; Douglas L. Welch (1992).
2465:
382:(motion toward or away from the Sun). The former is determined by plotting the changing position of the stars over many years, while the latter comes from measuring the
2998:
Cepheids in particular are massive stars, with short lifetimes, so they will only be found in places where stars have very recently been formed. Consequently, because
1059:
3034:
relationship between distance and the speed with which a galaxy is moving away from us, usually referred to as redshift, is a product of the cosmic distance ladder.
5686:
4525:
1311:. In 2008, galaxy diameters have been proposed as a possible standard ruler for cosmological parameter determination. More recently the physical scale imprinted by
1297:
1266:
976:
952:
803:
256:
measurements of distances to Venus and other nearby planets and asteroids, and by tracking interplanetary spacecraft in their orbits around the Sun through the
4859:
Ngeow, C.; Kanbur, S. M. (2006). "The Hubble Constant from Type Ia Supernovae Calibrated with the Linear and Nonlinear Cepheid Period-Luminosity Relations".
3125:
2912:. It is important to describe exactly what D represents, in order to understand this method. It is, more precisely, the galaxy's angular diameter out to the
312:. As the Earth orbits the Sun, the position of nearby stars will appear to shift slightly against the more distant background. These shifts are angles in an
450:
stars is an additional unknown. When applied to samples of multiple stars, the uncertainty can be reduced; the uncertainty is inversely proportional to the
4607:
1202:
There are other considerations that limit the accuracy of this distance, besides detector calibration. Fortunately, gravitational waves are not subject to
5679:
504:). The distance estimate comes from computing how far the object must be to make its observed absolute velocity appear with the observed angular motion.
2056:
We can assume that a supernova expands in a spherically symmetric manner. If the supernova is close enough such that we can measure the angular extent,
777:
of the pair, and the resultant shrinking of their orbits is directly observable as an increase in the frequency of the emitted gravitational waves. To
2362:
4084:
64:
51:
3776:
471:
Other individual objects can have fundamental distance estimates made for them under special circumstances. If the expansion of a gas cloud, like a
2770:
2073:
5864:
2715:
2569:
2506:
704:
A significant issue with standard candles is the recurring question of how standard they are. For example, all observations seem to indicate that
3003:
an intensive monitoring programâand luck during that programâis needed to gather enough novae in the target galaxy for a good distance estimate.
2261:
2991:
enough with reliable trigonometric parallax to calibrate the indicator. For example, Cepheid variables, one of the best indicators for nearby
1887:
1775:
483:
distance to that cloud can be estimated. Those measurements however suffer from uncertainties in the deviation of the object from sphericity.
5621:
5594:
5560:
3401:
3245:
3703:
Sneppen, Albert; Watson, Darach; Bauswein, Andreas; Just, Oliver; Kotak, Rubina; Nakar, Ehud; Poznanski, Dovi; Sim, Stuart (February 2023).
521:
Almost all astronomical objects used as physical distance indicators belong to a class that has a known brightness. By comparing this known
2759:
1416:
109:
3588:
221:
3647:
Linden, S.; Virey, J.-M.; Tilquin, A. (2009). "Cosmological parameter extraction and biases from type Ia supernova magnitude evolution".
3006:
Because the more distant steps of the cosmic distance ladder depend upon the nearer ones, the more distant steps include the effects of
2545:
Based on the method of comparing the luminosities of globular clusters (located in galactic halos) from distant galaxies to that of the
1323:
method requires an extensive galaxy survey in order to make this scale visible, but has been measured with percent-level precision (see
429:
2134:
169:
shows the use of parallax to measure distance. It is made from parts of the YaleâColumbia Refractor telescope (1924) damaged when the
5481:
Gilfanov, Marat; BogdĂĄn, Ăkos (2010). "An upper limit on the contribution of accreting white dwarfs to the type Ia supernova rate".
2741:
2532:
460:
is a technique where the motions of individual stars in a nearby star cluster can be used to find the distance to the cluster. Only
4339:
Kervella, Pierre; Mérand, Antoine; Szabados, Låszló; Fouqué, Pascal; Bersier, David; Pompei, Emanuela; Perrin, Guy (2 March 2008).
141:
distance measurement of an astronomical object is possible only for those objects that are "close enough" (within about a thousand
2922:
437:
The motion of the Sun through space provides a longer baseline that will increase the accuracy of parallax measurements, known as
5311:
1522:
4777:
4752:
2863:
Where N(M) is number of planetary nebula, having absolute magnitude M. M* is equal to the nebula with the brightest magnitude.
2345:
refining of this method. The current uncertainty approaches a mere 5%, corresponding to an uncertainty of just 0.1 magnitudes.
2206:
problem is further exacerbated by core-collapse supernova. All of these factors contribute to the distance error of up to 25%.
559:
5879:
2884:
2719:
2510:
1428:
1196:
92:
3451:
328:), and thereafter decreasing in angular amount as the distance increases. Astronomers usually express distances in units of
3421:
4314:
4178:
3031:
2550:
1422:
212:
Direct distance measurements are based upon the astronomical unit (AU), which is defined as the mean distance between the
31:
4463:"The long-period Galactic Cepheid RS Puppis. III. A geometric distance from HST polarimetric imaging of its light echoes"
2560:
Baum used just a single globular cluster, but individual formations are often poor standard candles. Canadian astronomer
1363:
361:
arcseconds, enabling reliable distance measurements up to 5,000 parsecs (16,000 ly) for small numbers of stars. The
5833:
5717:
4750:"Hubble Space Telescope Fine Guidance Sensor Parallaxes of Galactic Cepheid Variable Stars: Period-Luminosity Relations"
3078:
2125:
is angular extent. In order to get an accurate measurement, it is necessary to make two observations separated by time Î
1324:
1312:
1215:
3625:
1769:
stars. The following relation can be used to calculate the distance to Galactic and extragalactic classical Cepheids:
1410:
616:
4318:
3343:
2704:
2495:
1688:, and galaxies as a whole. Other methods are based more on the statistics and probabilities of things such as entire
1477:
228:
of the orbit sizes of objects orbiting the Sun, but provide no measurement of the overall scale of the orbit system.
3494:
Bartel, N.; et al. (1994). "The shape, expansion rate and distance of supernova 1993J from VLBI measurements".
2871:
433:
Hubble Space Telescope precision stellar distance measurement has been extended 10 times further into the Milky Way.
4467:
4409:
4345:
4252:
3649:
3217:
3011:
1762:
1219:
927:{\displaystyle {\frac {df}{dt}}={\frac {96\pi ^{8/3}(G{\mathcal {M}})^{\frac {5}{3}}f^{\frac {11}{3}}}{5\,c^{5}}},}
4541:
3121:
2723:
2708:
2514:
2499:
1471:
56:
5869:
5552:
4248:"Geometrical tests of cosmological models. I. Probing dark energy using the kinematics of high redshift galaxies"
3330:
3277:
1723:
1452:. Low-mass X-ray binaries are very faint in the optical, making their distances extremely difficult to determine.
1316:
413:
Parallax measurements may be an important clue to understanding three of the universe's most elusive components:
174:
5056:
Madore, Barry F.; Freedman, Wendy L. (2009). "Concerning the Slope of the Cepheid PeriodâLuminosity Relation".
3178:
321:
2467:
is the time derivative of the nova's mag, describing the average rate of decline over the first 2 magnitudes.
1758:
1715:
1701:
1998:
photometric contamination (blending) and a changing (typically unknown) extinction law on Cepheid distances.
4035:
1746:
457:
308:
170:
1348:
class of objects is homogeneous enough that its members can be used for meaningful estimation of distance.
544:. This quantity is derived from the logarithm of its luminosity as seen from a distance of 10 parsecs. The
4568:
4058:
3148:
1719:
1518:
1512:
1377:
1373:
1333:
can be also used as standard rulers, although it is challenging to correctly measure the source geometry.
1203:
955:
720:
That this is not merely a philosophical issue can be seen from the history of distance measurements using
682:
534:
350:
338:
4662:
4519:
2888:
1537:
465:
270:
5434:"The Dn-sigma relation for bulges of disk galaxies - A new, independent measure of the Hubble constant"
3910:; Berman, David (12 June 1967). "Gravitational-Radiation Detection Range for Binary Stellar Systems".
1489:
and are useful in determining extragalactic distances up to a few hundred Mpc. A notable exception is
5502:
5445:
5399:
5345:
5271:
5218:
5181:
5134:
5075:
5038:
4985:
4931:
4878:
4841:
4713:
4671:
4626:
4560:
4539:
Bonanos, A. Z. (2006). "Eclipsing Binaries: Tools for Calibrating the Extragalactic Distance Scale".
4486:
4428:
4364:
4271:
4210:
4182:
4127:
4050:
3984:
3919:
3871:
3809:
3726:
3668:
3552:
3505:
3362:
3296:
3226:
3187:
1707:
1445:
773:, have the useful property that energy emitted as gravitational radiation comes exclusively from the
391:
387:
295:
4573:
4063:
3473:
Popowski, P.; Gould, A. (1998). "Mathematics of Statistical Parallax and the Local Distance Scale".
1458:
can be used to derive distances to galactic and some extragalactic objects that have maser emission.
985:
137:) is the succession of methods by which astronomers determine the distances to celestial objects. A
84:
5859:
5838:
5730:
2909:
2225:
2219:
1455:
1013:
714:
354:
2765:
Astronomer George Howard Jacoby and his colleagues later proposed that the PNLF function equaled:
1183:(rate of energy emission) of the gravitational waves. Thus, such a gravitational wave source is a
689:, and astronomical distances, especially intergalactic ones, are sometimes tabulated in this way.
5526:
5492:
5415:
5389:
5361:
5335:
5287:
5261:
5234:
5208:
5171:
5124:
5091:
5065:
5028:
5001:
4975:
4947:
4921:
4894:
4868:
4831:
4799:
4731:
4642:
4616:
4586:
4550:
4476:
4418:
4354:
4261:
4200:
4159:
4117:
4076:
4016:
3974:
3833:
3799:
3758:
3716:
3684:
3658:
3521:
3474:
3352:
3312:
3286:
2913:
2214:
1735:
1731:
1355:
1211:
758:
738:
730:
710:
545:
541:
526:
367:
space mission provided similarly accurate distances to most stars brighter than 15th magnitude.
313:
113:
5019:
Majaess, D.; Turner, D.; Lane, D. (2009). "Type II Cepheids as Extragalactic Distance Candles".
