Edmund Stoner

Thomas

Keywords: total energy methods, electron theory, electron energy states (user supplied): cosmology, white dwarfs, Chandrasekhar mass

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Introduction

The Biographical Memoires of Fellows of the Royal Society [1] tell us that Edmund Clifton Stoner was born in Esher in the county of Surrey in 1899. When he was in his teens his parents moved to Bolton in Lancashire where he won a scholarship that enabled him to attend Bolton Grammar School. From there he entered Cambridge University in 1918 graduating in 1921 with a first-class degree in the Natural Sciences Tripos. Surprisingly, although the physics lectures given to undergraduates were quite mathematical in content there was no mathematics teaching in the syllabus, students being expected to pick up what they needed on their own. Stoner was conscious of this lack of formal mathematics training throughout his career.

Between 1921 and 1924 Stoner worked for a PhD in the Cavendish laboratory with Ernest Rutherford as his official supervisor. His research topic was the measurement of the absorption of X-rays by a selected group of elements. He obtained his PhD in 1924, despite a prolonged period in hospital after being diagnosed with diabetes. Towards the end of his time at the Cavendish he wrote a paper entitled The distribution of electrons among atomic energy levels. This paper, published in the Philosophical Magazine [START_REF] Stoner | The Distribution of Electrons among Atomic Levels[END_REF] preceded Pauli's enunciation of his exclusion principle and played a significant role in its creation: in effect it contained an explicit statement of the exclusion principle. It should be noted that the exclusion principle was to play a crucial role in the theory of white dwarfs.

After deciding that he did not wish to remain in Cambridge, Stoner successfully applied for a lectureship in the Physics Department of the University of Leeds. There, where he remained until his retirement in 1963, he switched to research in theoretical physics and embarked on pioneering work to understand the magnetic properties of matter: this is the work for which he is best known. His interest in white dwarfs was sparked by Fowler's paper On Dense Matter [3]. Over the period 1929 to 1932 he published the pioneering papers on white dwarfs, which are the subject of this article. In particular he deduced the existence of a limiting mass for white dwarfs now known as the Chandrasekhar mass.

White dwarfs

White dwarfs are hot, very compact, stars that have ceased nuclear burning. Their high temperature and low luminosity places them in the bottom left-hand corner of the Hertzprung-Russell diagram [4]. The white dwarf, 40 Eridani B, recorded by William Herschel in 1783 was the first to be detected [5]. By the late 1920s, when Stoner started his investigations, the masses, luminosities and surface temperatures of a small number of white dwarfs had been measured. The best-known example, Sirius B, is the faint companion of Sirius, the brightest star in the night sky. The observational data at this time for Sirius B gave its mass to be 0.85 solar masses, its surface temperature 8000 K and its radius 21 000 km. These figures imply a mean density of about 5 X 10 7 kg m -3 . At the time this density exceeded that of any other known object by several orders of magnitude. For example the mean density of the Sun is about 1 400 kg m -3 and the density of osmium metal, the densest element, is 2.2 X 10 4 kg m -3 . It is interesting to note that, with these values for the mass and radius of Sirius B, the General Theory of Relativity predicts that light coming from its surface would have a red shift of 20 km s -1 . A measurement of this red shift published by W S Adams [6] produced a value of 19 km s -1 . However, we now know that this apparent agreement between The General Theory of Relativity and the data on Sirius B, which so delighted Eddington, is coincidental. Adam's measurement of redshift is too low by a factor of 4 and the radius is too large by a factor of almost 4. The modern values for the mass, temperature, radius and red shift of Sirius B are 0.98M solar , 22 000 K and 6 000 km respectively and the red shift, measured using the Hubble space telescope, is 80 km s -1 [7]. With these measurements the density of Sirius B becomes 2 X 10 9 kg m -3 or 2 tonnes cm -3 . Such high densities seemed at the time to raise a paradox: matter compressed to this degree must be completely ionised so what happens to the star as its reaches the end of its lifetime and starts to cool? In the words of Sir Arthur Eddington [8] 'The star could not stop losing heat, but it would not have enough energy to cool down'.

Eddington supposed that for the star to reach a stable cold state, its nuclei and electrons needed to recombine into atoms. But to do this the star needed to regain normal density by expanding against the force of gravity, which it could not do. According to classical physics, a cold ionised gas cannot exert any pressure so the star was doomed to continue contracting indefinitely, a situation that Eddington regarded as absurd.

