Period-luminosity relation

Period-luminosity relation[edit]

Period-Luminosity Relation for Cepheids

A classical Cepheid's luminosity is directly related to its period of variation. The longer the pulsation period, the more luminous the star. The period-luminosity relation for classical Cepheids was discovered in 1908 by Henrietta Swan Leavitt in an investigation of thousands of variable stars in the Magellanic Clouds.^[20] She published it in 1912^[21] with further evidence. Once the period-luminosity relationship is calibrated, the luminosity of a given Cepheid whose period is known can be established. Their distance is then found from their apparent brightness. The period-luminosity relationship has been calibrated by many astronomers throughout the twentieth century, beginning with Hertzsprung.^[22] Calibrating the period-luminosity relation has been problematic; however, a firm Galactic calibration was established by Benedict et al. 2007 using precise HST parallaxes for 10 nearby classical Cepheids.^[23] Also, in 2008, ESO astronomers estimated with a precision within 1% the distance to the Cepheid RS Puppis, using light echos from a nebula in which it is embedded.^[24] However, that latter finding has been actively debated in the literature.^[25]

The following relationship between a Population I Cepheid's period P and its mean absolute magnitude M_v was established from Hubble Space Telescope trigonometric parallaxes for 10 nearby Cepheids:

${\displaystyle M_{\mathrm {v} }=(-2.43\pm 0.12)\left(\log _{10}P-1\right)-(4.05\pm 0.02)\,}$

with P measured in days. ^[19][23] The following relations can also be used to calculate the distance d to classical Cepheids:

${\displaystyle 5\log _{10}{d}=V+3.34\log _{10}{P}-2.45(V-I)+7.52\,.}$^[23]

or

${\displaystyle 5\log _{10}{d}=V+3.37\log _{10}{P}-2.55(V-I)+7.48\,.}$^[26]

I and V represent near infrared and visual apparent mean magnitudes, respectively.

Classical Cepheid variable

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Hertzsprung–Russell diagram showing the location of several types of variable stars superimposed on a display of the different luminosity classes.

Classical Cepheids (also known as Population I Cepheids, Type I Cepheids, or Delta Cepheid variables) are a type of Cepheid variable star. They are population I variable stars that exhibit regular radial pulsations with periods of a few days to a few weeks and visual amplitudes from a few tenths of a magnitude to about 2 magnitudes.

There exists a well-defined relationship between a classical Cepheid variable's luminosity and pulsation period,^[1][2] securing Cepheids as viable standard candles for establishing the galactic and extragalactic distance scales.^[3][4][5][6] HST observations of classical Cepheid variables have enabled firmer constraints on Hubble's law.^{[3][4][6][7][8]} Classical Cepheids have also been used to clarify many characteristics of our galaxy, such as the Sun's height above the galactic plane and the Galaxy's local spiral structure.^[5]

Around 800 classical Cepheids are known in the Milky Way Galaxy, out of an expected total of over 6,000. Several thousand more are known in the Magellanic Clouds, with more known in other galaxies.^[9] The Hubble Space Telescope has identified classical Cepheids in NGC 4603, which is 100 million light years distant.^[10]

Dynamics of the pulsation

Dynamics of the pulsation[edit]

The accepted explanation for the pulsation of Cepheids is called the Eddington valve,^[44] or κ-mechanism, where the Greek letter κ (kappa) denotes gas opacity. Helium is the gas thought to be most active in the process. Doubly ionized helium (helium whose atoms are missing both electrons) is more opaque than singly ionized helium. The more helium is heated, the more ionized it becomes. At the dimmest part of a Cepheid's cycle, the ionized gas in the outer layers of the star is opaque, and so is heated by the star's radiation, and due to the increased temperature, begins to expand. As it expands, it cools, and so becomes less ionized and therefore more transparent, allowing the radiation to escape. Then the expansion stops, and reverses due to the star's gravitational attraction. The process then repeats.

The mechanics of the pulsation as a heat-engine was proposed in 1917 by Arthur Stanley Eddington^[45] (who wrote at length on the dynamics of Cepheids), but it was not until 1953 that S. A. Zhevakin identified ionized helium^[46] as a likely valve for the engine.