3962:
2441:
1485:
that have a very well-determined maximum absolute magnitude as a function of the shape of their
285:
5775:
5617:
5590:
5580:
5556:
5518:
5463:
4228:
4151:
4008:
3958:
3935:
3907:
3889:
3855:
3825:
3750:
3742:
3570:
3397:
3174:"Astronomical constants and planetary ephemerides deduced from radar and optical observations"
3068:
3023:
2999:
2905:
1482:
1393:
1385:
705:
472:
446:
403:
394:
method is useful for measuring the distances of bright stars beyond 50 parsecs and giant
363:
317:
207:
1227:
orbital plane, so face-on signals are intrinsically stronger and the most commonly observed.
464:
are near enough for this technique to be useful. In particular the distance obtained for the
5510:
5483:
5453:
5407:
5353:
5306:
5279:
5275:
5226:
5222:
5142:
5083:
4993:
4939:
4886:
4721:
4679:
4634:
4578:
4494:
4490:
4436:
4432:
4372:
4368:
4279:
4275:
4218:
4191:
4143:
4135:
4068:
4054:
4000:
3992:
3927:
3879:
3817:
3734:
3676:
3672:
3621:
3560:
3513:
3496:
3372:
3367:
3304:
3230:
3195:
3073:
2901:
2755:
2006:
1766:
1739:
1727:
1681:
1461:
1407:
The following four indicators all use stars in the old stellar populations (Population II):
1401:
1381:
1342:
721:
686:
476:
438:
399:
245:
2680:. There is no universal globular cluster luminosity function that applies to all galaxies.
1275:
1244:
248:
were crucial in determining the AU; in the first half of the 20th century, observations of
4819:
4781:
4756:
4341:"The long-period Galactic Cepheid RS Puppis I. A geometric distance from its light echoes"
4004:
1180:
782:
379:
371:
345:
mission obtained parallaxes for over a hundred thousand stars with a precision of about a
252:
were also important. Presently the orbit of Earth is determined with high precision using
5199:
Tammann, G. A.; Sandage, A.; Reindl, B. (2008). "The expansion field: The value of H 0".
1500:
1239:
1230:
If the binary consists of a pair of neutron stars, their merger will be accompanied by a
552:
is used), can be measured and used with the absolute magnitude to calculate the distance
525:
to an object's observed brightness, the distance to the object can be computed using the
5506:
5449:
5403:
5357:
5349:
5283:
5185:
5138:
5079:
5042:
4989:
4935:
4882:
4845:
4717:
4675:
4630:
4564:
4214:
4131:
3988:
3923:
3875:
3813:
3730:
3556:
3509:
3300:
3191:
3102:
fundamental origin in the scale-free nature of gravity in Einstein's general relativity.
2673:
is the magnitude of the Virgo cluster, and sigma is the dispersion ~ 1.4 mag.
281:
Stellar parallax motion from annual parallax. Half the apex angle is the parallax angle.
5702:
5087:
4775:"New Evidence Supporting Membership for TW Nor in LyngÄ 6 and the Centaurus Spiral Arm"
3083:
3019:
2676:
It is assumed that globular clusters all have roughly the same luminosities within the
2561:
1689:
1308:
979:
961:
937:
788:
774:
508:
501:
409:
374:, about 30,000 light years away. Stars have a velocity relative to the Sun that causes
27:
Succession of methods by which astronomers determine the distances to celestial objects
5162:
Majaess, D. (2010). "The Cepheids of Centaurus A (NGC 5128) and Implications for H0".
4139:
3308:
2005:
Cepheid variable stars were the key instrument in Edwin Hubble's 1923 conclusion that
5853:
5639:
5576:
5365:
5291:
5147:
5110:
4646:
4072:
3762:
3704:
2992:
2546:
1561:
1526:
1437:
1359:
1351:
Physical distance indicators, used on progressively larger distance scales, include:
778:
762:
395:
383:
375:
346:
99:
5671:
5419:
5238:
5095:
4951:
4898:
4774:
4749:
4590:
4405:"On geometric distance determination to the Cepheid RS Puppis from its light echoes"
4163:
3688:
3316:
5756:
5750:
5608:
The Globular Cluster Luminosity Function as a Distance Indicator: Dynamical Effects
5530:
5005:
4080:
4020:
3837:
3525:
3060:
3035:
1533:
1449:
1441:
766:
725:
461:
257:
4499:
4462:
4441:
4404:
3680:
3377:
3334:
5634:
5584:
4377:
4340:
4284:
3443:
1172:{\displaystyle {\mathcal {M}}={\frac {(m_{1}m_{2})^{3/5}}{(m_{1}+m_{2})^{1/5}}}.}
4700:"A critical review of selected techniques for measuring extragalactic distances"
3931:
2693:
2484:
2065:
1711:
1486:
1369:
734:
698:
492:
488:
484:
451:
418:
414:
3738:
3417:
3010:
in the nearer steps, both systematic and statistical ones. The result of these
17:
5811:
5805:
5799:
5787:
5230:
4582:
3050:
1330:
1007:
770:
522:
325:
195:
146:
5650:
5467:
4155:
4012:
3939:
3893:
3821:
3746:
3574:
3043:
and distant galaxies in which individual distance indicators cannot be seen.
5781:
5769:
5706:
5661:
5656:
2978:
where C is a constant which depends on the distance to the galaxy clusters.
2045:
1490:
1397:
1235:
549:
442:
342:
299:
249:
178:
161:
5522:
4232:
3829:
3754:
3613:
42:
5646:
The Hubble Space Telescope Key Project on the Extragalactic Distance Scale
4310:
2553:
carries an uncertainty of distance of about 20% (or 0.4 magnitudes).
1315:(BAO) in the early universe has been used. In the early universe (before
697:
Two problems exist for any class of standard candle. The principal one is
5394:
5326:
Coelho, R.; et al. (2015). "Standardization of type Ia supernovae".
5111:"The effect of metallicity on Cepheid magnitudes and the distance to M33"
4926:
4873:
4836:
4804:
4621:
4555:
3479:
2982:
such as Cepheids. Instead, calibration is done using more crude methods.
2677:
2029:
2025:
1496:
1269:
1231:
1223:
1179:
By observing the waveform, the chirp mass can be computed and thence the
746:
497:
422:
349:, providing useful distances for stars out to a few hundred parsecs. The
303:
182:
5514:
4735:
4660:
Colgate, S. A. (1979). "Supernovae as a standard candle for cosmology".
4223:
4187:"A gravitational-wave standard siren measurement of the Hubble constant"
4186:
4147:
3246:"NASA's Hubble Extends Stellar Tape Measure 10 Times Farther Into Space"
2557:
Baum has assumed a direct correlation and estimated Virgo A's distance.
1988:{\displaystyle 5\log _{10}{d}=V+(3.37)\log _{10}{P}-(2.55)(V-I)+7.48\,.}
1876:{\displaystyle 5\log _{10}{d}=V+(3.34)\log _{10}{P}-(2.45)(V-I)+7.52\,.}
1517:
When the absolute magnitude for a group of stars is plotted against the
330:
277:
5666:
3963:"Determining the Hubble constant from gravitational wave observations"
3884:
3859:
3804:
5793:
3996:
3517:
3040:
1685:
1565:
153:
that can be used to determine the distances at the next higher rung.
142:
5645:
5252:
Freedman, Wendy L.; Madore, Barry F. (2010). "The Hubble Constant".
2188:
is the supernova's ejecta's radial velocity (it can be assumed that
5458:
5433:
5411:
4997:
4943:
4890:
4726:
4699:
4683:
4638:
4205:
3721:
3565:
3540:
3357:
3215:
Perryman, M. A. C.; et al. (1999). "The HIPPARCOS Catalogue".
3200:
3173:
2754:
Like the GCLF method, a similar numerical analysis can be used for
2040:, The Hubble Key Project Team, and The High-Z Supernova Search Team
2021:
1493:, the "Champagne Supernova", a Type Ia supernova of unusual nature.
5497:
5340:
5266:
5213:
5176:
5129:
5070:
5033:
4980:
4481:
4423:
4359:
4266:
4122:
3979:
3777:"When Neutron Stars Collide, the Explosion is Perfectly Spherical"
3663:
3291:
3007:
2870:
2020:
428:
408:
284:
276:
253:
229:
225:
213:
160:
2431:{\displaystyle \ M_{V}^{\max }=-9.96-2.31\log _{10}{\dot {x}}\,.}
3249:
2353:
2033:
1677:
1465:
548:, the magnitude as seen by the observer (an instrument called a
468:
has historically been an important step in the distance ladder.
390:
of the proper motions relative to their radial velocities. This
5675:
5109:
Scowcroft, V.; Bersier, D.; Mould, J. R.; Wood, P. R. (2009).
3339:
Data Release 2: Summary of the contents and survey properties"
2687:
2478:
2037:
334:(parallax arcseconds); light-years are used in popular media.
217:
36:
5616:, Richard de Grijs, Chichester: John Wiley & Sons, 2011,
3444:"Hubble stretches the stellar tape measure ten times further"
3418:"Hubble finds Universe may be expanding faster than expected"
2856:{\displaystyle \ N(M)\propto e^{0.307M}(1-e^{3(M^{*}-M)})\,.}
2110:{\displaystyle \omega ={\frac {\Delta \theta }{\Delta t}}\,,}
540:
The brightness of an object can be expressed in terms of its
2652:{\displaystyle \ \Phi (m)=Ae^{(m-m_{0})^{2}/2\sigma ^{2}}\,}
1065:
991:
867:
2313:{\displaystyle \ M_{B}\approx M_{V}\approx -19.3\pm 0.3\,.}
1734:. The distance to the star can then be calculated from its
2908:, relates the angular diameter (D) of the galaxy to its
5589:(4th ed.). Cambridge: Cambridge University Press.
5570:
Measuring the Universe The Cosmological Distance Ladder
4705:
Publications of the Astronomical Society of the Pacific
3705:"Spherical symmetry in the kilonova AT2017gfo/GW170817"
3545:
Publications of the Astronomical Society of the Pacific
122:
Dashed black lines: Uncertain calibration ladder step.
30:
For various definitions of distance in cosmology, see
3541:"The Period-Luminosity Relation: A Historical Review"
2925:
2773:
2572:
2444:
2365:
2264:
2228:
2137:
2076:
1890:
1778:
1362:
to measure the mass of the system, and hence use the
1278:
1247:
1062:
1016:
1006:
is a single (therefore computable) number called the
988:
964:
940:
811:
791:
619:
562:
357:
has the potential to provide a precision of 20 to 40
5614:
An Introduction to Distance Measurement in Astronomy
4403:
Bond, Howard E.; Sparks, William B. (4 March 2009).
3027:
useful simply because there is no real alternative.
1307:
Another class of physical distance indicator is the
709:
Ia supernovae is crucial in determining the correct
370:Distances can be measured within 10% as far as the
341:of the measurement. In the 1990s, for example, the
3589:"Finding the Distance to Stars - Distance Modulus"
3333:; et al. (Gaia collaboration) (August 2018).