The resolution of this paradox was provided by the newly discovered quantum mechanics and the Pauli Exclusion Principle. The Cambridge theoretician R H Fowler realised that even at the absolute zero of temperature a very dense electron gas will exert a large pressure and that this pressure could support a white dwarf star against collapse [3]. Fowler assumed that the electrons occupied the lowest energy states allowed by the Pauli Principle, which states that only two electrons, one in each of two spin states, can occupy a state of volume h 3 in six-dimensional phase space, where h is the Planck constant. Such a gas is said to be degenerate; the uppermost filled state has a momentum p 0 which, as a function of number density of electrons n, is given, at absolute zero, by

3 / 1 3 / 1 0 8 3 n h p       = π . ( 1
)
So a high density of electrons in physical space implies that they have high momenta and hence that they exert a large pressure which, for non-relativistic electrons, is given as a function of the mass density n m H e µ ρ = by the expression

3 / 5 3 / 2 2 8 3 5               = H e e m m h P µ ρ π , (2)
where m e is the mass of an electron, m H the mass of hydrogen and e µ the average atomic weight per electron. At the time, white dwarfs, being highly evolved stars, were thought to be composed of heavy elements so the value of e µ was taken to be about 2.5. We now know that white dwarfs are composed mainly of carbon and oxygen so the correct value of e µ is very close to 2. The nuclei being much more massive than the electrons do not contribute significantly to the pressure.

Stoner's papers on white dwarfs

In the first of three papers that appeared in the Philosophical Magazine [9], Stoner took Fowler's idea that white dwarfs are supported by electron degeneracy pressure and asked the question: is there a limit to the density of such stars? He modelled a star as a sphere of ionised gas having uniform density at zero temperature. Then, from the kinetic energy E K of the electrons and the gravitational binding energy E G of the star, the density is obtained by minimising the total energy with respect to the number density n, that is he imposed the condition

( ) 0 = + G K E E dn d . (3)
From this condition he obtained the following expression for the mean mass density of a star of mass M:

2 9 10 85 . 3         × = solar M M ρ kg m -3 . ( 4
3 / 4 3 / 1 8 3 4               = H e m ch P µ ρ π . ( 5
)
Stoner used his energy minimisation method with the constant density approximation and his relativistic equation of state to obtain the mass of a star for any value of the electron density n, or, equivalently, ρ . He found that as the density increases, the mass of the star increases more and more slowly, eventually reaching a limiting value given by the expression

( ) 2 2 / 3 1 2 5 16 3 H e L m G hc M µ π       = . ( 6
)
With

K G E E <
the star will expand until the dependence of E K on R alters allowing equilibrium to be reached. The limiting mass will occur when the two terms are equal. This condition determines the value of N and hence the limiting mass of the star.

Stoner also looked for and failed to find a relationship between the measured masses and luminosities of the few known white dwarfs. We now know that there is no relationship because white dwarfs have ceased nuclear burning and are cooling down, so the luminosity of a white dwarf depends on its age.

Stoner's third paper [START_REF] Stoner | A Note on Condensed Stars[END_REF] written in collaboration with his research student Frank Tyler is a sequel to his two previous papers. In it he attempts to obtain the actual density distribution in order to determine to what extent his previous results based on the constant density approximation would need to be modified. In his own words:

'The problem has proved more intractable than was anticipated, and a completely satisfactory solution has not been reached'.

He also addressed the fate of a star whose mass lay above the limit. He concluded that the star would contract, heat up and gradually lose mass through radiation until its mass dropped below the limiting value, allowing it to stabilise. It should be born in mind that, in the early 1930s, nuclear physics was in its infancy so his conclusions were reasonable given the state of knowledge at the time. We now know that, rather than a slow contraction, a catastrophic collapse would occur initiated by the onset of the inverse beta decay reaction e n p e ν + → + -, which undermines the support of the star by removing electrons. The collapse gives rise to very rapid heating, causing thermonuclear burning of the carbon and oxygen. The resultant release of energy is rapid enough to blow the star apart -an event now identified as a supernova of type 1a. The name that is attached to the limiting mass of white dwarfs is that of the Indian astrophysicist Subramanian Chandrasekhar, who was also working on the structure of white dwarfs in the early 1930s. He read Stoner's paper-The limiting density of white dwarf stars[9] and realized that, with Fowler's equation of state [START_REF] Stoner | The Distribution of Electrons among Atomic Levels[END_REF] , the density as a function of radius is given by the solution of the Lane-Emden equation with polytropic index n = 3/2. This solution is given in tabulated form in Eddington's book The Internal Constitution of the Stars [8] so he was able to rederive Stoner's result for the dependence of density on the mass, i.e. Chandrasekhar applied the theory of polytropic gas spheres once again but this time with the equation of state (5) which applies only in the extreme relativistic limit. The solution of the stellar structure equations with this equation of state is a polytrope of index n = 3. This solution is also tabulated in Eddington's book, and yields the unique mass 0.91 solar masses (1.4 M solar with