Classical Cepheids

Classical Cepheids[edit]

Main article: Classical Cepheid variable

Classical Cepheids (also known as Population I Cepheids, type I Cepheids, or Delta Cepheid variables) undergo pulsations with very regular periods on the order of days to months. Classical Cepheids are Population I variable stars which are 4–20 times more massive than the Sun,^[9] and up to 100,000 times more luminous.^[10] These Cepheids are yellow bright giants and supergiants of spectral class F6 – K2 and their radii change by (~25% for the longer-period I Carinae) millions of kilometers during a pulsation cycle.^[11]

Classical Cepheids are used to determine distances to galaxies within the Local Group and beyond, and are a means by which the Hubble constant can be established.^{[3][4][6][12][13]}Classical Cepheids have also been used to clarify many characteristics of our galaxy, such as the Sun's height above the galactic plane and the Galaxy's local spiral structure.^[5]

A group of classical Cepheids with small amplitudes and sinusoidal light curves are often separated out as Small Amplitude Cepheids or s-Cepheids, many of them pulsating in the first overtone.

Type II Cepheids[edit]

Main article: Type II Cepheid

Type II Cepheids (also termed Population II Cepheids) are population II variable stars which pulsate with periods typically between 1 and 50 days.^[14][15] Type II Cepheids are typically metal-poor, old (~10 Gyr), low mass objects (~half the mass of the Sun). Type II Cepheids are divided into several subgroups by period. Stars with periods between 1 and 4 days are of the BL Her subclass, 10–20 days belong to the W Virginis subclass, and stars with periods greater than 20 days belong to the RV Tauri subclass.^[14][15]

Type II Cepheids are used to establish the distance to the Galactic Center, globular clusters, and galaxies.^{[5][16][17][18][19][20][21]}

Anomalous Cepheids[edit]

A group of pulsating stars on the instability strip have periods of less than 2 days, similar to RR Lyrae variables but with higher luminosities. Anomalous Cepheid variables have masses higher than type II Cepheids, RR Lyrae variables, and our sun. It is unclear whether they are young stars on a "turned-back" horizontal branch, blue stragglers formed through mass transfer in binary systems, or a mix of both.^[22][23]

Double-mode Cepheids[edit]

A small proportion of Cepheid variables have been observed to pulsate in two modes at the same time, usually the fundamental and first overtone, occasionally the second overtone.^[24] A very small number pulsate in three modes, or an unusual combination of modes including higher overtones.^[25]

Cepheid variable

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"Cepheid" redirects here. For other uses, see Cepheid (disambiguation).

RS Puppis is one of the brightest known Cepheid variable stars in the Milky Way galaxy; image, Hubble Space Telescope

A Cepheid variable (/ˈsɛfiːɪd, ˈsiːfiːɪd/) is a type of star that pulsates radially, varying in both diameter and temperature and producing changes in brightness with a well-defined stable period and amplitude.

A strong direct relationship between a Cepheid variable's luminosity and pulsation period^[1][2] established Cepheids as important indicators of cosmic benchmarks for scaling galactic and extragalactic distances.^[3][4][5][6] This robust characteristic of classical Cepheids was discovered in 1908 by Henrietta Swan Leavitt after studying thousands of variable stars in the Magellanic Clouds.^[7] This discovery allows one to know the true luminosity of a Cepheid by simply observing its pulsation period. This in turn allows one to determine the distance to the star, by comparing its known luminosity to its observed brightness.

The term Cepheid originates from Delta Cephei in the constellation Cepheus, identified by John Goodricke in 1784, the first of its type to be so identified.

Perturbation theory

Perturbation theory[edit]

Main article: Perturbation theory

Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly. (It is closely related to methods used in numerical analysis, which are ancient.) The earliest use of modern perturbation theory was to deal with the otherwise unsolveable mathematical problems of celestial mechanics: Newton's solution for the orbit of the Moon, which moves noticeably differently from a simple Keplerian ellipse because of the competing gravitation of the Earth and the Sun.

Perturbation methods start with a simplified form of the original problem, which is carefully chosen to be exactly solvable. In celestial mechanics, this is usually a Keplerian ellipse, which is correct when there are only two gravitating bodies (say, the Earth and the Moon), or a circular orbit, which is only correct in special cases of two-body motion, but is often close enough for practical use.

The solved, but simplified problem is then "perturbed" to make its time-rate-of-change equations for the object's position closer to the values from the real problem, such as including the gravitational attraction of a third, more distant body (the Sun). The slight changes that result from the terms in the equations – which themselves may have been simplified yet again – are used as corrections to the original solution. Because simplifications are made at every step, the corrections are never perfect, but even one cycle of corrections often provides a remarkably better approximate solution to the real problem.