2970:
2892:which is directly related to a galaxy's distance.
2855:
2651:
2459:
2430:
2312:
2244:
2173:
2109:
2002:supplying a precise value of the Hubble constant.
1987:
1875:
1291:
1260:
1171:
1048:
998:
970:
946:
926:
797:
664:
605:
112:technique is applicable to all populations of the
5116:Monthly Notices of the Royal Astronomical Society
3860:"Measuring cosmic distances with standard sirens"
3172:Ash, M. E.; Shapiro, I. I.; Smith, W. B. (1967).
1206:due to an intervening absorbing medium. But they
105:Light Purple boxes: Geometric distance technique.
3244:Harrington, J. D.; Villard, R. (10 April 2014).
2379:
2174:{\displaystyle \ d={\frac {V_{ej}}{\omega }}\,,}
2048:can be used to measure extragalactic distances.
324:for an object at 1 parsec's distance (3.26
529:. These objects of known brightness are termed
445:disk, this corresponds to a mean baseline of 4
119:Solid black lines: Well calibrated ladder step.
4034:Hendry, Martin; Woan, Graham (February 2007).
3030:The observational result of Hubble's Law, the
2044:There are several different methods for which
5687:
5547:Carroll, Bradley W.; Ostlie, Dale A. (2014).
4311:"Light echoes whisper the distance to a star"
8:
5875:Length, distance, or range measuring devices
4788:, Volume 741, Issue 2, article id. L2 (2011)
2971:{\displaystyle \log(D)=1.333\log(\Sigma )+C}
5586:The Cosmos: Astronomy in the New Millennium
5254:Annual Review of Astronomy and Astrophysics
5201:Annual Review of Astronomy and Astrophysics
4763:, Volume 133, Issue 4, pp. 1810â1827 (2007)
4608:The Astrophysical Journal Supplement Series
4524:: CS1 maint: numeric names: authors list (
3953:
3951:
3949:
3849:
3847:
2887:(SBF) method takes advantage of the use of
2722:. Unsourced material may be challenged and
2513:. Unsourced material may be challenged and
1440:(thermonuclear flashes on the surface of a
1214:, in the same way as light. If a signal is
1010:of the system, a combination of the masses
91:Light green boxes: Technique applicable to
5694:
5680:
5672:
2181:where d is the distance to the supernova,
1396:â used for measuring distances within the
98:Light blue boxes: Technique applicable to
5496:
5457:
5393:
5339:
5265:
5212:
5175:
5146:
5128:
5069:
5032:
4979:
4925:
4872:
4835:
4803:
4725:
4620:
4572:
4554:
4498:
4480:
4440:
4422:
4376:
4358:
4283:
4265:
4222:
4204:
4121:
4062:
3978:
3883:
3803:
3720:
3662:
3591:. Australia Telescope National Facility.
3564:
3478:
3376:
3366:
3356:
3290:
3199:
3079:Orders of magnitude (length)#Astronomical
2924:
2849:
2829:
2818:
2796:
2772:
2742:Learn how and when to remove this message
2648:
2640:
2628:
2622:
2612:
2598:
2571:
2533:Learn how and when to remove this message
2446:
2445:
2443:
2424:
2413:
2412:
2403:
2378:
2373:
2364:
2306:
2285:
2272:
2263:
2236:
2227:
2167:
2153:
2147:
2136:
2103:
2083:
2075:
1981:
1943:
1934:
1907:
1898:
1889:
1869:
1831:
1822:
1795:
1786:
1777:
1283:
1277:
1252:
1246:
1153:
1149:
1139:
1126:
1107:
1103:
1093:
1083:
1073:
1064:
1063:
1061:
1037:
1024:
1015:
990:
989:
987:
963:
939:
912:
907:
891:
876:
866:
865:
849:
845:
835:
812:
810:
790:
652:
630:
618:
573:
561:
2341:, the peak magnitude can be determined.
1547:
606:{\displaystyle 5\cdot \log _{10}d=m-M+5}
83:
67:of all important aspects of the article.
3113:
3094:
2028:(bright spot on the lower left) in the
5549:An Introduction to Modern Astrophysics
4517:
3424:from the original on 11 September 2018
63:Please consider expanding the lead to
3628:from the original on 29 February 2012
3454:from the original on October 30, 2017
3256:from the original on 17 February 2019
2867:Surface brightness fluctuation method
1618:Globular Cluster Luminosity Function
1604:Planetary Nebula Luminosity Function
556:to the object in parsecs as follows:
479:, can be observed over time, then an
7:
2760:planetary nebula luminosity function
2720:adding citations to reliable sources
2684:Planetary nebula luminosity function
2551:globular cluster luminosity function
2511:adding citations to reliable sources
2475:Globular cluster luminosity function
1722:. Many stars have features in their
1557:Uncertainty for Single Galaxy (mag)
1423:Globular cluster luminosity function
1417:Planetary nebula luminosity function
110:planetary nebula luminosity function
5610:, Ostriker and Gnedin, May 5, 1997.
5284:10.1146/annurev-astro-082708-101829
4448:distance to this important Cepheid.
3854:Holz, Daniel E.; Hughes, Scott A.;
2052:Measuring a supernova's photosphere
765:of compact binary systems, such as
4383:we derive a geometric distance of
4246:Marinoni, C.; et al. (2008).
2956:
2576:
2094:
2086:
1549:Extragalactic distance indicators
25:
1197:gravitational wave interferometer
5640:The Extragalactic Distance Scale
5312:NASA/IPAC Extragalactic Database
5148:10.1111/j.1365-2966.2009.14822.x
4073:10.1111/j.1468-4004.2007.48110.x
3396:. PHI Learning Private Limited.
3053:
2692:
2483:
2349:Novae in distance determinations
1761:, the next method relies on the
1632:Surface Brightness Fluctuations
665:{\displaystyle d=10^{(m-M+5)/5}}
378:(transverse across the sky) and
269:This section is an excerpt from
41:
5572:, Stephen Webb, copyright 2001.
5307:"Calibration and Uncertainties"
4321:from the original on 2015-09-24
4292:from the original on 2022-01-26
4090:from the original on 2012-12-22
3595:from the original on 2020-11-07
3539:Fernie, J. D. (December 1969).
3394:An Introduction to Astrophysics
3128:from the original on 9 May 2021
676:is the apparent magnitude, and
294:The most important fundamental
55:may be too short to adequately
5865:Physical cosmological concepts
4005:11858/00-001M-0000-0013-73C1-2
2959:
2953:
2938:
2932:
2885:surface brightness fluctuation
2846:
2841:
2822:
2805:
2786:
2780:
2619:
2599:
2585:
2579:
1972:
1960:
1957:
1951:
1927:
1921:
1860:
1848:
1845:
1839:
1815:
1809:
1429:Surface brightness fluctuation
1146:
1119:
1100:
1076:
1043:
1017:
999:{\displaystyle {\mathcal {M}}}
873:
859:
649:
631:
244:Historically, observations of
189:At the base of the ladder are
65:provide an accessible overview
1:
5635:The ABC's of distances (UCLA)
5358:10.1088/0143-0807/36/1/015007
4315:European Southern Observatory
4179:LIGO Scientific Collaboration
2758:within far off galaxies. The
2245:{\displaystyle 1.4M_{\odot }}
1358:, uses orbital parameters of
1049:{\displaystyle (m_{1},m_{2})}
32:Distance measures (cosmology)
5834:Orders of magnitude (length)
5718:Astronomical system of units
5088:10.1088/0004-637X/696/2/1498
4542:Proceedings of IAU Symposium
4177:Abbott, B. P.; et al. (
4036:"Gravitational astrophysics"
3344:Astronomy & Astrophysics
3039:estimating the distances of
2323:multicolor light curve shape
1544:Extragalactic distance scale
1366:to determine the luminosity
1337:Galactic distance indicators
1325:baryon acoustic oscillations
1313:baryon acoustic oscillations
135:extragalactic distance scale
5328:European Journal of Physics
4500:10.1051/0004-6361/201424395
4442:10.1051/0004-6361:200810280
4140:10.1088/0004-637X/725/1/496
3932:10.1103/PhysRevLett.18.1071
3681:10.1051/0004-6361/200912811
3378:10.1051/0004-6361/201833051
3309:10.1088/0004-637X/785/2/161
2904:(or ÎŁ-D relation), used in
2666:is the turnover magnitude,
2202:if spherically symmetric).
2129:. Subsequently, we can use
2051:
1532:In a gravitationally-bound
1523:HertzsprungâRussell diagram
1411:Tip of the red-giant branch
5896:
5657:NASA Cosmic Distance Scale
5551:. Harlow, United Kingdom:
4748:Benedict, G. Fritz et al.
4468:Astronomy and Astrophysics
4410:Astronomy and Astrophysics
4378:10.1051/0004-6361:20078961
4346:Astronomy and Astrophysics
4285:10.1051/0004-6361:20077116
4253:Astronomy and Astrophysics
4043:Astronomy & Geophysics
3739:10.1038/s41586-022-05616-x
3650:Astronomy and Astrophysics
3218:Astronomy and Astrophysics
2460:{\displaystyle {\dot {x}}}
2068:, we can use the equation
1763:period-luminosity relation
1699:
1658:
1644:
1630:
1616:
1602:
1588:
1574:
1552:
1510:
1413:(TRGB) distance indicator.
1340:
500:(usually obtained via the
268:
205:
29:
5821:
5726:
5713:
5667:The Astrophysical Journal
5662:PNLF information database
5653:, a historical discussion
5604:The Astrophysical Journal
5553:Pearson Education Limited
5438:The Astrophysical Journal
5231:10.1007/s00159-008-0012-y
5058:The Astrophysical Journal
4968:The Astrophysical Journal
4914:The Astrophysical Journal
4861:The Astrophysical Journal
4583:10.1017/S1743921307003845
4110:The Astrophysical Journal
3693:(And references therein.)
3278:The Astrophysical Journal
1718:uses the effect known as
737:content than the distant
175:Mount Stromlo Observatory
4761:The Astronomical Journal
4505:We obtain a distance of
3822:10.1103/PhysRevD.49.2658
3179:The Astronomical Journal
3149:"Cosmic distance ladder"
1757:Beyond the reach of the
1364:massâluminosity relation
1268:. This was the case for
302:come from trigonometric
5432:Dressler, Alan (1987).
5276:2010ARA&A..48..673F
5223:2008A&ARv..15..289T
4491:2014A&A...572A...7K
4433:2009A&A...495..371B
4369:2008A&A...480..167K
4276:2008A&A...478...43M
4055:2007A&G....48a..10H
3912:Physical Review Letters
3673:2009A&A...506.1095L
3392:B., Baidyanath (2003).
