Hayashi limit

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

The Hayashi limit is a theoretical constraint upon the maximum radius of a star for a given mass. When a star is fully within hydrostatic equilibrium—a condition where the inward force of gravity is matched by the outward pressure of the gas—the star can not exceed the radius defined by the Hayashi limit. This has important implications for the evolution of a star, both during the formulative contraction period and later when the star has consumed most of its hydrogen supply through nuclear fusion.

A Hertzsprung-Russell diagram displays a plot of a star's surface temperature against the luminosity. On this diagram, the Hayashi limit forms a nearly vertical line at about 3,500 K. The outer layers of low temperature stars are always convective, and models of stellar structure for fully convective stars do not provide a solution to the right of this line. Thus in theory, stars are constrained to remain to the left of this limit during all periods when they are in hydrostatic equilibrium, and the region to the right of the line forms a type of "forbidden zone". Note, however, that there are exceptions to the Hayashi limit. These include collapsing protostars, as well as stars with magnetic fields that interfere with the internal transport of energy through convection.

Red giants are stars that have expanded their outer envelope in order to support the nuclear fusion of helium. This moves them up and to the right on the H-R diagram. However, they are constrained by the Hayashi limit not to expand beyond a certain radius. Stars that find themselves across the Hayashi limit have large convection currents in their interior driven by massive temperature gradients. Additionally, those stars states are unstable so the stars rapidly adjust their states, moving in the Hertzprung-Russel diagram until they reach the Hayashi limit.

When lower mass stars in the main sequence start expanding and becoming a red giant the stars revisit the Hayashi track. The Hayashi limit constrains the asymptotic giant branch evolution of stars which is important in the late evolution of stars and can be observed, for example, in the ascending branches of the Hertzsprung–Russell diagrams of globular clusters, which have stars of approximately the same age and composition.

The Hayashi limit is named after Chūshirō Hayashi, a Japanese astrophysicist.

Despite its importance to protostars and late stage main sequence stars, the Hayashi limit was only recognized in Hayashi’s paper in 1961. This late recognition may be because the properties of the Hayashi track required numerical calculations that were not fully developed before. 

Derivation of the limit

We can derive the relation between the luminosity, temperature and pressure for a simple model for a fully convective star and from the form of this relation we can infer the Hayashi limit. This is an extremely crude model of what occurs in convective stars, but it has good qualitative agreement with the full model with less complications. We follow the derivation in Kippenhahn, Weigert, and Weiss in Stellar Structure and Evolution.

Nearly all of the interior part of convective stars has an adiabatic stratification (corrections to this are small for fully convective regions), such that

${\displaystyle \frac{\delta lnT}{\delta lnP}={\mathrm{\nabla }}_{adiabatic}=0.4}$, which holds for an adiabatic expansion of an ideal gas.

We assume that this relation holds from the interior to the surface of the star—the surface is called photosphere. We assume \grad_{adiabatic} to be constant throughout the interior of the star with value 0.4. However, we obtain the correct distinctive behavior.

For the interior we consider a simple polytropic relation between P and T:

${\displaystyle P=C{T}^{(1+n)}}$

With the index ${\displaystyle n=3/2}$.

We assume the relation above to hold until the photosphere where we assume to have a simple absorption law

${\displaystyle \kappa ={\kappa }_0{P}^a{T}^b}$

Then, we use the hydrostatic equilibrium equation and integrate it with respect to the radius to give us

${\displaystyle P_0=Constant\ast {\left(\frac{M}{R^2}{T}_{eff}^{-b}\right)}^{\frac{1}{1+a}}}$

For the solution in the interior we set ${\displaystyle P=P_0}$ ; ${\displaystyle T=T_{eff}}$ in the P-T relation and then eliminate pressure of this equation. Luminosity is given by the Stephan-Boltzmann law applied to a perfect black body:

${\displaystyle L=4\pi R^2\sigma T_{eff}^4}$.

Thus, any value of R corresponds to a certain point in the Hertzsprung–Russell diagram.

Finally, after some algebra this is the equation for the Hayashi limit in the Hertzsprung–Russell diagram:

${\displaystyle \log (T_{e}ff)=A\log (L)+B\log (M)+constant}$ 

With coefficients

${\displaystyle A=\frac{0.75a-0.25}{b-5.5a+1.5}}$, ${\displaystyle B=\frac{0.5a-1.5}{b-5.5a+1.5}}$

Takeaways from plugin in ${\displaystyle a\approx 1}$ and ${\displaystyle b\approx 3}$ for a cool hydrogen ion dominated atmosphere opacity model (${\displaystyle T<5000K}$):

• The Hayashi limit must be far to the right in the Hertzsprung–Russell diagram which means temperatures have to be low.
• The Hayashi limit must be very steep. The gradient of Luminosity with respect to temperature has to be large.
• The Hayashi limit shifts slightly to the left in the Hertzsprung–Russell diagram for increasing M.

These predictions are supported by numerical simulations of stars. 

What happens when stars cross the limit

Until now we have made no claims on the stability of locale to the left, right or at the Hayashi limit in the Hertzsprung–Russell diagram. To the left of the Hayashi limit, we have ${\displaystyle \mathrm{\nabla }<\mathrm{\nabla }adiabatic}$ and some part of the model is radiative. The model is fully convective at the Hayashi limit with ${\displaystyle \mathrm{\nabla }=\mathrm{\nabla }adiabatic}$. Models to the right of the Hayashi limit should have ${\displaystyle \mathrm{\nabla }>{\mathrm{\nabla }}_{adiabatic}}$.