There is no requirement to stop at only one cycle of corrections. A partially corrected solution can be re-used as the new starting point for yet another cycle of perturbations and corrections. In principle, for most problems the recycling and refining of prior solutions to obtain a new generation of better solutions could continue indefinitely, to any desired finite degree of accuracy.

The common difficulty with the method is that the corrections usually progressively make the new solutions very much more complicated, so each cycle is much more difficult to manage than the previous cycle of corrections. Newton is reported to have said, regarding the problem of the Moon's orbit "It causeth my head to ache."^[1]

This general procedure – starting with a simplified problem and gradually adding corrections that make the starting point of the corrected problem closer to the real situation – is a widely used mathematical tool in advanced sciences and engineering. It is the natural extension of the "guess, check, and fix" method used anciently with numbers.

See also

Albert Einstein

Albert Einstein[edit]

Albert Einstein (14 March 1879–18 April 1955) explained the anomalous precession of Mercury's perihelion in his 1916 paper The Foundation of the General Theory of Relativity. This led astronomers to recognize that Newtonian mechanics did not provide the highest accuracy. Binary pulsars have been observed, the first in 1974, whose orbits not only require the use of General Relativity for their explanation, but whose evolution proves the existence of gravitational radiation, a discovery that led to the 1993 Nobel Physics Prize.

Simon Newcomb

Simon Newcomb[edit]

Simon Newcomb (12 March 1835–11 July 1909) was a Canadian-American astronomer who revised Peter Andreas Hansen's table of lunar positions. In 1877, assisted by George William Hill, he recalculated all the major astronomical constants. After 1884, he conceived with A. M. W. Downing a plan to resolve much international confusion on the subject. By the time he attended a standardisation conference in Paris, France in May 1886, the international consensus was that all ephemerides should be based on Newcomb's calculations. A further conference as late as 1950 confirmed Newcomb's constants as the international standard.

Joseph-Louis Lagrange

Joseph-Louis Lagrange[edit]

After Newton, Lagrange (25 January 1736–10 April 1813) attempted to solve the three-body problem, analyzed the stability of planetary orbits, and discovered the existence of the Lagrangian points. Lagrange also reformulated the principles of classical mechanics, emphasizing energy more than force and developing a method to use a single polar coordinate equation to describe any orbit, even those that are parabolic and hyperbolic. This is useful for calculating the behaviour of planets and comets and such. More recently, it has also become useful to calculate spacecraft trajectories.

Isaac Newton

Isaac Newton[edit]

Isaac Newton (25 December 1642–31 March 1727) is credited with introducing the idea that the motion of objects in the heavens, such as planets, the Sun, and the Moon, and the motion of objects on the ground, like cannon balls and falling apples, could be described by the same set of physical laws. In this sense he unified celestial and terrestrial dynamics. Using Newton's law of universal gravitation, proving Kepler's Laws for the case of a circular orbit is simple. Elliptical orbits involve more complex calculations, which Newton included in his Principia.

Johannes Kepler

For detailed treatments of how his laws of planetary motion can be used, see Kepler's laws of planetary motion and Keplerian problem. Johannes Kepler (1571–1630) was the first to closely integrate the predictive geometrical astronomy, which had been dominant from Ptolemy in the 2nd century to Copernicus, with physical concepts to produce a New Astronomy, Based upon Causes, or Celestial Physics in 1609. His work led to the modern laws of planetary orbits, which he developed using his physical principles and the planetary observations made by Tycho Brahe. Kepler's model greatly improved the accuracy of predictions of planetary motion, years before Isaac Newton developed his law of gravitation in 1686.

Celestial mechanics

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Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to produce ephemeris data.

As an astronomical field of study, celestial mechanics includes the sub-fields of orbital mechanics, which deals with the launching and orbits artificial satellites, and lunar theory, a specialty which deals with the complications of the orbit of the Moon. Modern celestial mechanics tends to divide between five broad fields of study:

• trajectories of artificial satellites (astrodynamics)
• motions of major planets, minor planets, and natural satellites in the Solar system and other stellar systems and widely spaced multiple-star systems (planetary dynamics)
• the motion of component stars and their planetary systems in closely spaced multiple-star systems and globular clusters (astrodynamics and stellar dynamics)
• flow of stars within and among the bodies of large galaxies, dwarf galaxies, and globular clusters (stellar dynamics and galactic dynamics)
• motions of galaxies and intergalactic dust and gas within galaxy clusters (computational astrophysics)

All of the above fields overlap, but are sometimes treated as separate, especially the study of the motion of stars within galaxies and interactions between whole galaxies, which both tend to rely heavily on fluid mechanics (whole stars being particles of the