Chandrasekhar and Landau

2 = e µ
) which Chandrasekhar took, without proof, to be the maximum mass [START_REF] Chandrasekhar | The Maximum Mass of Ideal White Dwarfs[END_REF] of a white dwarf. By comparison, Stoner's maximum mass, arrived at with the constant-density approximation, was about 20% higher than this. It seems that at this stage Chandrasekhar did not understand the full significance of his result [START_REF] Weart | an interview with Chandrasekhar[END_REF]. Finally four years later Chandrasekhar succeeded in solving the stellar structure equations with the Stoner-Anderson equation of state for the mass-radius relation for all stellar masses up to the limiting mass [START_REF] Chandrasekhar | The Highly Collapsed Configurations of a Stellar Mass (second paper)[END_REF]. The solutions show that, as the limit is approached, the radius of the star shrinks to zero and the density goes to infinity. In practice the equation of state ceases to be valid when the electron energies exceed the threshold for inverse beta decay to occur, so the limiting mass is never reached.

It is interesting to note that L Landau also obtained a value of

solar M M 5 . 1 ~
for the limiting mass of stars supported by relativistic electron degeneracy pressure [START_REF] Landau | On the Theory of Stars[END_REF]. But then, surprisingly, he went on to interpret this limit as arising from a failure of quantum mechanics:

'As in reality such masses exist quietly as stars and do not show any such ridiculous tendencies we must conclude that all stars heavier than 1.5M solar certainly possess regions in which the laws of quantum mechanics (and therefore of quantum statistics) are violated'.

Discussion

As has often been the case in the history of science an important discovery is not solely due to one person. Such is the case with the discovery of a limiting mass for white dwarf stars.

Stoner's paper The equilibrium of Dense Stars [11] was the first paper to give a convincing demonstration of the existence of a limiting mass. So why is it that Chandrasekhar's name is the one that has subsequently come to be associated with this discovery and why has the important role played by Stoner has been largely forgotten? There seem to be several reasons for this neglect of Stoner's work. Chandrasekhar pursued the study of white dwarfs further than Stoner and was the first person to carry out the accurate numerical integration of the stellar structure equations and to obtain the value 1.4 solar masses [START_REF] Weart | an interview with Chandrasekhar[END_REF]. Also he tended to give rather sparing acknowledgment of Stoner's pioneering work in his papers and in the numerous interviews he gave later in his life. For a more detailed account of this matter see the article by M Nauenberg [START_REF] Nauenberg | Stoner and the Discovery of the Maximum Mass of White Dwarfs' The Journal for the History of Astronomy[END_REF]. A conspicuous example of this neglect occurred In 1983 when Chandrasekhar was awarded a Nobel Prize for his work on stars. His Nobel lecture entitled On Stars and their Stability recounted in some detail the steps leading to the discovery of a limiting mass for white dwarfs. Surprisingly, in this account, acknowledgement of Stoner's work is completely absent.

Finally there is the often recounted story of the dramatic meeting of the Royal Astronomical Society that took place on 11 th January 1935 [START_REF] Miller | Empire of the Stars[END_REF] in which Chandrasekhar presented his results, only for Eddington to get up and ridicule them with the words 'The star has to go on radiating and radiating and contracting and contracting until I suppose it gets down to a few kilometres radius when gravity becomes strong enough to hold in the radiation and the star can at last find peace…. I think there should be a law of nature to prevent a star behaving in this absurd way'.

Stoner's name does not appear in this famous story and he does not appear to have taken part in the subsequent debate with Eddington over the existence of a limiting mass for white dwarfs.

Such was Eddington's preeminent position as the world's leading astrophysicist that, although many of the leading physicists of the day, such as Fowler, Rosenfeld, Bohr, Dirac and Pauli were convinced that Eddington was wrong, none of them were prepared to challenge him openly [START_REF] Miller | Empire of the Stars[END_REF] [21]. Most probably it was Eddington's unyielding opposition to the idea of gravitational collapse that delayed the exploration of its full implications until the 1960s when research into black holes began.

F

I

N white dwarf" stars the last stages of the ionization process are reached, so that at least in the central parts the molecules consist almost entirely of atomic nuclei and free electrons. Under these conditions very high densities are possible. The mean densities deduced from observations, though large (of the order 50 to 100 thousand), fall very far short o[ those corresponding to a "closepacked" arrangement, if a naive view of the "sizes" o[ electrons and nuclei is taken. It was suggested that stone limitation might be introduced by the " jamming" of a few remaining atoms with K-ring electrons.

In a previous paper ~', however, it was shown that a ]imitation is imposed by the exclusion principle applying to free electrons, as embodied in the Fermi statistics. The number o[ electrons with momenta within a definite range cannot exceed a certain maximum.