If a star is formed such that some region in its deep interior has large ${\displaystyle \mathrm{\nabla }-{\mathrm{\nabla }}_{adiabatic}>0}$ large convective fluxes with velocities ${\displaystyle v_{convective}\approx (\mathrm{\nabla }-{\mathrm{\nabla }}_{adiabatic})/2}$. The convective fluxes of energy cooldown the interior rapidly until ${\displaystyle \mathrm{\nabla }={\mathrm{\nabla }}_{adiabatic}}$ and the star has moved to the Hayashi limit. In fact, it can be shown from the mixing length model that even a small excess can transport energy from the deep interior to the surface by convective fluxes. This will happen within the short timescale for the adjustment of convection which is still larger than timescales for non-equilibrium processes in the star such as hydrodynamic adjustment associated with the thermal time scale. Hence, the limit between an “allowed” stable region (left) and a “forbidden” unstable region (right) for stars of given M and composition that are in hydrostatic equilibrium and have a fully adjusted convection is the Hayashi limit.

Eddington luminosity

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

The Eddington luminosity, also referred to as the Eddington limit, is the maximum luminosity a body (such as a star) can achieve when there is balance between the force of radiation acting outward and the gravitational force acting inward. The state of balance is called hydrostatic equilibrium. When a star exceeds the Eddington luminosity, it will initiate a very intense radiation-driven stellar wind from its outer layers. Since most massive stars have luminosities far below the Eddington luminosity, their winds are driven mostly by the less intense line absorption.  The Eddington limit is invoked to explain the observed luminosities of accreting black holes such as quasars.

Originally, Sir Arthur Eddington took only the electron scattering into account when calculating this limit, something that now is called the classical Eddington limit. Nowadays, the modified Eddington limit also takes into account other radiation processes such as bound–free and free–free radiation interaction.

Derivation

The Eddington limit is obtained by setting the outward radiation pressure equal to the inward gravitational force. Both forces decrease by inverse-square laws, so once equality is reached, the hydrodynamic flow is the same throughout the star.

From Euler's equation in hydrostatic equilibrium, the mean acceleration is zero,

$${\displaystyle \frac{du}{dt}=-\frac{\mathrm{\nabla }p}{\rho }-\mathrm{\nabla }\mathrm{\Phi }=0}$$

where ${\displaystyle u}$ is the velocity, ${\displaystyle p}$ is the pressure, ${\displaystyle \rho }$ is the density, and ${\displaystyle \mathrm{\Phi }}$ is the gravitational potential. If the pressure is dominated by radiation pressure associated with an irradiance ${\displaystyle F_{\mathrm{r}\mathrm{a}\mathrm{d}}}$,

$${\displaystyle -\frac{\mathrm{\nabla }p}{\rho }=\frac{\kappa }{c}{F}_{\mathrm{r}\mathrm{a}\mathrm{d}}.}$$

Here ${\displaystyle \kappa }$ is the opacity of the stellar material, defined as the fraction of radiation energy flux absorbed by the medium per unit density and unit length. For ionized hydrogen, ${\displaystyle \kappa ={\sigma }_{\mathrm{T}}/m_{\mathrm{p}}}$, where ${\displaystyle {\sigma }_{\mathrm{T}}}$ is the Thomson scattering cross-section for the electron and ${\displaystyle m_{\mathrm{p}}}$ is the mass of a proton. Note that ${\displaystyle F_{\mathrm{r}\mathrm{a}\mathrm{d}}=d^{2}E/dAdt}$ is defined as the energy flux over a surface, which can be expressed with the momentum flux using ${\displaystyle E=pc}$ for radiation. Therefore, the rate of momentum transfer from the radiation to the gaseous medium per unit density is ${\displaystyle \kappa F_{\mathrm{r}\mathrm{a}\mathrm{d}}/c}$, which explains the right-hand side of the above equation.

The luminosity of a source bounded by a surface ${\displaystyle S}$ may be expressed with these relations as

$${\displaystyle L={\int }_S{F}_{\mathrm{r}\mathrm{a}\mathrm{d}}\cdot dS={\int }_S\frac{c}{\kappa }\mathrm{\nabla }\mathrm{\Phi }\cdot dS.}$$

Now assuming that the opacity is a constant, it can be brought outside the integral. Using Gauss's theorem and Poisson's equation gives

$${\displaystyle L=\frac{c}{\kappa }{\int }_S\mathrm{\nabla }\mathrm{\Phi }\cdot dS=\frac{c}{\kappa }{\int }_V{\mathrm{\nabla }}^2\mathrm{\Phi }dV=\frac{4\pi Gc}{\kappa }{\int }_V\rho dV=\frac{4\pi GMc}{\kappa }}$$

where ${\displaystyle M}$ is the mass of the central object. This result is called the Eddington luminosity. For pure ionized hydrogen,

$${\displaystyle \begin{array}{rl}{L}_{\mathrm{E}\mathrm{d}\mathrm{d}}& =\frac{4\pi GM{m}_{\mathrm{p}}c}{{\sigma }_{\mathrm{T}}}\\ & \cong 1.26\times {10}^{31}\left(\frac{M}{M_{⨀}}\right)\mathrm{W}=1.26\times {10}^{38}\left(\frac{M}{M_{⨀}}\right)\mathrm{e}\mathrm{r}\mathrm{g}/\mathrm{s}=3.2\times {10}^4\left(\frac{M}{M_{⨀}}\right)L_{⨀}\end{array}}$$

where ${\displaystyle M_{⨀}}$ is the mass of the Sun and ${\displaystyle L_{⨀}}$ is the luminosity of the Sun.

The maximum possible luminosity of a source in hydrostatic equilibrium is the Eddington luminosity. If the luminosity exceeds the Eddington limit, then the radiation pressure drives an outflow.