3368:2018A&A...616A...1G
3231:1997A&A...323L..49P
1747:interstellar extinction
1519:spectral classification
1436:In galactic astronomy,
733:stars with much higher
458:Moving cluster parallax
309:stellar parallax method
171:2003 Canberra bushfires
5829:Cosmic distance ladder
2972:
2876:
2857:
2653:
2461:
2432:
2314:
2246:
2175:
2111:
2041:
1989:
1877:
1730:, that indicate their
1720:spectroscopic parallax
1706:Discovered in 1956 by
1513:Spectroscopic parallax
1478:FaberâJackson relation
1378:Small Magellanic Cloud
1374:Large Magellanic Cloud
1293:
1262:
1173:
1050:
1000:
972:
956:gravitational constant
948:
928:
799:
666:
607:
535:Henrietta Swan Leavitt
434:
426:
351:Hubble Space Telescope
291:
282:
186:
131:cosmic distance ladder
126:
5880:Concepts in astronomy
5382:Astrophysical Journal
4663:Astrophysical Journal
4185:) (16 October 2017).
3961:(25 September 1986).
2973:
2874:
2858:
2654:
2462:
2433:
2315:
2247:
2176:
2121:is angular velocity,
2112:
2024:
1990:
1878:
1507:Main sequence fitting
1472:TullyâFisher relation
1294:
1292:{\displaystyle H_{0}}
1263:
1261:{\displaystyle H_{0}}
1212:gravitational lensing
1174:
1051:
1001:
973:
949:
929:
800:
761:originating from the
667:
608:
432:
412:
296:distance measurements
288:
280:
271:Parallax in astronomy
164:
93:star-forming galaxies
87:
4317:. 11 February 2008.
2923:
2771:
2716:improve this section
2570:
2507:improve this section
2442:
2363:
2262:
2226:
2210:Type Ia light curves
2135:
2074:
1888:
1884:
1776:
1772:
1446:Eddington luminosity
1276:
1245:
1060:
1014:
986:
962:
938:
809:
789:
617:
560:
454:of the sample size.
392:statistical parallax
388:statistical analysis
306:, as applied in the
5839:Conversion of units
5651:The Hubble Constant
5515:10.1038/nature08685
5507:2010Natur.463..924G
5450:1987ApJ...317....1D
5404:2001ApJ...546..681T
5350:2015EJPh...36a5007C
5186:2010AcA....60..121M
5139:2009MNRAS.396.1287S
5080:2009ApJ...696.1498M
5043:2009AcA....59..403M
4990:2008ApJ...684..102B
4936:2006ApJ...652.1133M
4883:2006ApJ...642L..29N
4846:2001AcA....51..221U
4718:1992PASP..104..599J
4676:1979ApJ...232..404C
4631:2000ApJS..128..431F
4565:2007IAUS..240...79B
4224:10.1038/nature24471
4215:2017Natur.551...85A
4183:Virgo Collaboration
4132:2010ApJ...725..496N
3989:1986Natur.323..310S
3924:1967PhRvL..18.1071F
3876:2018PhT....71l..34H
3814:1994PhRvD..49.2658C
3779:. 17 February 2023.
3731:2023Natur.614..436S
3614:"Type Ia Supernova"
3557:1969PASP...81..707F
3510:1994Natur.368..610B
3301:2014ApJ...785..161R
3192:1967AJ.....72..338A
3000:elliptical galaxies
2986:Overlap and scaling
2917:between D and ÎŁ is
2910:velocity dispersion
2906:elliptical galaxies
2383:
2220:Chandrasekhar limit
1759:WilsonâBappu effect
1716:WilsonâBappu effect
1702:WilsonâBappu effect
1696:WilsonâBappu effect
1660:Type Ia Supernovae
1576:Classical Cepheids
1550:
1456:Interstellar masers
1187:of known loudness.
1056:of the two objects
759:Gravitational waves
441:. For stars in the
355:Wide Field Camera 3
290:would be minuscule.
133:(also known as the
108:Light Red box: The
4780:2017-03-10 at the
4755:2016-01-23 at the
3959:Schutz, Bernard F.
3908:Forward, Robert L.
3856:Schutz, Bernard F.
3012:propagating errors
2968:
2914:surface brightness
2877:
2853:
2649:
2457:
2428:
2369:
2310:
2242:
2215:Type Ia supernovae
2171:
2107:
2042:
1985:
1873:
1753:Classical Cepheids
1736:apparent magnitude
1732:absolute magnitude
1548:
1521:of the star, in a
1483:Type Ia supernovae
1394:RR Lyrae variables
1370:Eclipsing binaries
1356:Dynamical parallax
1289:
1258:
1169:
1046:
996:
968:
944:
924:
795:
711:cosmological model
706:Type Ia supernovae
662:
603:
546:apparent magnitude
542:absolute magnitude
527:inverse-square law
481:expansion parallax
435:
427:
404:RR Lyrae variables
314:isosceles triangle
292:
283:
187:
157:Direct measurement
127:
114:Virgo Supercluster
5847:
5846:
5776:Astronomical unit
5622:978-0-470-51180-0
5596:978-1-107-68756-1
5562:978-1-292-02293-2
4313:(Press release).
3973:(6086): 310â311.
3918:(24): 1071â1074.
3885:10.1063/PT.3.4090
3858:(December 2018).
3792:Physical Review D
3715:(7948): 436â439.
3504:(6472): 610â613.
3448:ESA/Hubble Images
3403:978-81-203-1121-3
3124:. 16 April 2013.
3069:Araucaria Project
2776:
2756:planetary nebulae
2752:
2751:
2744:
2575:
2543:
2542:
2535:
2454:
2421:
2368:
2267:
2165:
2140:
2101:
2032:galaxy. Image by
1682:globular clusters
1673:
1672:
1646:Sigma-D relation
1402:globular clusters
1386:Triangulum Galaxy
1164:
971:{\displaystyle c}
947:{\displaystyle G}
919:
899:
884:
830:
798:{\displaystyle f}
722:Cepheid variables
715:density parameter
473:supernova remnant
208:Astronomical unit
202:Astronomical unit
82:
81:
16:(Redirected from
5887:
5870:Standard candles
5696:
5689:
5682:
5673:
5600:
5566:
5535:
5534:
5500:
5478:
5472:
5471:
5461:
5429:
5423:
5422:
5397:
5395:astro-ph/0011223
5376:
5370:
5369:
5343:
5323:
5317:
5316:
5302:
5296:
5295:
5269:
5249:
5243:
5242:
5216:
5196:
5190:
5189:
5179:
5164:Acta Astronomica
5159:
5153:
5152:
5150:
5132:
5106:
5100:
5099:
5073:
5064:(2): 1498â1501.
5053:
5047:
5046:
5036:
5021:Acta Astronomica
5016:
5010:
5009:
4983:
4962:
4956:
4955:
4929:
4927:astro-ph/0608211
4920:(2): 1133â1149.
4909:
4903:
4902:
4876:
4874:astro-ph/0603643
4856:
4850:
4849:
4839:
4837:astro-ph/0109446
4824:Acta Astronomica
4816:
4810:
4809:
4807:
4805:astro-ph/9909346
4795:
4789:
4770:
4764:
4746:
4740:
4739:
4729:
4712:(678): 599â662.
4694:
4688:
4687:
4657:
4651:
4650:
4624:
4622:astro-ph/9910501
4601:
4595:
4594:
4576:
4558:
4556:astro-ph/0610923
4536:
4530:
4529:
4523:
4515:
4512:
4510:
4502:
4484:
4457:
4451:
4450:
4444:
4426:
4400:
4394:
4393:
4390:
4388:
4380:
4362:
4336:
4330:
4329:
4327:
4326:
4307:
4301:
4300:
4298:
4297:
4287:
4269:
4243:
4237:
4236:
4226:
4208:
4174:
4168:
4167:
4125:
4105:
4099:
4098:
4096:
4095:
4089:
4066:
4049:(1): 1.10â1.17.
4040:
4031:
4025:
4024:
3997:10.1038/323310a0
3982:
3955:
3944:
3943:
3904:
3898:
3897:
3887:
3851:
3842:
3841:
3807:
3798:(6): 2658â2697.
3787:
3781:
3780:
3773:
3767:
3766:
3724:
3700:
3694:
3692:
3666:
3657:(3): 1095â1105.
3644:
3638:
3637:
3635:
3633:
3610:
3604:
3603:
3601:
3600:
3585:
3579:
3578:
3568:
3536:
3530:
3529:
3518:10.1038/368610a0
3491:
3485:
3484:
3482:
3480:astro-ph/9703140
3470:
3464:
3463:
3461:
3459:
3440:
3434:
3433:
3431:
3429:
3414:
3408:
3407:
3389:
3383:
3382:
3380:
3370:
3360:
3327:
3321:
3320:
3294:
3272:
3266:
3265:
3263:
3261:
3241:
3235:
3234:
3212:
3206:
3205:
3203:
3169:
3163:
3162:
3160:
3158:
3153:
3144:
3138:
3137:
3135:
3133:
3122:"The Astronomer"
3118:
3103:
3099:
3074:Distance measure
3063:
3058:
3057:
3056:
2977:
2975:
2974:
2969:
2902:Sigma-D relation
2896:Sigma-D relation
2862:
2860:
2859:
2854:
2845:
2844:
2834:
2833:
2804:
2803:
2774:
2747:
2740:
2736:
2733:
2727:
2696:
2688:
2658:
2656:
2655:
2650:
2647:
2646:
2645:
2644:
2632:
2627:
2626:
2617:
2616:
2573:
2538:
2531:
2527:
2524:
2518:
2487:
2479:
2466:
2464:
2463:
2458:
2456:
2455:
2447:
2437:
2435:
2434:
2429:
2423:
2422:
2414:
2408:
2407:
2382:
2377:
2366:
2319:
2317:
2316:
2311:
2290:
2289:
2277:
2276:
2265:
2251:
2249:
2248:
2243:
2241:
2240:
2180:
2178:
2177:
2172:
2166:
2161:
2160:
2148:
2138:
2116:
2114:
2113:
2108:
2102:
2100:
2092:
2084:
1994:
1992:
1991:
1986:
1947:
1939:
1938:
1911:
1903:
1902:
1882:
1880:
1879:
1874:
1835:
1827:
1826:
1799:
1791:
1790:
1767:Cepheid variable
1740:distance modulus
1712:M.K. Vainu Bappu
1551:
1382:Andromeda Galaxy
1343:Distance measure
1298:
1296:
1295:
1290:
1288:
1287:
1267:
1265:
1264:
1259:
1257:
1256:
1178:
1176:
1175:
1170:
1165:
1163:
1162:
1161:
1157:
1144:
1143:
1131:
1130:
1117:
1116:
1115:
1111:
1098:
1097:
1088:
1087:
1074:
1069:
1068:
1055:
1053:
1052:
1047:
1042:
1041:
1029:
1028:
1005:
1003:
1002:
997:
995:
994:
977:
975:
974:
969:
953:
951:
950:
945:
933:
931:
930:
925:
920:
918:
917:
916:
902:
901:
900:
892:
886:
885:
877:
871:
870:
858:
857:
853:
836:
831:
829:
821:
813:
804:
802:
801:
796:
724:. In the 1950s,
687:distance modulus
671:
669:
668:
663:
661:
660:
656:
612:
610:
609:
604:
578:
577:
531:standard candles
517:Standard candles
477:planetary nebula
439:secular parallax
240:
238:
224:provide precise
77:
74:
68:
45:
37:
21:
5895:
5894:
5890:
5889:
5888:
5886:
5885:
5884:
5850:
5849:
5848:
5843:
5817:
5765:
5762:
5745:
5739:
5736:
5722:
5709:
5703:Units of length
5700:
5631:
5597:
5577:Pasachoff, J.M.