Any increase in density involves an increase of energy. In the limiting case, at absolute zero, the star can contract until the decrease in gravitational energy becomes insufficient to balance the increase of kinetic energy oE the electrons. With a few simplifying assmnptions, an expression was derived which indicated that the maximum density varied as the square of the mass of the star. It has been pointed out by Anderson ++ that in this calculation the effect of the relativity change of mass was neglected. He has taken this effect into account, but in a manner which seems open to criticism. His general conclusions, which seem to be correct, are that the simple expression holds provided the electron densities are not too large, but their the mass corresponding to large electron densities is smaller than that previously calculated, and that it roaches a limit. The correction becomes important for stars of mass about half that of the Sun, and so for the white dwarfs which were actually considered. The main purpose of this paper is to calculate the effect of the relativity change ot mass, using a method which seems more rigorous than , Communicated by Prof. R. Whiddington, F.R.S. The Eq,dhbrlton oJ Dense Stars. 945 that of Anderson. The conclusion drawn is that th, range in which the simple expression holds approximately is wider than that indicated by Anderson's results, the "limiting mass" being somewhat greater, so that there is still an equilibrimn state, under the particular conditions, for the stars under consideration. The significance of" the results for very dense stars is discussed, and the effect of the electron gas following the Fermi statistics at the smaller densities found in normal stars is briefly considered.

Ea~eet of the Relativity Change of 3lass on the

Zero-point Energy of Electron Gas.

Let E~ be the gravitational energy of the idealized star, EK the total kinetic energy of the electrons, n the number of electrons per unit volume. Then, as previously shown, the equilibrium condition is given by d (E~+EK)=O. Lot ~ be tile mean kinetic energy of tlle electrons at absolute zero, V the volume of the star, supposed to be of uniform density. (The distribution el density will be considered later.) When the relativity change of mass is neglected, the total kinetic energy at absolute zero is given by

Es__nV~=nV 3 (3 ~ 2/a h'n~/3 ~r,, "o " (2)
It is this expression which has to be modified. In Anderson's treatment m0 is replaced by m, where m is derived from the equation 7,, .....

(a)

It is thus assumed that the total kinetic energy can be expressed in the alternative forms -~, 3 t 3 \2/a hln2/a .w(,,,-.,0) and o 7,, whore m has the same value in both expressions, being a " mean" mass; further, that this mean mttss can be employed in subsequent calculations in place of m0. The value of a mean mass, however, depends on the manner in which the mass" enters into the particular expressions for which the appropriate mean value is being derived, and i~ is not legitimate to use one particular mean wtlue indiscriminately .Phil. Mag. S. 7. Vol. 9. No. 60.3la~ 1930.

In the completely condensed state there are two electrons in each cell of the 6-dimensional phase-space (of 6-volume £3), so that the number of electrons in a volume V with momenta between p and p+dp is equal to

2 x 4~rp~dp x V. h 3
Thus for the total energy EK of the electrons in a condensed state with maximum momentum P0

Since nV is the total number of electrons (double the number of cells), the maximum momentum Po is related to n by mo--~ -,~ \ ~-~ ] " The energy is less than that given by (2). In the limit when x is large (x>>l, n>>5"88 × 10~9),

8~-Vmo4C5 x 4 (E~)~>>I--... ha 4' 2~rVc ~ ,, 3 -----~, po--nv~(3)l/ahcn 1/3. ( 12a
)
These expressions may be compared with (10a)and (10b). They show that when n is large the mean kinetic energy increases as n 1/3 (instead of n 2 ). For a constant total number of electrons (nV constant), Ex then varies as nil 3, as does also --Eo. When this is so there will obviously be no equilibrium under the conditions imposed.

The Equilibrium Co~editio~.

If the conditions are such that an equilibrium state is possible, there will be a limiting densit.y--"limiting" because the calculation refers to a sphere or" uniform density at zero temperature. The limiting density is that corresponding to the value of n when d(EK+Eo)=0 ...... [START_REF] Stoner | A Note on Condensed Stars[END_REF] It is more convenient, for the present application, to treat x, defined by (8), as the w~riable, so that the equilibrium condition becomes d f EI~ + E~) = (14) 0. 15) and ( 16) in ( 14), the resuR is The final result is

• .

(19)

By means of equations (18 a) and ( 19) the mass may be found correspondi,Jg to any value of x, and so to any limiting electron concentration. Since the mean molecular weight is about 2"5 m~, the limiting density is given by po=2"5 mHn=4"15× 10-24n.

. The method of procedure adopted is to calculate F(x) for different values of x, from which a F(x), .v curve may be plotted. From this, using (18 a), an M, x curve may be derived. If desired, this may be converted into an M, n curve, using the relation giving n-----5'876x 1029x s or x=l'!94x 10-1°n 1/~. . (21)

The full expression for F(x) ( 19) is inconvenient to use when .v is small. The expression may then be expanded, with the result 1 1 s 1 5

......