The mass of the proton appears because, in the typical environment for the outer layers of a star, the radiation pressure acts on electrons, which are driven away from the center. Because protons are negligibly pressured by the analog of Thomson scattering, due to their larger mass, the result is to create a slight charge separation and therefore a radially directed electric field, acting to lift the positive charges, which, under the conditions in stellar atmospheres, typically are free protons. When the outward electric field is sufficient to levitate the protons against gravity, both electrons and protons are expelled together.

Different limits for different materials

The derivation above for the outward light pressure assumes a hydrogen plasma. In other circumstances the pressure balance can be different from what it is for hydrogen.

In an evolved star with a pure helium atmosphere, the electric field would have to lift a helium nucleus (an alpha particle), with nearly 4 times the mass of a proton, while the radiation pressure would act on 2 free electrons. Thus twice the usual Eddington luminosity would be needed to drive off an atmosphere of pure helium.

At very high temperatures, as in the environment of a black hole or neutron star, high-energy photons can interact with nuclei, or even with other photons, to create an electron–positron plasma. In that situation the combined mass of the positive–negative charge carrier pair is approximately 918 times smaller (half of the proton-to-electron mass ratio), while the radiation pressure on the positrons doubles the effective upward force per unit mass, so the limiting luminosity needed is reduced by a factor of ≈ 918×2.

The exact value of the Eddington luminosity depends on the chemical composition of the gas layer and the spectral energy distribution of the emission. A gas with cosmological abundances of hydrogen and helium is much more transparent than gas with solar abundance ratios. Atomic line transitions can greatly increase the effects of radiation pressure, and line-driven winds exist in some bright stars (e.g., Wolf–Rayet and O-type stars).

Super-Eddington luminosities

The role of the Eddington limit in today's research lies in explaining the very high mass loss rates seen in, for example, the series of outbursts of η Carinae in 1840–1860. The regular, line-driven stellar winds can only explain a mass loss rate of around 10⁻⁴–10⁻³ solar masses per year, whereas losses of up to 0.5 solar masses per year are needed to understand the η Carinae outbursts. This can be done with the help of the super-Eddington broad spectrum radiation driven winds.

Gamma-ray bursts, novae and supernovae are examples of systems exceeding their Eddington luminosity by a large factor for very short times, resulting in short and highly intensive mass loss rates. Some X-ray binaries and active galaxies are able to maintain luminosities close to the Eddington limit for very long times. For accretion-powered sources such as accreting neutron stars or cataclysmic variables (accreting white dwarfs), the limit may act to reduce or cut off the accretion flow, imposing an Eddington limit on accretion corresponding to that on luminosity. Super-Eddington accretion onto stellar-mass black holes is one possible model for ultraluminous X-ray sources (ULXs).

For accreting black holes, not all the energy released by accretion has to appear as outgoing luminosity, since energy can be lost through the event horizon, down the hole. Such sources effectively may not conserve energy. Then the accretion efficiency, or the fraction of energy actually radiated of that theoretically available from the gravitational energy release of accreting material, enters in an essential way.

Other factors

The Eddington limit is not a strict limit on the luminosity of a stellar object. The limit does not consider several potentially important factors, and super-Eddington objects have been observed that do not seem to have the predicted high mass-loss rate. Other factors that might affect the maximum luminosity of a star include:

• Porosity. A problem with steady winds driven by broad-spectrum radiation is that both the radiative flux and gravitational acceleration scale with r ⁻². The ratio between these factors is constant, and in a super-Eddington star, the whole envelope would become gravitationally unbound at the same time. This is not observed. A possible solution is introducing an atmospheric porosity, where we imagine the stellar atmosphere to consist of denser regions surrounded by regions of lower-density gas. This would reduce the coupling between radiation and matter, and the full force of the radiation field would be seen only in the more homogeneous outer, lower-density layers of the atmosphere.
• Turbulence. A possible destabilizing factor might be the turbulent pressure arising when energy in the convection zones builds up a field of supersonic turbulence. The importance of turbulence is being debated, however.
• Photon bubbles. Another factor that might explain some stable super-Eddington objects is the photon bubble effect. Photon bubbles would develop spontaneously in radiation-dominated atmospheres when the radiation pressure exceeds the gas pressure. We can imagine a region in the stellar atmosphere with a density lower than the surroundings, but with a higher radiation pressure. Such a region would rise through the atmosphere, with radiation diffusing in from the sides, leading to an even higher radiation pressure. This effect could transport radiation more efficiently than a homogeneous atmosphere, increasing the allowed total radiation rate. Accretion discs may exhibit luminosities as high as 10–100 times the Eddington limit without experiencing instabilities.

Humphreys–Davidson limit

The upper H–R diagram with the empirical Humphreys-Davidson limit marked (green line). Stars are observed above the limit only during brief outbursts.

Observations of massive stars show a clear upper limit to their luminosity, termed the Humphreys–Davidson limit after the researchers who first wrote about it. Only highly unstable objects are found, temporarily, at higher luminosities. Efforts to reconcile this with the theoretical Eddington limit have been largely unsuccessful.

The H-D limit for cool supergiants is placed at around 316,000 L_☉.