5575:
5563:
5546:
5543:
5538:
5480:
5479:
5475:
5431:
5430:
5426:
5378:
5377:
5373:
5325:
5324:
5320:
5304:
5303:
5299:
5251:
5250:
5246:
5198:
5197:
5193:
5161:
5160:
5156:
5108:
5107:
5103:
5055:
5054:
5050:
5018:
5017:
5013:
4964:
4963:
4959:
4911:
4910:
4906:
4858:
4857:
4853:
4818:
4817:
4813:
4797:
4796:
4792:
4782:Wayback Machine
4771:
4767:
4757:Wayback Machine
4747:
4743:
4697:
4695:
4691:
4659:
4658:
4654:
4603:
4602:
4598:
4574:10.1.1.254.2692
4538:
4537:
4533:
4516:
4508:
4506:
4475:: A7 (13 pp.).
4459:
4458:
4454:
4402:
4401:
4397:
4386:
4384:
4338:
4337:
4333:
4324:
4322:
4309:
4308:
4304:
4295:
4293:
4245:
4244:
4240:
4199:(7678): 85â88.
4176:
4175:
4171:
4107:
4106:
4102:
4093:
4091:
4087:
4064:10.1.1.163.5500
4038:
4033:
4032:
4028:
3957:
3956:
3947:
3906:
3905:
3901:
3853:
3852:
3845:
3789:
3788:
3784:
3775:
3774:
3770:
3702:
3701:
3697:
3646:
3645:
3641:
3631:
3629:
3612:
3611:
3607:
3598:
3596:
3587:
3586:
3582:
3538:
3537:
3533:
3493:
3492:
3488:
3472:
3471:
3467:
3457:
3455:
3442:
3441:
3437:
3427:
3425:
3416:
3415:
3411:
3404:
3391:
3390:
3386:
3331:Brown, A. G. A.
3329:
3328:
3324:
3274:
3273:
3269:
3259:
3257:
3243:
3242:
3238:
3214:
3213:
3209:
3171:
3170:
3166:
3156:
3154:
3151:
3146:
3145:
3141:
3131:
3129:
3120:
3119:
3115:
3111:
3106:
3100:
3096:
3092:
3059:
3054:
3052:
3049:
2993:spiral galaxies
2988:
2921:
2920:
2898:
2869:
2825:
2814:
2792:
2769:
2768:
2748:
2737:
2731:
2728:
2713:
2697:
2686:
2672:
2665:
2636:
2618:
2608:
2594:
2568:
2567:
2539:
2528:
2522:
2519:
2504:
2488:
2477:
2440:
2439:
2399:
2361:
2360:
2351:
2332:Similarly, the
2281:
2268:
2260:
2259:
2232:
2224:
2223:
2212:
2200:
2193:
2186:
2149:
2133:
2132:
2093:
2085:
2072:
2071:
2054:
2019:
1930:
1894:
1886:
1885:
1883:
1818:
1782:
1774:
1773:
1755:
1704:
1698:
1690:galaxy clusters
1546:
1515:
1509:
1360:visual binaries
1345:
1339:
1305:
1279:
1274:
1273:
1248:
1243:
1242:
1240:Hubble constant
1216:strongly lensed
1145:
1135:
1122:
1118:
1099:
1089:
1079:
1075:
1058:
1057:
1033:
1020:
1012:
1011:
984:
983:
960:
959:
936:
935:
908:
903:
887:
872:
841:
837:
822:
814:
807:
806:
787:
786:
756:
695:
626:
615:
614:
569:
558:
557:
519:
514:
513:
509:interferometers
487:which are both
380:radial velocity
372:Galactic Center
274:
266:
236:
234:
210:
204:
173:burned out the
159:
125:
78:
72:
69:
62:
50:This article's
46:
35:
28:
23:
22:
18:Standard sirens
15:
12:
11:
5:
5893:
5891:
5883:
5882:
5877:
5872:
5867:
5862:
5852:
5851:
5845:
5844:
5842:
5841:
5836:
5831:
5826:
5822:
5819:
5818:
5816:
5815:
5809:
5803:
5797:
5791:
5785:
5779:
5773:
5767:
5763:
5760:
5754:
5748:
5743:
5737:
5734:
5727:
5724:
5723:
5721:
5720:
5714:
5711:
5710:
5701:
5699:
5698:
5691:
5684:
5676:
5670:
5669:
5664:
5659:
5654:
5648:
5643:
5637:
5630:
5629:External links
5627:
5626:
5625:
5611:
5601:
5595:
5581:Filippenko, A.
5573:
5567:
5561:
5542:
5539:
5537:
5536:
5491:(3): 924â925.
5473:
5459:10.1086/165251
5424:
5412:10.1086/318301
5388:(2): 681â693,
5371:
5318:
5297:
5244:
5191:
5154:
5101:
5048:
5011:
4998:10.1086/589965
4957:
4944:10.1086/508530
4904:
4891:10.1086/504478
4851:
4811:
4790:
4765:
4741:
4727:10.1086/133035
4689:
4684:10.1086/157300
4670:(1): 404â408.
4652:
4639:10.1086/313391
4615:(2): 431â459.
4596:
4531:
4452:
4417:(2): 371â377.
4395:
4353:(1): 167â178.
4331:
4302:
4238:
4169:
4116:(1): 496â514.
4100:
4026:
3945:
3899:
3843:
3782:
3768:
3695:
3639:
3605:
3580:
3566:10.1086/128847
3531:
3486:
3465:
3435:
3409:
3402:
3384:
3322:
3267:
3236:
3207:
3201:10.1086/110230
3164:
3139:
3112:
3110:
3107:
3105:
3104:
3093:
3091:
3088:
3087:
3086:
3084:Standard ruler
3081:
3076:
3071:
3065:
3064:
3048:
3045:
2987:
2984:
2967:
2964:
2961:
2958:
2955:
2952:
2949:
2946:
2943:
2940:
2937:
2934:
2931:
2928:
2897:
2894:
2875:Galaxy cluster
2868:
2865:
2852:
2848:
2843:
2840:
2837:
2832:
2828:
2824:
2821:
2817:
2813:
2810:
2807:
2802:
2799:
2795:
2791:
2788:
2785:
2782:
2779:
2750:
2749:
2700:
2698:
2691:
2685:
2682:
2670:
2663:
2643:
2639:
2635:
2631:
2625:
2621:
2615:
2611:
2607:
2604:
2601:
2597:
2593:
2590:
2587:
2584:
2581:
2578:
2541:
2540:
2491:
2489:
2482:
2476:
2473:
2453:
2450:
2427:
2420:
2417:
2411:
2406:
2402:
2398:
2395:
2392:
2389:
2386:
2381:
2376:
2372:
2350:
2347:
2339:Stretch Factor
2334:stretch method
2309:
2305:
2302:
2299:
2296:
2293:
2288:
2284:
2280:
2275:
2271:
2239:
2235:
2231:
2211:
2208:
2198:
2191:
2184:
2170:
2164:
2159:
2156:
2152:
2146:
2143:
2106:
2099:
2096:
2091:
2088:
2082:
2079:
2053:
2050:
2018:
2015:
1984:
1980:
1977:
1974:
1971:
1968:
1965:
1962:
1959:
1956:
1953:
1950:
1946:
1942:
1937:
1933:
1929:
1926:
1923:
1920:
1917:
1914:
1910:
1906:
1901:
1897:
1893:
1872:
1868:
1865:
1862:
1859:
1856:
1853:
1850:
1847:
1844:
1841:
1838:
1834:
1830:
1825:
1821:
1817:
1814:
1811:
1808:
1805:
1802:
1798:
1794:
1789:
1785:
1781:
1754:
1751:
1728:calcium K-line
1726:, such as the
1700:Main article:
1697:
1694:
1671:
1670:
1667:
1664:
1661:
1657:
1656:
1653:
1650:
1647:
1643:
1642:
1639:
1636:
1633:
1629:
1628:
1625:
1622:
1619:
1615:
1614:
1611:
1608:
1605:
1601:
1600:
1597:
1594:
1591:
1587:
1586:
1583:
1580:
1577:
1573:
1572:
1569:
1558:
1555:
1545:
1542:
1511:Main article:
1508:
1505:
1504:
1503:
1494:
1480:
1474:
1468:
1459:
1453:
1450:X-ray binaries
1434:
1433:
1432:
1426:
1420:
1414:
1405:
1400:and in nearby
1391:
1390:
1389:
1338:
1335:
1309:standard ruler
1304:
1303:Standard ruler
1301:
1286:
1282:
1255:
1251:
1185:standard siren
1168:
1160:
1156:
1152:
1148:
1142:
1138:
1134:
1129:
1125:
1121:
1114:
1110:
1106:
1102:
1096:
1092:
1086:
1082:
1078:
1072:
1067:
1045:
1040:
1036:
1032:
1027:
1023:
1019:
993:
980:speed of light
967:
943:
923:
915:
911:
906:
898:
895:
890:
883:
880:
875:
869:
864:
861:
856:
852:
848:
844:
840:
834:
828:
825:
820:
817:
794:
783:rate of change
775:orbital energy
763:inspiral phase
755:
754:Standard siren
752:
745:Most recently
694:
691:
659:
655:
651:
648:
645:
642:
639:
636:
633:
629:
625:
622:
602:
599:
596:
593:
590:
587:
584:
581:
576:
572:
568:
565:
518:
515:
502:Doppler effect
396:variable stars
347:milliarcsecond
275:
267:
265:
262:
246:Venus transits
206:Main article:
203:
200:
167:The Astronomer
165:The sculpture
158:
155:
124:
123:
120:
117:
106:
103:
96:
88:
80:
79:
59:the key points
49:
47:
40:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5892:
5881:
5878:
5876:
5873:
5871:
5868:
5866:
5863:
5861:
5858:
5857:
5855:
5840:
5837:
5835:
5832:
5830:
5827:
5824:
5823:
5820:
5813:
5810:
5807:
5804:
5801:
5798:
5795:
5792:
5789:
5786:
5783:
5780:
5777:
5774:
5771:
5768:
5758:
5755:
5752:
5749:
5746:
5732:
5729:
5728:
5725:
5719:
5716:
5715:
5712:
5708:
5704:
5697:
5692:
5690:
5685:
5683:
5678:
5677:
5674:
5668:
5665:
5663:
5660:
5658:
5655:
5652:
5649:
5647:
5644:
5641:
5638:
5636:
5633:
5632:
5628:
5623:
5619:
5615:
5612:
5609:
5605:
5602:
5598:
5592:
5588:
5587:
5582:
5578:
5574:
5571:
5568:
5564:
5558:
5554:
5550:
5545:
5544:
5540:
5532:
5528:
5524:
5520:
5516:
5512:
5508:
5504:
5499:
5494:
5490:
5486:
5485:
5477:
5474:
5469:
5465:
5460:
5455:
5451:
5447:
5443:
5439:
5435:
5428:
5425:
5421:
5417:
5413:
5409:
5405:
5401:
5396:
5391:
5387:
5383:
5375:
5372:
5367:
5363:
5359:
5355:
5351:
5347:
5342:
5337:
5334:(1): 015007.