C19a)

When x is small enough for tho first term in the expansion to be sufficient, the same result is obtained as before, namely, or, substituting numerical values from (18a), [START_REF] Miller | Empire of the Stars[END_REF], and (21), n=2"396 x 10-~M 2, ) p=9"95x10-61M 2. ~ . . .

. (22)*

The first two terms in (19a) give F(x) correct to less than 1 per cent. up to x ='5, and form the most convenient expression for this range.

When x is large, as may be seen from ( 19), The value obtained by Anderson for M0 is 1"37 × 1033, so that the range of mass in which this particular type of equilibrium can occur is shown by this method to be considerably greater than that indicated by Anderson's approximate treatment.

IF(,)] ~ = ~ (2.~ ~-3x), 3 = "2500--~ ..... ( 19

A series of corresponding vaIues of x and F(x) calculated from (19), supplemented by (19a) and (19b), is shown in Table 1. The relation required is that between n and 3I, related to x and F(x) by (21) and (18a). It is convenient

]ogl0M----3~t'2432 + 3 log F(x).

(

) 25
The conversion is then readily carried out. Corresponding values of n and M are shown in the last two columns of the table. It is unnecessary to give values for ]og M less than 32, as no stars are known of mass less than a tenth that of the Sun (log M~=33"3010), and in any case, for stars of small mass, the approximate expression (22) will be sufficiently accurate. For any value of 3:[ the maximum value of logn can be found by interpolation, and from that The rehltion between log M and log~ is shown by the curve in the figure. The "positions" of a number o[ stars are indicated. The straight line is the result found by the original approximation, in which the relativity effect was neglected (giving n varying as M~"). The main conclusion is ~hat for stairs for which this equilibrium can occur the limiting density is somewhat less than that calculated by Anderson, and that the "limiting mass" is larger; his results are indicated by the dotted line.

WMte D,'arfs.

The number of stars knowl~ to be of the white dwarf type is sm~dl~ but this does not necessarily indicate that stars of very high density are uncommon. Dense stars of ordinary mass will have a small radius, and so will be faint objects unless they ttre near the Sun (on the stellar scale of nearness) or have a high-surt'ace temperature. "Black dwarfs" (to use Fowler's term) would not be observed. According to Jeans e, four white dwarfs are known with certainty, but for one of these--the companion of the red giant o Ceti--he does not give any quantitative data. Procyon B is a possible white dwarf, but its spectral type is unknown. In the followin~z table a~'e collected the relevant observational data derived from Jeans I". Proeyon B is included because its mass is known. The four columns give the spectral type, the absolut, e visual and bolometrie magnitudes, and the m~lss in terms of the Sun's mass (.M~). From these observational data deductions may be made as to other characteristics of the stars. From the spectral type the surface temperature (T~) may be estimated, and from the effective temperature and the absolute magnitudes the radius of the star. The following formulm are given by Jeans tbr the radius in terms of that of the sun (r/r, denoted by R) :- (29) (The sun's mass is taken as 2"0 × 10 aa grams.)

The mean density is given by

P = ~ = \MJ \ r / 47rr, 3' = (M/M,)(1/R)ap~= 1.42 (M/M~) (1/R)~.. (30)
The values given in Table IIl. have been calculated from these equations. It may be noted ttlat for Sirius B the radius deduced from the relativity shift is "03, ill remarkable agreement with the value "~1312 calculated in the way indicated.

(In general the values are not significant to more than two figures.) Tile available data for Procyon B and Van l~Iaanen's Star are insufficient for the calculation

955

of p or E' ; these stars will be discussed later. The value of M/M, is included in the table, and also the value of the limiting density, p,, calculated as already described.

The calculated limiting densities are considmably greater than those obtained by the simple formula ( H a ) (used in I.), which gives for stars of the mass of Sirius B and o, Eridani B the values p = 2.8 x l o 6 and 7.5 x lo5. The order of magnitude of the difference for stars of different mass will be apparent from the figure.

The calculated radiation per gram (E') of o, Eridani B is about 10 times greater than the value given by Jeans.

This will be discussed later. The mean density for Sirius B is somewhat smaller than that usually given (:rbout 5 x 104).

This results from the use of a slightly larger d u e for the radius. Taking the relativity shift value (K='03), the mean density comes out as 4.5 x lo5.

The mean molecular weight has been taken as 2.5mH. This is the value for completely ionized atoms of lead (El). For atoms of' lower atomic number the value is slightly smaller, and for those of high atomic number, larger. Jeans derives a value 2.6 nl, on the assumption, for which there is strong evidence, that in many stars atoms of the atomic number equal to that of uranium, and even higher, are present. I n condensed stars of the type considered, however, most of these atoms will have been transformed into atoms of an inert type, and the estimate 2.5 ? n H is probably sufficiently accurate. I n any case, a small change will not affect the nun~erical results appreciably.