Arthur Eddington

From Wikipedia, the free encyclopedia

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

 

Sir Arthur Eddington OM FRS
Born	Arthur Stanley Eddington 28 December 1882 Kendal, Westmorland, England
Died	22 November 1944 (aged 61) Cambridge, Cambridgeshire, England
Alma mater	University of Manchester Trinity College, Cambridge
Known for	Arrow of time Eddington approximation Eddington experiment Eddington's affine geometry Eddington limit Eddington number Eddington valve Eddington–Dirac number Eddington–Finkelstein coordinates Eddington stellar model Eddington–Sweet circulation
Awards	Royal Society Royal Medal (1928) Smith's Prize (1907) RAS Gold Medal (1924) Henry Draper Medal (1924) Bruce Medal (1924) Knight Bachelor (1930) Order of Merit (1938)
Scientific career
Fields	Astrophysics
Institutions	Trinity College, Cambridge
Academic advisors	E. T. Whittaker Alfred North Whitehead Ernest William Barnes Robert Alfred Herman
Doctoral students	Subrahmanyan Chandrasekhar Leslie Comrie Hermann Bondi
Other notable students	Georges Lemaître Vibert Douglas George C. McVittie

Sir Arthur Stanley Eddington OM FRS (28 December 1882 – 22 November 1944) was an English astronomer, physicist, and mathematician. He was also a philosopher of science and a populariser of science. The Eddington limit, the natural limit to the luminosity of stars, or the radiation generated by accretion onto a compact object, is named in his honour.

Around 1920, he foreshadowed the discovery and mechanism of nuclear fusion processes in stars, in his paper "The Internal Constitution of the Stars". At that time, the source of stellar energy was a complete mystery; Eddington was the first to correctly speculate that the source was fusion of hydrogen into helium.

Eddington wrote a number of articles that announced and explained Einstein's theory of general relativity to the English-speaking world. World War I had severed many lines of scientific communication, and new developments in German science were not well known in England. He also conducted an expedition to observe the solar eclipse of 29 May 1919 on the Island of Príncipe that provided one of the earliest confirmations of general relativity, and he became known for his popular expositions and interpretations of the theory.

Early years

Eddington was born 28 December 1882 in Kendal, Westmorland (now Cumbria), England, the son of Quaker parents, Arthur Henry Eddington, headmaster of the Quaker School, and Sarah Ann Shout.

His father taught at a Quaker training college in Lancashire before moving to Kendal to become headmaster of Stramongate School. He died in the typhoid epidemic which swept England in 1884. His mother was left to bring up her two children with relatively little income. The family moved to Weston-super-Mare where at first Stanley (as his mother and sister always called Eddington) was educated at home before spending three years at a preparatory school. The family lived at a house called Varzin, 42 Walliscote Road, Weston-super-Mare. A commemorative plaque on the building explains Sir Arthur's contribution to science.

In 1893 Eddington entered Brynmelyn School. He proved to be a most capable scholar, particularly in mathematics and English literature. His performance earned him a scholarship to Owens College, Manchester (what was later to become the University of Manchester), in 1898, which he was able to attend, having turned 16 that year. He spent the first year in a general course, but he turned to physics for the next three years. Eddington was greatly influenced by his physics and mathematics teachers, Arthur Schuster and Horace Lamb. At Manchester, Eddington lived at Dalton Hall, where he came under the lasting influence of the Quaker mathematician J. W. Graham. His progress was rapid, winning him several scholarships, and he graduated with a BSc in physics with First Class Honours in 1902.

Based on his performance at Owens College, he was awarded a scholarship to Trinity College, Cambridge, in 1902. His tutor at Cambridge was Robert Alfred Herman and in 1904 Eddington became the first ever second-year student to be placed as Senior Wrangler. After receiving his M.A. in 1905, he began research on thermionic emission in the Cavendish Laboratory. This did not go well, and meanwhile he spent time teaching mathematics to first year engineering students. This hiatus was brief. Through a recommendation by E. T. Whittaker, his senior colleague at Trinity College, he secured a position at the Royal Observatory, Greenwich, where he was to embark on his career in astronomy, a career whose seeds had been sown even as a young child when he would often "try to count the stars".

Plaque at 42 Walliscote Road, Weston-super-Mare

Eddington, right, on a toy donkey; possibly during the Fifth Conference of the International Union for Co-operation in Solar Research, held in Bonn, Germany, 1913

Astronomy

In January 1906, Eddington was nominated to the post of chief assistant to the Astronomer Royal at the Royal Greenwich Observatory. He left Cambridge for Greenwich the following month. He was put to work on a detailed analysis of the parallax of 433 Eros on photographic plates that had started in 1900. He developed a new statistical method based on the apparent drift of two background stars, winning him the Smith's Prize in 1907. The prize won him a fellowship of Trinity College, Cambridge. In December 1912, George Darwin, son of Charles Darwin, died suddenly, and Eddington was promoted to his chair as the Plumian Professor of Astronomy and Experimental Philosophy in early 1913. Later that year, Robert Ball, holder of the theoretical Lowndean chair, also died, and Eddington was named the director of the entire Cambridge Observatory the next year. In May 1914, he was elected a fellow of the Royal Society: he was awarded the Royal Medal in 1928 and delivered the Bakerian Lecture in 1926.

Eddington also investigated the interior of stars through theory, and developed the first true understanding of stellar processes. He began this in 1916 with investigations of possible physical explanations for Cepheid variable stars. He began by extending Karl Schwarzschild's earlier work on radiation pressure in Emden polytropic models. These models treated a star as a sphere of gas held up against gravity by internal thermal pressure, and one of Eddington's chief additions was to show that radiation pressure was necessary to prevent collapse of the sphere. He developed his model despite knowingly lacking firm foundations for understanding opacity and energy generation in the stellar interior. However, his results allowed for calculation of temperature, density and pressure at all points inside a star (thermodynamic anisotropy), and Eddington argued that his theory was so useful for further astrophysical investigation that it should be retained despite not being based on completely accepted physics. James Jeans contributed the important suggestion that stellar matter would certainly be ionized, but that was the end of any collaboration between the pair, who became famous for their lively debates.