5333:
5329:
5322:
5319:
5314:
5313:
5308:
5301:
5298:
5293:
5289:
5285:
5281:
5277:
5273:
5268:
5263:
5259:
5255:
5248:
5245:
5240:
5236:
5232:
5228:
5224:
5220:
5215:
5210:
5206:
5202:
5195:
5192:
5187:
5183:
5178:
5173:
5169:
5165:
5158:
5155:
5149:
5144:
5140:
5136:
5131:
5126:
5122:
5118:
5117:
5112:
5105:
5102:
5097:
5093:
5089:
5085:
5081:
5077:
5072:
5067:
5063:
5059:
5052:
5049:
5044:
5040:
5035:
5030:
5026:
5022:
5015:
5012:
5007:
5003:
4999:
4995:
4991:
4987:
4982:
4977:
4973:
4969:
4961:
4958:
4953:
4949:
4945:
4941:
4937:
4933:
4928:
4923:
4919:
4915:
4908:
4905:
4900:
4896:
4892:
4888:
4884:
4880:
4875:
4870:
4866:
4862:
4855:
4852:
4847:
4843:
4838:
4833:
4829:
4825:
4821:
4815:
4812:
4806:
4801:
4794:
4791:
4787:
4783:
4779:
4776:
4769:
4766:
4762:
4758:
4754:
4751:
4745:
4742:
4737:
4733:
4728:
4723:
4719:
4715:
4711:
4707:
4706:
4701:
4696:Adapted from
4693:
4690:
4685:
4681:
4677:
4673:
4669:
4665:
4664:
4656:
4653:
4648:
4644:
4640:
4636:
4632:
4628:
4623:
4618:
4614:
4610:
4609:
4600:
4597:
4592:
4588:
4584:
4580:
4575:
4570:
4566:
4562:
4557:
4552:
4548:
4544:
4543:
4535:
4532:
4527:
4521:
4514:
4501:
4496:
4492:
4488:
4483:
4478:
4474:
4470:
4469:
4464:
4456:
4453:
4449:
4443:
4438:
4434:
4430:
4425:
4420:
4416:
4412:
4411:
4406:
4399:
4396:
4392:
4379:
4374:
4370:
4366:
4361:
4356:
4352:
4348:
4347:
4342:
4335:
4332:
4320:
4316:
4312:
4306:
4303:
4291:
4286:
4281:
4277:
4273:
4268:
4263:
4259:
4255:
4254:
4249:
4242:
4239:
4234:
4230:
4225:
4220:
4216:
4212:
4207:
4202:
4198:
4194:
4193:
4188:
4184:
4180:
4173:
4170:
4165:
4161:
4157:
4153:
4149:
4145:
4141:
4137:
4133:
4129:
4124:
4119:
4115:
4111:
4104:
4101:
4086:
4082:
4078:
4074:
4070:
4065:
4060:
4056:
4052:
4048:
4044:
4037:
4030:
4027:
4022:
4018:
4014:
4010:
4006:
4002:
3998:
3994:
3990:
3986:
3981:
3976:
3972:
3968:
3964:
3960:
3954:
3952:
3950:
3946:
3941:
3937:
3933:
3929:
3925:
3921:
3917:
3913:
3909:
3903:
3900:
3895:
3891:
3886:
3881:
3877:
3873:
3870:(12): 34â40.
3869:
3865:
3864:Physics Today
3861:
3857:
3850:
3848:
3844:
3839:
3835:
3831:
3827:
3823:
3819:
3815:
3811:
3806:
3805:gr-qc/9402014
3801:
3797:
3793:
3786:
3783:
3778:
3772:
3769:
3764:
3760:
3756:
3752:
3748:
3744:
3740:
3736:
3732:
3728:
3723:
3718:
3714:
3710:
3706:
3699:
3696:
3690:
3686:
3682:
3678:
3674:
3670:
3665:
3660:
3656:
3652:
3651:
3643:
3640:
3627:
3623:
3619:
3615:
3609:
3606:
3594:
3590:
3584:
3581:
3576:
3572:
3567:
3562:
3558:
3554:
3550:
3546:
3542:
3535:
3532:
3527:
3523:
3519:
3515:
3511:
3507:
3503:
3499:
3498:
3490:
3487:
3481:
3476:
3469:
3466:
3453:
3449:
3445:
3439:
3436:
3423:
3419:
3413:
3410:
3405:
3399:
3395:
3388:
3385:
3379:
3374:
3369:
3364:
3359:
3354:
3350:
3346:
3345:
3340:
3338:
3332:
3326:
3323:
3318:
3314:
3310:
3306:
3302:
3298:
3293:
3288:
3284:
3280:
3279:
3271:
3268:
3255:
3251:
3247:
3240:
3237:
3232:
3228:
3224:
3220:
3219:
3211:
3208:
3202:
3197:
3193:
3189:
3185:
3181:
3180:
3175:
3168:
3165:
3150:
3147:Terence Tao.
3143:
3140:
3127:
3123:
3117:
3114:
3108:
3098:
3095:
3089:
3085:
3082:
3080:
3077:
3075:
3072:
3070:
3067:
3066:
3062:
3051:
3046:
3044:
3042:
3037:
3033:
3028:
3025:
3021:
3015:
3013:
3009:
3004:
3001:
2996:
2994:
2985:
2983:
2979:
2965:
2962:
2950:
2947:
2944:
2941:
2935:
2929:
2926:
2918:
2915:
2911:
2907:
2903:
2895:
2893:
2890:
2886:
2881:
2873:
2866:
2864:
2850:
2838:
2835:
2830:
2826:
2819:
2815:
2811:
2808:
2800:
2797:
2793:
2789:
2783:
2777:
2766:
2763:
2761:
2757:
2746:
2743:
2735:
2725:
2721:
2717:
2711:
2710:
2706:
2701:This section
2699:
2695:
2690:
2689:
2683:
2681:
2679:
2674:
2669:
2662:
2641:
2637:
2633:
2629:
2623:
2613:
2609:
2605:
2602:
2595:
2591:
2588:
2582:
2565:
2563:
2558:
2554:
2552:
2548:
2547:Virgo Cluster
2537:
2534:
2526:
2516:
2512:
2508:
2502:
2501:
2497:
2492:This section
2490:
2486:
2481:
2480:
2474:
2472:
2468:
2451:
2448:
2425:
2418:
2415:
2409:
2404:
2400:
2396:
2393:
2390:
2387:
2384:
2374:
2370:
2358:
2355:
2348:
2346:
2342:
2340:
2335:
2330:
2328:
2324:
2307:
2303:
2300:
2297:
2294:
2291:
2286:
2282:
2278:
2273:
2269:
2257:
2256:magnitude of
2253:
2237:
2233:
2229:
2221:
2216:
2209:
2207:
2203:
2201:
2194:
2187:
2168:
2162:
2157:
2154:
2150:
2144:
2141:
2130:
2128:
2124:
2120:
2104:
2097:
2089:
2080:
2077:
2069:
2067:
2063:
2059:
2049:
2047:
2039:
2035:
2031:
2027:
2023:
2016:
2014:
2010:
2008:
2003:
1999:
1995:
1982:
1978:
1975:
1969:
1966:
1963:
1954:
1948:
1944:
1940:
1935:
1931:
1924:
1918:
1915:
1912:
1908:
1904:
1899:
1895:
1891:
1870:
1866:
1863:
1857:
1854:
1851:
1842:
1836:
1832:
1828:
1823:
1819:
1812:
1806:
1803:
1800:
1796:
1792:
1787:
1783:
1779:
1770:
1768:
1765:of classical
1764:
1760:
1752:
1750:
1748:
1743:
1741:
1737:
1733:
1729:
1725:
1721:
1717:
1713:
1709:
1703:
1695:
1693:
1691:
1687:
1683:
1679:
1668:
1665:
1662:
1659:
1654:
1651:
1648:
1645:
1640:
1637:
1634:
1631:
1626:
1623:
1620:
1617:
1612:
1609:
1606:
1603:
1598:
1595:
1592:
1589:
1584:
1581:
1578:
1575:
1570:
1567:
1563:
1562:Virgo Cluster
1559:
1556:
1553:
1543:
1541:
1539:
1535:
1530:
1528:
1527:main sequence
1524:
1520:
1514:
1506:
1502:
1498:
1495:
1492:
1488:
1484:
1481:
1479:
1475:
1473:
1469:
1467:
1463:
1460:
1457:
1454:
1451:
1447:
1443:
1439:
1435:
1430:
1427:
1424:
1421:
1418:
1415:
1412:
1409:
1408:
1406:
1403:
1399:
1395:
1392:
1387:
1383:
1379:
1375:
1371:
1368:
1367:
1365:
1361:
1357:
1354:
1353:
1352:
1349:
1344:
1336:
1334:
1332:
1328:
1326:
1320:
1318:
1317:recombination
1314:
1310:
1302:
1300:
1284:
1280:
1271:
1253:
1249:
1241:
1237:
1233:
1228:
1225:
1221:
1217:
1213:
1209:
1205:
1200:
1198:
1192:
1188:
1186:
1182:
1166:
1158:
1154:
1150:
1140:
1136:
1132:
1127:
1123:
1112:
1108:
1104:
1094:
1090:
1084:
1080:
1070:
1038:
1034:
1030:
1025:
1021:
1009:
981:
965:
957:
941:
921:
913:
909:
904:
896:
893:
888:
881:
878:
862:
854:
850:
846:
842:
838:
832:
826:
823:
818:
815:
792:
785:of frequency
784:
780:
779:leading order
776:
772:
768:
767:neutron stars
764:
760:
753:
751:
748:
743:
740:
739:population II
736:
732:
727:
723:
718:
716:
712:
707:
702:
700:
692:
690:
688:
684:
679:
675:
657:
653:
646:
643:
640:
637:
634:
627:
623:
620:
600:
597:
594:
591:
588:
585:
582:
579:
574:
570:
566:
563:
555:
551:
547:
543:
538:
536:
532:
528:
524:
516:
510:
505:
503:
499:
494:
493:spectroscopic
490:
486:
482:
478:
474:
469:
467:
463:
462:open clusters
459:
455:
453:
448:
444:
440:
431:
424:
420:
416:
411:
407:
405:
401:
397:
393:
389:
385:
384:Doppler shift
381:
377:
376:proper motion
373:
368:
366:
365:
360:
356:
352:
348:
344:
340:
335:
333:
332:
327:
323:
319:
315:
311:
310:
305:
301:
297:
287:
279:
272:
263:
261:
259:
255:
251:
247:
242:
233:100 billion (
231:
227:
223:
222:Kepler's laws
219:
215:
209:
201:
199:
197:
192:
184:
180:
176:
172:
168:
163:
156:
154:
150:
148:
144:
140:
136:
132:
121:
118:
115:
111:
107:
104:
101:
100:population II
97:
94:
90:
89:
86:
76:
66:
60:
58:
53:
48:
44:
39:
38:
33:
19:
5828:
5757:Solar radius
5751:Light-second
5741:
5731:Earth radius
5642:by Bill Keel
5613:
5607:
5603:
5585:
5569:
5548:
5541:Bibliography
5488:
5482:
5476:
5441:
5437:
5427:
5385:
5381:
5374:
5331:
5327:
5321:
5310:
5300:
5257:
5253:
5247:
5204:
5200:
5194:
5167:
5163:
5157:
5123:(3): 43â47.