I t might be thought that the change of mass of the electron would produce a not negligible effect, but i t may readily be shown that the effect will be very small. Let m be the mass of an electron with momentunl p. The molecular weight becomes 2.5 mH + p, where

From (8) the maximum momentum is given by

[ 3hSn'~ l/3 po= \-~1 , so that Po h ~ 3\~/'~ 13 - - - =' [--/ n/, moc moC \ 8~r / ~-1"194 X 10-1°n 1/3 .... (32)
For an electron concentration n = 10 ~ (corresponding to a density greater than 109) tt~e nmximum molecular weight is 2"5 ,~.+~=,~(2"5+'0065).

.

The correction is thus quite negligible for all the densities which come into consideration.

Mean ~ensity.

In the foregoing treatment the star has been idealized as having a uniform density. This seems legitimate when the aim has been to obtain an estimate of the maximum density under gravitational-kinetic equilibrium. In an actual star, however, the density will not be unitbrm, even in the condensed limit. When n is small the energy per unit volume is proportional to ~.~/3 (eq. 10 b), and when n is large, to n ~¢3 (eq. 12a). Since the pressure is proportional to the energy per unit volume, and the density is proportional to n, the following relations hold between the pressure and density :-n << 5"9 x1029 , p<<2"4x106 , p=•ps/3, .

n>>5"9× 1029 , p:>>2"4×10% p=tcp4/3. . (33b)

In an ideal condensed star the distribution will thus be Dolvtropic, *he relation between pressure and density obeying This table certainly indicates that the maximum mean density may be considerably less than the maximum central density, and it might seem that all that is necessary to convert the previously calculated results for limiting densities to mean densities is to divide by the appropriate ratio. There are a number of other factors, however, which have to be taken into consideration. The gravitation'tl energy of a star is given by *

3 M ~ E~=--~ G --- (34)
For uniform density the coefficient is 3/5, the value used (eq. 15 a). For 7=5/3 the factor becomes 6/7, so that for a given M the calculated n would be increased. The whole II. the radius ,nay be calculated (eq. 27), the result being R='00955, taking the surface temperature as 7;',00 corresponding to an F-type star. The mass is unknown, and since the star is not a binary component, there is no method available by which it may be estimated. It is, however, possible to calculate compatible values of the mass (giving the mean density) and of the radiation per gram (E') from the known data. From (29), log E'= --5"72 + 2'22 + log (M./M), E'(M/MD= 3"162 × 10-~.

(35) As will be seen from the above table, the values which Jeans suggests as a pure guess* for M/]V[, and E'--'2 and • 00055--are not compatible with each other. The interesting point which emerges, however, is that unless the nua, erical results of the equilibrium theory are very wide of the mark owing to the essential factors being ignored, a lower limit for the mass of Van Maanen's Star can be roughly estimated. The mean density cannot be greater than the calculated limiting density, and it is probably several times smaller. For Procyon B the spectral type is unknown, though it is probably M or K a. It is, however, of interest to examine how far the star approaches the condensed type if a reasonable value of the surface temperature is assumed. :For assmned temperatures of 3000 (M type) and 4000 (K type) the estimated correction? to convert M~i~. to Mbol. may be applied, and using ( 27), (29), and (30), the following results are obtained :-- OE course nothing definite can be deduced from these result% as in tne case of Vail Maauen's Star. They simply serve to show that unless the surface temperature is considerably greater than 3000, Procyon B does not approach the limiting condensed state, the mean density being much smaller than that of other white dwarfs.

Generation of .Energy.

Some of the more important observed and deduced data for the three white dwarfs about which most is known are collected together, for convenience in comparison, in Table VII. The total energy radiated per second is symbolized by E, the energy per gram by E'.