Eddington defended his method by pointing to the utility of his results, particularly his important mass–luminosity relation. This had the unexpected result of showing that virtually all stars, including giants and dwarfs, behaved as ideal gases. In the process of developing his stellar models, he sought to overturn current thinking about the sources of stellar energy. Jeans and others defended the Kelvin–Helmholtz mechanism, which was based on classical mechanics, while Eddington speculated broadly about the qualitative and quantitative consequences of possible proton–electron annihilation and nuclear fusion processes.

Around 1920, he anticipated the discovery and mechanism of nuclear fusion processes in stars, in his paper "The Internal Constitution of the Stars". At that time, the source of stellar energy was a complete mystery; Eddington correctly speculated that the source was fusion of hydrogen into helium, liberating enormous energy according to Einstein's equation E = mc². This was a particularly remarkable development since at that time fusion and thermonuclear energy, and even the fact that stars are largely composed of hydrogen (see metallicity), had not yet been discovered. Eddington's paper, based on knowledge at the time, reasoned that:

1. The leading theory of stellar energy, the contraction hypothesis (cf. the Kelvin–Helmholtz mechanism), should cause stars' rotation to visibly speed up due to conservation of angular momentum. But observations of Cepheid variable stars showed this was not happening.
2. The only other known plausible source of energy was conversion of matter to energy; Einstein had shown some years earlier that a small amount of matter was equivalent to a large amount of energy.
3. Francis Aston had also recently shown that the mass of a helium atom was about 0.8% less than the mass of the four hydrogen atoms which would, combined, form a helium atom, suggesting that if such a combination could happen, it would release considerable energy as a byproduct.
4. If a star contained just 5% of fusible hydrogen, it would suffice to explain how stars got their energy. (We now know that most "ordinary" stars contain far more than 5% hydrogen.)
5. Further elements might also be fused, and other scientists had speculated that stars were the "crucible" in which light elements combined to create heavy elements, but without more-accurate measurements of their atomic masses nothing more could be said at the time.

All of these speculations were proven correct in the following decades.

With these assumptions, he demonstrated that the interior temperature of stars must be millions of degrees. In 1924, he discovered the mass–luminosity relation for stars (see Lecchini in § Further reading). Despite some disagreement, Eddington's models were eventually accepted as a powerful tool for further investigation, particularly in issues of stellar evolution. The confirmation of his estimated stellar diameters by Michelson in 1920 proved crucial in convincing astronomers unused to Eddington's intuitive, exploratory style. Eddington's theory appeared in mature form in 1926 as The Internal Constitution of the Stars, which became an important text for training an entire generation of astrophysicists.

Eddington's work in astrophysics in the late 1920s and the 1930s continued his work in stellar structure, and precipitated further clashes with Jeans and Edward Arthur Milne. An important topic was the extension of his models to take advantage of developments in quantum physics, including the use of degeneracy physics in describing dwarf stars.

Dispute with Chandrasekhar on the mass limit of stars

Main article: Chandrasekhar–Eddington dispute

The topic of extension of his models precipitated his dispute with Subrahmanyan Chandrasekhar, who was then a student at Cambridge. Chandrasekhar's work presaged the discovery of black holes, which at the time seemed so absurdly non-physical that Eddington refused to believe that Chandrasekhar's purely mathematical derivation had consequences for the real world. Eddington was wrong and his motivation is controversial. Chandrasekhar's narrative of this incident, in which his work is harshly rejected, portrays Eddington as rather cruel and dogmatic. Chandra benefited from his friendship with Eddington. It was Eddington and Milne who put up Chandra's name for the fellowship for the Royal Society which Chandra obtained. An FRS meant he was at the Cambridge high-table with all the luminaries and a very comfortable endowment for research. Eddington's criticism seems to have been based partly on a suspicion that a purely mathematical derivation from relativity theory was not enough to explain the seemingly daunting physical paradoxes that were inherent to degenerate stars, but to have "raised irrelevant objections" in addition, as Thanu Padmanabhan puts it.

Relativity

During World War I, Eddington was secretary of the Royal Astronomical Society, which meant he was the first to receive a series of letters and papers from Willem de Sitter regarding Einstein's theory of general relativity. Eddington was fortunate in being not only one of the few astronomers with the mathematical skills to understand general relativity, but owing to his internationalist and pacifist views inspired by his Quaker religious beliefs, one of the few at the time who was still interested in pursuing a theory developed by a German physicist. He quickly became the chief supporter and expositor of relativity in Britain. He and Astronomer Royal Frank Watson Dyson organized two expeditions to observe a solar eclipse in 1919 to make the first empirical test of Einstein's theory: the measurement of the deflection of light by the Sun's gravitational field. In fact, Dyson's argument for the indispensability of Eddington's expertise in this test was what prevented Eddington from eventually having to enter military service.

When conscription was introduced in Britain on 2 March 1916, Eddington intended to apply for an exemption as a conscientious objector. Cambridge University authorities instead requested and were granted an exemption on the ground of Eddington's work being of national interest. In 1918, this was appealed against by the Ministry of National Service. Before the appeal tribunal in June, Eddington claimed conscientious objector status, which was not recognized and would have ended his exemption in August 1918. A further two hearings took place in June and July, respectively. Eddington's personal statement at the June hearing about his objection to war based on religious grounds is on record. The Astronomer Royal, Sir Frank Dyson, supported Eddington at the July hearing with a written statement, emphasising Eddington's essential role in the solar eclipse expedition to Príncipe in May 1919. Eddington made clear his willingness to serve in the Friends' Ambulance Unit, under the jurisdiction of the British Red Cross, or as a harvest labourer. However, the tribunal's decision to grant a further twelve months' exemption from military service was on condition of Eddington continuing his astronomy work, in particular in preparation for the Príncipe expedition. The war ended before the end of his exemption.