5120:
5114:
5104:
5061:
5057:
5051:
5024:
5020:
5014:
4971:
4967:
4960:
4917:
4913:
4907:
4864:
4860:
4854:
4827:
4823:
4814:
4793:
4785:
4768:
4760:
4744:
4709:
4703:
4692:
4667:
4661:
4655:
4612:
4606:
4599:
4546:
4540:
4534:
4520:cite journal
4504:
4472:
4466:
4455:
4446:
4414:
4408:
4398:
4382:
4350:
4344:
4334:
4323:. Retrieved
4305:
4294:. Retrieved
4260:(1): 43â55.
4257:
4251:
4241:
4196:
4190:
4172:
4148:1721.1/60956
4113:
4109:
4103:
4092:. Retrieved
4046:
4042:
4029:
3970:
3966:
3915:
3911:
3902:
3867:
3863:
3795:
3791:
3785:
3771:
3712:
3708:
3698:
3654:
3648:
3642:
3630:. Retrieved
3618:Weekly Topic
3617:
3608:
3597:. Retrieved
3583:
3551:(483): 707.
3548:
3544:
3534:
3501:
3495:
3489:
3468:
3456:. Retrieved
3447:
3438:
3426:. Retrieved
3412:
3393:
3387:
3348:
3342:
3336:
3325:
3282:
3276:
3270:
3258:. Retrieved
3239:
3222:
3216:
3210:
3183:
3177:
3167:
3155:. Retrieved
3142:
3130:. Retrieved
3116:
3097:
3061:Space portal
3036:Edwin Hubble
3032:proportional
3029:
3016:
3005:
2997:
2989:
2980:
2919:
2899:
2882:
2878:
2767:
2764:
2753:
2738:
2729:
2714:Please help
2702:
2675:
2667:
2660:
2566:
2559:
2555:
2544:
2529:
2520:
2505:Please help
2493:
2469:
2359:
2352:
2343:
2338:
2333:
2331:
2326:
2322:
2258:
2254:
2213:
2204:
2196:
2189:
2182:
2131:
2126:
2122:
2118:
2070:
2061:
2057:
2055:
2043:
2011:
2004:
2000:
1996:
1771:
1756:
1744:
1705:
1674:
1571:Range (Mpc)
1560:Distance to
1536:such as the
1534:star cluster
1531:
1516:
1501:Hubble's law
1442:neutron star
1438:X-ray bursts
1350:
1346:
1329:
1321:
1306:
1229:
1220:weak lensing
1207:
1201:
1193:
1189:
1184:
805:is given by
757:
744:
731:population I
726:Walter Baade
719:
703:
696:
677:
673:
553:
539:
533:, coined by
530:
520:
485:Binary stars
480:
470:
456:
436:
398:, including
369:
362:
358:
336:
329:
307:
293:
258:Solar System
243:
211:
190:
188:
166:
151:
138:
134:
130:
128:
73:January 2023
70:
54:
52:lead section
5260:: 673â710.
4820:Udalski, A.
4786:ApJ Letters
3225:: L49âL52.
2562:René Racine
2066:photosphere
1708:Olin Wilson
1666:19.4 ± 5.0
1652:16.8 ± 2.4
1638:15.9 ± 0.9
1624:18.8 ± 3.8
1610:15.4 ± 1.1
1596:21.1 ± 3.9
1487:light curve
1331:Light echos
1210:subject to
771:black holes
699:calibration
512:themselves.
452:square root
419:dark energy
415:dark matter
326:light-years
191:fundamental
5860:Astrometry
5854:Categories
5812:gigaparsec
5806:megaparsec
5800:kiloparsec
5788:light-year
5305:Welch, D.
5207:(4): 289.
5170:(2): 121.
5027:(4): 403.
4974:(1): 102.
4867:(1): L29.
4511:80 pc
4389:28 pc
4325:2015-10-18
4296:2019-10-23
4206:1710.05835
4094:2019-08-04
3722:2302.06621
3632:30 January
3599:2020-11-02
3358:1804.09365
3285:(2): 161.
3260:17 October
3109:References
2064:), of its
2046:supernovae
2017:Supernovae
1738:using the
1669:> 1000
1341:See also:
1204:extinction
1008:chirp mass
683:extinction
523:luminosity
196:astrometry
147:luminosity
5782:terametre
5770:gigametre
5707:Astronomy
5498:1002.3359
5468:0004-637X
5366:119096479
5341:1411.3596
5292:119263173
5267:1004.1856
5214:0806.3018
5177:1006.2458
5130:0903.4088
5071:0902.3747
5034:0909.0181
4981:0805.1592
4647:121612286
4569:CiteSeerX
4549:: 79â87.
4482:1408.1697
4424:0811.2943
4391:to RS Pup
4360:0802.1501
4267:0710.0759
4156:0004-637X
4123:0904.1017
4059:CiteSeerX
4013:0028-0836
3980:1307.2638
3940:0031-9007
3894:0031-9228
3763:256846834
3747:1476-4687
3664:0907.4495
3575:0004-6280
3458:April 12,
3292:1401.0484
3090:Footnotes
3024:evolution
3020:formation
2957:Σ
2951:
2930:
2836:−
2831:∗
2812:−
2790:∝
2732:July 2022
2703:does not
2638:σ
2606:−
2577:Φ
2523:July 2022
2494:does not
2452:˙
2419:˙
2410:
2394:−
2388:−
2301:±
2295:−
2292:≈
2279:≈
2238:⊙
2163:ω
2095:Δ
2090:θ
2087:Δ
2078:ω
1967:−
1949:−
1941:
1905:
1855:−
1837:−
1829:
1793:
1655:> 100
1497:Redshifts
1491:SN 2003fg
1236:hypernova
1224:redshifts
843:π
638:−
592:−
580:
567:⋅
550:bolometer
443:Milky Way
423:neutrinos
343:Hipparcos
339:precision
322:arcsecond
316:, with 2
300:astronomy
250:asteroids
179:Questacon
102:galaxies.
57:summarize
5825:See also
5764:☉
5705:used in
5583:(2013).
5523:20164924
5420:17628238
5239:18463474
5096:16325249
4952:15728812
4899:17860528
4778:Archived
4753:Archived
4736:40679907
4591:18827791
4319:Archived
4290:Archived
4233:29094696
4164:14028891
4085:Archived
3830:10017261
3755:36792736
3689:15178494
3626:Archived
3593:Archived
3452:Archived
3422:Archived
3317:55928992
3254:Archived
3126:Archived
3047:See also
2678:universe
2325:method (
2030:NGC 4526
2026:SN 1994D
1462:Cepheids
1270:GW170817
1232:kilonova
747:kilonova
693:Problems
498:velocity
402:and the
400:Cepheids
304:parallax
264:Parallax
216:and the
183:Canberra
5531:4389201
5503:Bibcode
5446:Bibcode
5400:Bibcode
5346:Bibcode
5272:Bibcode
5219:Bibcode
5182:Bibcode
5135:Bibcode
5076:Bibcode
5039:Bibcode
5006:6275274
4986:Bibcode
4932:Bibcode
4879:Bibcode
4842:Bibcode
4830:: 221.
4714:Bibcode
4672:Bibcode
4627:Bibcode
4561:Bibcode
4487:Bibcode
4429:Bibcode
4365:Bibcode
4272:Bibcode
4211:Bibcode
4128:Bibcode
4081:8926158
4051:Bibcode
4021:4327285
3985:Bibcode
3920:Bibcode
3872:Bibcode
3838:5808548
3810:Bibcode
3727:Bibcode
3669:Bibcode
3553:Bibcode
3526:4316734
3506:Bibcode
3363:Bibcode
3297:Bibcode
3227:Bibcode
3188:Bibcode
3186:: 338.
3041:quasars
2724:removed
2709:sources
2515:removed
2500:sources
2195:equals
1724:spectra
1686:nebulae
1554:Method
1380:(SMC),
1376:(LMC),
978:is the
954:is the
331:parsecs
143:parsecs
5794:parsec
5620:
5593:
5559:
5529:
5521:
5484:Nature
5466:
5418:
5364:
5290:
5237:
5094:
5004:
4950:
4897:
4734:
4645:
4589:
4571:
4513:(4.2%)
4231:
4192:Nature
4181:&
4162:
4154:
4079:
4061:
4019:
4011:
3967:Nature
3938:
3892:
3836:
3828:
3761:
3753:
3745:
3709:Nature
3687:
3622:Caglow
3573:
3524:
3497:Nature
3428:3 June
3400:
3351:. A1.