As was to be expected, the calculated limiting density P0 is considerably greater than the "observed " mean density. The polytrope ratios (Table IV According to Jeans's hypothesis, which, in spite o~ its speculative nature, cerLainly satisfies most of the conditions, the energy generation in stars is to be traced to electronproton annihilation occurring in hyper-uranium atoms as a result of one of the extra-nuclear electrons of tile atom falling into the nucleus. Ill completely ionized stars of the white dwarf type this process could not occur, so that the small radiafioa per gram of these stars is in complete accord° anco with the hypothesis. The radiation wouht be mainly due to the atoms in the outer layers ot~ the star, so that there is not necessarily any simple relation between either the mass or the mean density and the amount el radiation per gram. It, does none the less remain peculiar that o~ Eridani B, which approaches the condensed state, should have a much higher surface temperature than Sirius B, and that it should generate more enecgy per gram. The m~lterial available in this connexion, however, is too slight to permit of any inductive genel:alizations being m~de. The densities of white dwarfs are very much greater than those of other stars, in which the densities are too low and the temperatures too high for the electron gas to approach the condensed state. According to Jeans's theory oR stellar evolution (outlined in I.), the tenanted portions oR the Russell temperatm'e-lumiuosity diagram correspond to stable states iu which the pressures exceed considerably the perfect gaspressures. Thele will be a range of stability for each successive stage oR ionization of the atoms, :rod this~ on the liquid-star hypothesis, corresponds to the "jamming" of the incompletely-ioMzed atoms. The difficulty about this hypothesis is that the size which must be attributed to the atoms is very much greater than that indicated by other evidence. ~ow, with the Fermi statistics the pressure is always greater than that oR a perfect gas, and it is interesting to note that there will be a maximum deviation eorrespon(ling to each stage of ioniz~ltiou. When with increasing temperature there is :m increase i~L the number of electrons through iouizatio,~, there will be a deviation from the perfect gas relatioa between pressure and number of molecules. Thls deviation will decrease as the temperature increases, and will iucrease again when the next stage o[ ionization will increase. The deviations required, however, are much greater than those indicated by the Fermi statistics. This m~y be illustrated by taking a particular case and making the most favourat)le estimates. To ensure stability at the centre of the star, according to Jeans, the pressure must be at least 1"107 times the normal pressure*. The pressure oR electron gas is given by

{ i nh 3 } p = nkT 1 + 16 ×~ (~rmkT) :~' 2 .... ( 36
)
Inserting numerical values = nkT 1+3"62 x 10-17,1,~2 (371 P Taking p as 2"5 nmH, Table VIII. gives the corresponding values oR T, n, and p for the pressure to exceed the normal by 10 per cent.

In stars generally, for a given central temperafure ~he central density is of order oR ten times less than tha~ i~ The pressure is thus about ~ per cent. above normal instead of the required 10 per cent. it would seem, therefore, that unless there are other parts or" the theory requiring modification, a simple application of the Fermi statistics is unable to remove the difficulties, although it does lead to deviations from the perfect gas laws of the required type.

T~tBLE VIII.

Corresponding Values for the Pressure to exceed normal by 10 per cent.

Conclusion.

It is generally agreed that the dense dwarf stars represent an advanced stage of stellar evolution. It is reasonable to suppose that the w]dte dwarf stage, corresponding to high surface temperature, will be a comparatively short one, and since the stars are small, it is probable that most condensed stars are too faint to be observed. From the mass luminosity relation, moreover, even treating this as a purely empirical generalization, the masses of condensed stars will in general be small. There is therefore a strong probability that stars of the type in which the gravitational kinetic equilibrium discussed in this paper is of importance are much more numerous than is suggested by the number of known examples. For these the theory may be said to account in a general way for the order of magnitude of the densities deduced from the observational data.

For more normal stars the application of the Fermi statistics shows that there will be an appreciable deviation from the perfect gas laws for the electron gas. Although the deviations will be of the kind required in Jeans's theory */Sid. p. 104. In a previous paper the conclusion was reached that there was a limiting density for stars in which the atoms were completely ionized, varying as the square of the mass of the star. The limiting st~,te occurs when the decrease in gravitational energy on contraction is equal to the increase in the total kinetic energy of the electron gas. In the treatment the relativity change of mass with velocity was neglected. Some approximate calculations by Anderson indicate the general effect: of this change of mass, which necessitates a modification of the previous conclusions when the mass of the star becomes comparable with that of th~ sun.

In the present paper the effect of tile relativity change of mass is worked out with more rigour for the idealized case for a sphere of uniform density. For spheres of increasing mass the limiting density varies at first as the square of the, mass, and then more rapidly, there being a limiLing mass (2"19 x 10 aa) ab,ve which the gravitational kinetic equilibrium considered will not occur. Tables and curves are given showing the relation between mass and limiting density.

It is shown that the distribution of density in condensed stars will be polytropic, and rough estimates are made of the ratio of the central to the mean density.

The observational and deduced data for known white dwarfs are considered in some detail. The density of Sirius B is well below the limit calculated, while that of o~ Eridani B approaches it. The theory enables a lower limit for the mass and density of Van Maanen's Star to be roughly estimated. The observational, deduced, and calculated data for these three stars, and also for Procyon B, are given in tables.

The deduced data indicate that the generation of energy per gram in known dense stars is not simply related to the mass of the star.

The application of the Fermi statistics to the electron gas in norm:d stars is briefly discussed in connexion with Jeans's theory of stahility. There will be deviations from the perfect gas laws of the type required, but they are abou~ twenty rimes too small to satisfy the stability conditions.