One of Eddington's photographs of the total solar eclipse of 29 May 1919, presented in his 1920 paper announcing its success, confirming Einstein's theory that light "bends"

After the war, Eddington travelled to the island of Príncipe off the west coast of Africa to watch the solar eclipse of 29 May 1919. During the eclipse, he took pictures of the stars (several stars in the Hyades cluster, including Kappa Tauri of the constellation Taurus) whose line of sight from the Earth happened to be near the Sun's location in the sky at that time of year. This effect is noticeable only during a total solar eclipse when the sky is dark enough to see stars which are normally obscured by the Sun's brightness. According to the theory of general relativity, stars with light rays that passed near the Sun would appear to have been slightly shifted because their light had been curved by its gravitational field. Eddington showed that Newtonian gravitation could be interpreted to predict half the shift predicted by Einstein.

Eddington's observations published the next year allegedly confirmed Einstein's theory, and were hailed at the time as evidence of general relativity over the Newtonian model. The news was reported in newspapers all over the world as a major story. Afterward, Eddington embarked on a campaign to popularize relativity and the expedition as landmarks both in scientific development and international scientific relations.

It has been claimed that Eddington's observations were of poor quality, and he had unjustly discounted simultaneous observations at Sobral, Brazil, which appeared closer to the Newtonian model, but a 1979 re-analysis with modern measuring equipment and contemporary software validated Eddington's results and conclusions. The quality of the 1919 results was indeed poor compared to later observations, but was sufficient to persuade contemporary astronomers. The rejection of the results from the expedition to Brazil was due to a defect in the telescopes used which, again, was completely accepted and well understood by contemporary astronomers.

The minute book of Cambridge ∇²V Club for the meeting where Eddington presented his observations of the curvature of light around the Sun, confirming Einstein's theory of general relativity. They include the line "A general discussion followed. The President remarked that the 83rd meeting was historic".

Throughout this period, Eddington lectured on relativity, and was particularly well known for his ability to explain the concepts in lay terms as well as scientific. He collected many of these into the Mathematical Theory of Relativity in 1923, which Albert Einstein suggested was "the finest presentation of the subject in any language." He was an early advocate of Einstein's general relativity, and an interesting anecdote well illustrates his humour and personal intellectual investment: Ludwik Silberstein, a physicist who thought of himself as an expert on relativity, approached Eddington at the Royal Society's (6 November) 1919 meeting where he had defended Einstein's relativity with his Brazil-Príncipe solar eclipse calculations with some degree of scepticism, and ruefully charged Arthur as one who claimed to be one of three men who actually understood the theory (Silberstein, of course, was including himself and Einstein as the other). When Eddington refrained from replying, he insisted Arthur not be "so shy", whereupon Eddington replied, "Oh, no! I was wondering who the third one might be!"

Cosmology

Eddington was also heavily involved with the development of the first generation of general relativistic cosmological models. He had been investigating the instability of the Einstein universe when he learned of both Lemaître's 1927 paper postulating an expanding or contracting universe and Hubble's work on the recession of the spiral nebulae. He felt the cosmological constant must have played the crucial role in the universe's evolution from an Einsteinian steady state to its current expanding state, and most of his cosmological investigations focused on the constant's significance and characteristics. In The Mathematical Theory of Relativity, Eddington interpreted the cosmological constant to mean that the universe is "self-gauging".

Fundamental theory and the Eddington number

During the 1920s until his death, Eddington increasingly concentrated on what he called "fundamental theory" which was intended to be a unification of quantum theory, relativity, cosmology, and gravitation. At first he progressed along "traditional" lines, but turned increasingly to an almost numerological analysis of the dimensionless ratios of fundamental constants.

His basic approach was to combine several fundamental constants in order to produce a dimensionless number. In many cases these would result in numbers close to 10⁴⁰, its square, or its square root. He was convinced that the mass of the proton and the charge of the electron were a "natural and complete specification for constructing a Universe" and that their values were not accidental. One of the discoverers of quantum mechanics, Paul Dirac, also pursued this line of investigation, which has become known as the Dirac large numbers hypothesis. A somewhat damaging statement in his defence of these concepts involved the fine-structure constant, α. At the time it was measured to be very close to 1/136, and he argued that the value should in fact be exactly 1/136 for epistemological reasons. Later measurements placed the value much closer to 1/137, at which point he switched his line of reasoning to argue that one more should be added to the degrees of freedom, so that the value should in fact be exactly 1/137, the Eddington number. Wags at the time started calling him "Arthur Adding-one". This change of stance detracted from Eddington's credibility in the physics community. The current CODATA value is 1/137.035999177(21).

Eddington believed he had identified an algebraic basis for fundamental physics, which he termed "E-numbers" (representing a certain group – a Clifford algebra). These in effect incorporated spacetime into a higher-dimensional structure. While his theory has long been neglected by the general physics community, similar algebraic notions underlie many modern attempts at a grand unified theory. Moreover, Eddington's emphasis on the values of the fundamental constants, and specifically upon dimensionless numbers derived from them, is nowadays a central concern of physics. In particular, he predicted a number of hydrogen atoms in the Universe 136 × 2²⁵⁶ ≈ 1.57×10⁷⁹, or equivalently the half of the total number of particles protons + electrons. He did not complete this line of research before his death in 1944; his book Fundamental Theory was published posthumously in 1948.

Eddington number for cycling

Eddington is credited with devising a measure of a cyclist's long-distance riding achievements. The Eddington number in the context of cycling is defined as the maximum number E such that the cyclist has cycled at least E miles on at least E days.