3315:
3157:13 May
3008:errors
2775:
2659:where
2574:
2549:, the
2438:Where
2367:
2266:
2139:
2117:where
1714:, the
1590:Novae
1582:15â25
1538:Hyades
1425:(GCLF)
1419:(PNLF)
1398:galaxy
982:, and
934:where
781:, the
672:where
489:visual
466:Hyades
226:ratios
139:direct
5814:(Gpc)
5808:(Mpc)
5802:(kpc)
5527:S2CID
5493:arXiv
5444:: 1.
5416:S2CID
5390:arXiv
5362:S2CID
5336:arXiv
5288:S2CID
5262:arXiv
5235:S2CID
5209:arXiv
5172:arXiv
5125:arXiv
5092:S2CID
5066:arXiv
5029:arXiv
5002:S2CID
4976:arXiv
4948:S2CID
4922:arXiv
4895:S2CID
4869:arXiv
4832:arXiv
4800:arXiv
4732:JSTOR
4643:S2CID
4617:arXiv
4587:S2CID
4551:arXiv
4477:arXiv
4419:arXiv
4355:arXiv
4262:arXiv
4201:arXiv
4160:S2CID
4118:arXiv
4088:(PDF)
4077:S2CID
4039:(PDF)
4017:S2CID
3975:arXiv
3834:S2CID
3800:arXiv
3759:S2CID
3717:arXiv
3685:S2CID
3659:arXiv
3522:S2CID
3475:arXiv
3353:arXiv
3313:S2CID
3287:arXiv
3152:(PDF)
3132:9 May
2945:1.333
2798:0.307
2354:Novae
1678:stars
1663:0.10
1579:0.16
1466:novae
1431:(SBF)
1181:power
735:metal
359:micro
254:radar
230:Radar
214:Earth
177:; at
5796:(pc)
5790:(ly)
5784:(Tm)
5778:(au)
5772:(Gm)
5753:(ls)
5618:ISBN
5591:ISBN
5557:ISBN
5519:PMID
5464:ISSN
4526:link
4507:1910
4385:1992
4229:PMID
4152:ISSN
4009:ISSN
3936:ISSN
3890:ISSN
3826:PMID
3751:PMID
3743:ISSN
3634:2012
3571:ISSN
3460:2014
3430:2016
3398:ISBN
3337:Gaia
3262:2014
3250:NASA
3159:2023
3134:2021
3022:and
2900:The
2883:The
2707:any
2705:cite
2498:any
2496:cite
2397:2.31
2391:9.96
2327:MLCS
2298:19.3
2034:NASA
1979:7.48
1955:2.55
1925:3.37
1867:7.52
1843:2.45
1813:3.34
1710:and
1649:0.5
1635:0.3
1621:0.4
1607:0.3
1593:0.4
1499:and
1476:The
1470:The
1464:and
1384:and
491:and
421:and
364:Gaia
129:The
5740:or
5511:doi
5489:463
5454:doi
5442:317
5408:doi
5386:546
5354:doi
5280:doi
5227:doi
5143:doi
5121:396
5084:doi
5062:696
4994:doi
4972:684
4940:doi
4918:652
4887:doi
4865:642
4722:doi
4710:104
4680:doi
4668:232
4635:doi
4613:128
4579:doi
4547:240
4495:doi
4473:572
4437:doi
4415:495
4373:doi
4351:480
4280:doi
4258:478
4219:doi
4197:551
4144:hdl
4136:doi
4114:725
4069:doi
4001:hdl
3993:doi
3971:323
3928:doi
3880:doi
3818:doi
3735:doi
3713:614
3677:doi
3655:506
3561:doi
3514:doi
3502:368
3373:doi
3349:616
3305:doi
3283:785
3223:323
3196:doi
2948:log
2927:log
2889:CCD
2718:by
2509:by
2401:log
2380:max
2304:0.3
2230:1.4
2222:of
2038:ESA
2007:M31
1932:log
1896:log
1820:log
1784:log
1641:50
1627:50
1613:50
1599:20
1585:29
1566:Mpc
1208:are
769:or
613:or
571:log
475:or
353:'s
298:in
241:).
218:Sun
5856::
5738:đš
5606:,
5579:;
5555:.
5525:.
5517:.
5509:.
5501:.
5487:.
5462:.
5452:.
5440:.
5436:.
5414:,
5406:,
5398:,
5384:,
5360:.
5352:.
5344:.
5332:36
5330:.
5309:.
5286:.
5278:.
5270:.
5258:48
5256:.
5233:.
5225:.
5217:.
5205:15
5203:.
5180:.
5168:60
5166:.
5141:.
5133:.
5119:.
5113:.
5090:.
5082:.
5074:.
5060:.
5037:.
5025:59
5023:.
5000:.
4992:.
4984:.
4970:.
4946:.
4938:.
4930:.
4916:.
4893:.
4885:.
4877:.
4863:.
4840:.
4828:51
4826:.
4784:,
4759:,
4730:.
4720:.
4708:.
4702:.
4678:.
4666:.
4641:.
4633:.
4625:.
4611:.
4585:.
4577:.
4567:.
4559:.
4545:.
4522:}}
4518:{{
4503:.
4493:.
4485:.
4471:.
4465:.
4445:.
4435:.
4427:.
4413:.
4407:.
4381:.
4371:.
4363:.
4349:.
4343:.
4288:.
4278:.
4270:.
4256:.
4250:.
4227:.
4217:.
4209:.
4195:.
4189:.
4158:.
4150:.
4142:.
4134:.
4126:.
4112:.
4083:.
4075:.
4067:.
4057:.
4047:48
4045:.
4041:.
4015:.
4007:.
3999:.
3991:.
3983:.
3969:.
3965:.
3948:^
3934:.
3926:.
3916:18
3914:.
3888:.
3878:.
3868:71
3866:.
3862:.
3846:^
3832:.
3824:.
3816:.
3808:.
3796:49
3794:.
3757:.
3749:.
3741:.
3733:.
3725:.
3711:.
3707:.
3683:.
3675:.
3667:.
3653:.
3624:.
3620:.
3616:.
3569:.
3559:.
3549:81
3547:.
3543:.
3520:.
3512:.
3500:.
3450:.
3446:.
3420:.
3371:.
3361:.
3347:.
3341:.
3311:.
3303:.
3295:.
3281:.
3252:.
3248:.
3221:.
3194:.
3184:72
3182:.
3176:.
2405:10
2252:.
2192:ej
2185:ej
2036:,
1936:10
1900:10
1824:10
1788:10
1742:.
1692:.
1684:,
1680:,
1568:)
1299:.
1199:.
958:,
894:11
839:96
717:.
628:10
575:10
537:.
447:AU
417:,
406:.
318:AU
260:.
239:10
220:.
181:,
149:.
5766:)
5761:R
5759:(
5747:)
5744:E
5742:R
5735:R
5733:(
5695:e
5688:t
5681:v
5624:.
5599:.
5565:.
5533:.
5513::
5505::
5495::
5470:.
5456::
5448::
5410::
5402::
5392::
5368:.
5356::
5348::
5338::
5315:.
5294:.
5282::
5274::
5264::
5241:.
5229::
5221::
5211::
5188:.
5184::
5174::
5151:.
5145::
5137::
5127::
5098:.
5086::
5078::
5068::
5045:.
5041::
5031::
5008:.
4996::
4988::
4978::
4954:.
4942::
4934::
4924::
4901:.
4889::
4881::
4871::
4848:.
4844::
4834::
4808:.
4802::
4738:.
4724::
4716::
4686:.
4682::
4674::
4649:.
4637::
4629::
4619::
4593:.
4581::
4563::
4553::
4528:)
4509:±
4497::
4489::
4479::
4439::
4431::
4421::
4387:±
4375::
4367::
4357::
4328:.
4299:.
4282::
4274::
4264::
4235:.
4221::
4213::
4203::
4166:.
4146::
4138::
4130::
4120::
4097:.
4071::
4053::
4023:.
4003::
3995::
3987::
3977::
3942:.
3930::
3922::
3896:.
3882::
3874::
3840:.
3820::
3812::
3802::
3765:.
3737::
3729::
3719::
3691:.
3679::
3671::
3661::
3636:.
3602:.
3577:.
3563::
3555::
3528:.
3516::
3508::
3483:.
3477::
3462:.
3432:.
3406:.
3381:.
3375::
3365::
3355::
3335:"
3319:.
3307::
3299::
3289::
3264:.
3233:.
3229::
3204:.
3198::
3190::
3161:.
3136:.
2966:C
2963:+
2960:)
2954:(
2942:=
2939:)
2936:D
2933:(
2851:.
2847:)
2842:)
2839:M
2827:M
2823:(
2820:3
2816:e
2809:1
2806:(
2801:M
2794:e
2787:)
2784:M
2781:(
2778:N
2745:)
2739:(
2734:)
2730:(
2726:.
2712:.
2671:0
2668:M
2664:0
2661:m
2642:2
2634:2
2630:/
2624:2
2620:)
2614:0
2610:m
2603:m
2600:(
2596:e
2592:A
2589:=
2586:)
2583:m
2580:(
2536:)
2530:(
2525:)
2521:(
2517:.
2503:.
2449:x
2426:.
2416:x
2385:=
2375:V
2371:M
2308:.
2287:V
2283:M
2274:B
2270:M
2234:M
2199:Ξ
2197:V
2190:V
2183:V
2169:,
2158:j
2155:e
2151:V
2145:=
2142:d
2127:t
2123:Ξ
2119:Ï
2105:,
2098:t
2081:=
2062:t
2060:(
2058:Ξ
1983:.
1976:+
1973:)
1970:I
1964:V
1961:(
1958:)
1952:(
1945:P
1928:)
1922:(
1919:+
1916:V
1913:=
1909:d
1892:5
1871:.
1864:+
1861:)
1858:I
1852:V
1849:(
1846:)
1840:(
1833:P
1816:)
1810:(
1807:+
1804:V
1801:=
1797:d
1780:5
1564:(
1404:.
1285:0
1281:H
1254:0
1250:H
1234:/
1167:.
1159:5
1155:/
1151:1
1147:)
1141:2
1137:m
1133:+
1128:1
1124:m
1120:(
1113:5
1109:/
1105:3
1101:)
1095:2
1091:m
1085:1
1081:m
1077:(
1071:=
1066:M
1044:)
1039:2
1035:m
1031:,
1026:1
1022:m
1018:(
992:M
966:c
942:G
922:,
914:5
910:c
905:5
897:3
889:f
882:3
879:5
874:)
868:M
863:G
860:(
855:3
851:/
847:8
833:=
827:t
824:d
819:f
816:d
793:f
678:M
674:m
658:5
654:/
650:)
647:5
644:+
641:M
635:m
632:(
624:=
621:d
601:5
598:+
595:M
589:m
586:=
583:d
564:5
554:d
425:.
273:.
237:Ă
235:1
185:.
116:.
95:.
75:)
71:(
61:.
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