For example, an Eddington number of 70 would imply that the cyclist has cycled at least 70 miles in a day on at least 70 occasions. Achieving a high Eddington number is difficult, since moving from, say, 70 to 75 will (probably) require more than five new long-distance rides, since any rides shorter than 75 miles will no longer be included in the reckoning. Eddington's own life-time E-number was 84.

The Eddington number for cycling is analogous to the h-index that quantifies both the actual scientific productivity and the apparent scientific impact of a scientist.

Philosophy

Idealism

Eddington wrote in his book The Nature of the Physical World that "The stuff of the world is mind-stuff."

The mind-stuff of the world is, of course, something more general than our individual conscious minds ... The mind-stuff is not spread in space and time; these are part of the cyclic scheme ultimately derived out of it ... It is necessary to keep reminding ourselves that all knowledge of our environment from which the world of physics is constructed, has entered in the form of messages transmitted along the nerves to the seat of consciousness ... Consciousness is not sharply defined, but fades into subconsciousness; and beyond that we must postulate something indefinite but yet continuous with our mental nature ... It is difficult for the matter-of-fact physicist to accept the view that the substratum of everything is of mental character. But no one can deny that mind is the first and most direct thing in our experience, and all else is remote inference.

— Eddington, The Nature of the Physical World, 276–81.

The idealist conclusion was not integral to his epistemology but was based on two main arguments.

The first derives directly from current physical theory. Briefly, mechanical theories of the ether and of the behaviour of fundamental particles have been discarded in both relativity and quantum physics. From this, Eddington inferred that a materialistic metaphysics was outmoded and that, in consequence, since the disjunction of materialism or idealism are assumed to be exhaustive, an idealistic metaphysics is required. The second, and more interesting argument, was based on Eddington's epistemology, and may be regarded as consisting of two parts. First, all we know of the objective world is its structure, and the structure of the objective world is precisely mirrored in our own consciousness. We therefore have no reason to doubt that the objective world too is "mind-stuff". Dualistic metaphysics, then, cannot be evidentially supported.

But, second, not only can we not know that the objective world is nonmentalistic, we also cannot intelligibly suppose that it could be material. To conceive of a dualism entails attributing material properties to the objective world. However, this presupposes that we could observe that the objective world has material properties. But this is absurd, for whatever is observed must ultimately be the content of our own consciousness, and consequently, nonmaterial.

Eddington believed that physics cannot explain consciousness - "light waves are propagated from the table to the eye; chemical changes occur in the retina; propagation of some kind occurs in the optic nerves; atomic changes follow in the brain. Just where the final leap into consciousness occurs is not clear. We do not know the last stage of the message in the physical world before it became a sensation in consciousness".

Ian Barbour, in his book Issues in Science and Religion (1966), p. 133, cites Eddington's The Nature of the Physical World (1928) for a text that argues the Heisenberg uncertainty principle provides a scientific basis for "the defense of the idea of human freedom" and his Science and the Unseen World (1929) for support of philosophical idealism, "the thesis that reality is basically mental".

Charles De Koninck points out that Eddington believed in objective reality existing apart from our minds, but was using the phrase "mind-stuff" to highlight the inherent intelligibility of the world: that our minds and the physical world are made of the same "stuff" and that our minds are the inescapable connection to the world. As De Koninck quotes Eddington,

There is a doctrine well known to philosophers that the moon ceases to exist when no one is looking at it. I will not discuss the doctrine since I have not the least idea what is the meaning of the word existence when used in this connection. At any rate the science of astronomy has not been based on this spasmodic kind of moon. In the scientific world (which has to fulfill functions less vague than merely existing) there is a moon which appeared on the scene before the astronomer; it reflects sunlight when no one sees it; it has mass when no one is measuring the mass; it is distant 240,000 miles from the earth when no one is surveying the distance; and it will eclipse the sun in 1999 even if the human race has succeeded in killing itself off before that date.

— Eddington, The Nature of the Physical World, 226

Science

See also: Indeterminism and Indeterminism in science

Against Albert Einstein and others who advocated determinism, indeterminism—championed by Eddington—says that a physical object has an ontologically undetermined component that is not due to the epistemological limitations of physicists' understanding. The uncertainty principle in quantum mechanics, then, would not necessarily be due to hidden variables but to an indeterminism in nature itself. Eddington proclaimed "It is a consequence of the advent of the quantum theory that physics is no longer pledged to a scheme of deterministic law".

Eddington agreed with the tenet of logical positivism that "the meaning of a scientific statement is to be ascertained by reference to the steps which would be taken to verify it".

Popular and philosophical writings

Eddington wrote a parody of The Rubaiyat of Omar Khayyam, recounting his 1919 solar eclipse experiment. It contained the following quatrain:

Oh leave the Wise our measures to collate
           One thing at least is certain, LIGHT has WEIGHT,
One thing is certain, and the rest debate—
Light-rays, when near the Sun, DO NOT GO STRAIGHT.

In addition to his textbook The Mathematical Theory of Relativity, during the 1920s and 30s, Eddington gave numerous lectures, interviews, and radio broadcasts on relativity, and later, quantum mechanics. Many of these were gathered into books, including The Nature of the Physical World and New Pathways in Science. His use of literary allusions and humour helped make these difficult subjects more accessible.

Eddington's books and lectures were immensely popular with the public, not only because of his clear exposition, but also for his willingness to discuss the philosophical and religious implications of the new physics. He argued for a deeply rooted philosophical harmony between scientific investigation and religious mysticism, and also that the positivist nature of relativity and quantum physics provided new room for personal religious experience and free will. Unlike many other spiritual scientists, he rejected the idea that science could provide proof of religious propositions.

His popular writings made him a household name in Great Britain between the world wars.

Death

Eddington died of cancer in the Evelyn Nursing Home, Cambridge, on 22 November 1944. He was unmarried. His body was cremated at Cambridge Crematorium (Cambridgeshire) on 27 November 1944; the cremated remains were buried in the grave of his mother in the Ascension Parish Burial Ground in Cambridge.

Cambridge University's North West Cambridge development has been named Eddington in his honour.

Eddington was played by David Tennant in the television film Einstein and Eddington, with Einstein played by Andy Serkis. The film was notable for its groundbreaking portrayal of Eddington as a somewhat repressed gay man. It was first broadcast in 2008.

The actor Paul Eddington was a relative, mentioning in his autobiography (in light of his own weakness in mathematics) "what I then felt to be the misfortune" of being related to "one of the foremost physicists in the world".

Obituaries

• Obituary 1 by Henry Norris Russell, Astrophysical Journal 101 (1943–46) 133
• Obituary 2 by A. Vibert Douglas, Journal of the Royal Astronomical Society of Canada, 39 (1943–46) 1
• Obituary 3 by Harold Spencer Jones and E. T. Whittaker, Monthly Notices of the Royal Astronomical Society 105 (1943–46) 68
• Obituary 4 by Herbert Dingle, The Observatory 66 (1943–46) 1
• The Times, Thursday, 23 November 1944; pg. 7; Issue 49998; col D: Obituary (unsigned) – Image of cutting available at O'Connor, John J.; Robertson, Edmund F., "Arthur Eddington", MacTutor History of Mathematics Archive, University of St Andrews

Honours

Awards and honors

• Smith's Prize (1907)
• International Honorary Member of the American Academy of Arts and Sciences (1922)
• Bruce Medal of Astronomical Society of the Pacific (1924)
• Henry Draper Medal of the National Academy of Sciences (1924)
• Gold Medal of the Royal Astronomical Society (1924)
• International Member of the United States National Academy of Sciences (1925)
• Foreign membership of the Royal Netherlands Academy of Arts and Sciences (1926)
• Prix Jules Janssen of the Société astronomique de France (French Astronomical Society) (1928)
• Royal Medal of the Royal Society (1928)
• Knighthood (1930)
• International Member of the American Philosophical Society (1931)
• Order of Merit (1938)
• Honorary member of the Norwegian Astronomical Society (1939)
• Hon. Freeman of Kendal, 1930

Named after him

• Lunar crater Eddington
• asteroid 2761 Eddington
• Royal Astronomical Society's Eddington Medal
• Eddington mission, now cancelled
• Eddington Tower, halls of residence at the University of Essex
• Eddington Astronomical Society, an amateur society based in his hometown of Kendal
• Eddington, a house (group of students, used for in-school sports matches) of Kirkbie Kendal School
• Eddington, new suburb of North West Cambridge, opened in 2017

Service

• Gave the Swarthmore Lecture in 1929
• Chairman of the National Peace Council 1941–1943
• President of the International Astronomical Union; of the Physical Society, 1930–32; of the Royal Astronomical Society, 1921–23
• Romanes Lecturer, 1922
• Gifford Lecturer, 1927

In popular culture

• Eddington is a central figure in the short story "The Mathematician's Nightmare: The Vision of Professor Squarepunt" by Bertrand Russell, a work featured in The Mathematical Magpie by Clifton Fadiman.
• He was portrayed by David Tennant in the television film Einstein and Eddington, a co-production of the BBC and HBO, broadcast in the United Kingdom on Saturday, 22 November 2008, on BBC2.
• His thoughts on humour and religious experience were quoted in the adventure game The Witness, a production of the Thelka, Inc., released on 26 January 2016.
• Time placed him on the cover on 16 April 1934.
• The song “In Transit”, from the 2023 album Signs Of Life by Neil Gaiman and Fourplay String Quartet was written in memory of him.

Publications

• 1914. Stellar Movements and the Structure of the Universe. London: Macmillan.
• 1918. Report on the relativity theory of gravitation. London, Fleetway Press, Ltd.
• 1920. Space, Time and Gravitation: An Outline of the General Relativity Theory. Cambridge University Press. ISBN 0-521-33709-7
• 1922. The theory of relativity and its influence on scientific thought
• 1923. 1952. The Mathematical Theory of Relativity. Cambridge University Press.
• 1925. The Domain of Physical Science. 2005 reprint: ISBN 1-4253-5842-X
• 1926. Stars and Atoms. Oxford: British Association.
• 1926. The Internal Constitution of Stars. Cambridge University Press. ISBN 0-521-33708-9
• 1928. The Nature of the Physical World. MacMillan. 1935 replica edition: ISBN 0-8414-3885-4, University of Michigan 1981 edition: ISBN 0-472-06015-5 (1926–27 Gifford lectures)
• 1929. Science and the Unseen World. US Macmillan, UK Allen & Unwin. 1980 Reprint Arden Library ISBN 0-8495-1426-6. 2004 US reprint – Whitefish, Montana : Kessinger Publications: ISBN 1-4179-1728-8. 2007 UK reprint London, Allen & Unwin ISBN 978-0-901689-81-8 (Swarthmore Lecture), with a new foreword by George Ellis.
• 1930. Why I Believe in God: Science and Religion, as a Scientist Sees It. Arrow/scrollable preview.
• 1933. The Expanding Universe: Astronomy's 'Great Debate', 1900–1931. Cambridge University Press. ISBN 0-521-34976-1
• 1935. New Pathways in Science. Cambridge University Press.
• 1936. Relativity Theory of Protons and Electrons. Cambridge Univ. Press.
• 1939. Philosophy of Physical Science. Cambridge University Press. ISBN 0-7581-2054-0 (1938 Tarner lectures at Cambridge)
• 1946. Fundamental Theory. Cambridge University Press.