Physics Department, The University, Leeds. December 1929. 3R2

  Stoner's constant-density approximation [14]. The next year, following the publication of Stoner's second paper The Equilibrium of Dense Stars [11],

  "1" E. C. Stoner, Phil. Mug. vii. p. 63 (1929). (This paper will be referred to as I.) W. Anderson, Zeits. fi~r _Phys. lvi. p. 851 (1929).

  . C. Stoner on the in more general calculations. (Perhaps the simplest illustration is that the arithmetic mean is not in general equal to the root mean square. The objection to Anderson's treatment has a similar basis.) It seems more satisfactory to attack the problem of determining the zero-point energy without introducing the idea of mean mass at all. Let • be the kinetic energy of :in elec~rou, p the momentum, and let f~=v/c,

  As long as x is small (i. e.,po<<moc), f(~) may be readily evaluated by integration in series, EK--87r~m"%~ ,v 5 (1--limi~ with (2) for n small. The above expression may be used for po<<moc, corre-8~-(m0% ~ sponding to n<<~-\ h ] ' or, substituting numerical values, n<<5"88 x 10 =9 . For stars of mass "44: and "85 that o[ the Sun, the value of n came ou~ on the simple ~heory as about 1"86 x 10 ~9 and 6'9 x 102~. The approximation given bv series integration is thus inadequate, and it is necessary ~o use the complete expression for the integral f(x) = (1 +v~)~,~v ,d~, __ (l+v~)~av-(1+$1W ~d,J. 0 The second integral is known, and the first may be reduced by noting that ~{V(1 +$12) 112 } = ~(1 + .~]')3/2 3(1 1]2)1/2 + 3qe Downloaded By: [Davis, Edward] At: 18:18 4 January 2011 For m small this may be expanded (noting that the log is equal to giving the result (10) on substitution in (9 a). The complete'expression for the total kinetic energy of the electrons thus becomes EK= 8~-Vm¢¢h~ [~ x(1 +,*~)~,'~(1. + 2x')--~ xa -log {~ + (1 +.~)'/'-'}], . (12) with

From

  potential energy (as shown in I.) with _9"5 m~ as the mean molecular weight of the material of the star, EG= 3GM~(~rn) t'3(2"SmH)l/q5 M 1(12) E~ may be written as ~Ymo4C 5 ,, ~ -, Substituting M/(2"5 m~n) for V, and further substituting for n as above, E $Mmo c2 f~(x)

  ~f~(x)]dxd =F(x) =1"483 x 10-235I 2/a, • (18) M--1"751 x i03~ [F(x)] ~/~ ..... (18a) F(x) is obtained by straightforward differentiation of 1 fl(x), where f,(x) is the bracketed quantity in (12).

  . C. Stone," on t]~e _}Vumerical Results.

  b) This expression gives F(x) correct to 1 per cent. for x>10. It indicates, moreover, the limiting mass for which an * This is in agreement with Anderson's result. In I., as pointed out by Anderson, a slight numerical error was made, and the result was given a$n----2'31 × 10--aTM ~. Correspondingly, the result for the maximum density given as p ---3"85 x 10 ~ (M/Ms) ~, should be p ----3"977 × 106 (M/Ms) 2, with the Sun s mass, M,, taken as 2'0 × 1033,

  . C. Stoner on the to use logarithms (baee 10) when tile following equations are obtained : log10 n =2,9'7690+3 logx, ....

  electron concentration (n) with mass (M) in a sphere of uniform density. (1) Siru~ ]3 ........ log M =33'230 log n--30"748 (2) o 2 Eridani B .... log 3£ =32'944 1ogn=29"524 (3) Procyon B ...... log :M: =3:2'869 log n=29'313 (4) Limiting :~ .... log M.=33"340 (Mo----2'19×1033) :For the limiting density log Po = log n-24+0"618. The straight line corresponds to the formula in which the relativity effect is neglected. The dotted curve gives Anderson's results. the value of n. If the density is required it can be found from (20), while log P0 is given by log po=log n--24:+'6180 ..... (26)

  953

t

  Lee. cir. p. 59. I am indebted to Prof. Eddington for informing me that the value '21 given for the mass of 02 Eridani B in his ' Internal Constitution of the Stars'is based on au earlier determination of the orbit, and that the value "44 is presumably the more trustworthy. Also to Sir James Jeans for confirming" the value "44.

  . C. Stoner on the

  log R= -0'2 Mbol. --2 log T~+ 8"53, . (27a) 5880 logR= --0"2 hL~. + -wi~--0"01 .... (27b) It is further possible to calculate the total radiation of energy in ergs per second (E), log E = --0"4= Mbo~. + 35"52 ..... (28) TABLE III. Deduced Characteristics of Sirius B and o~ Eridani B. ..... "85 8,0U0 '0312 "0065 3"97 X 104 0"3 × 107 o~ Eridani B... "44 11,200 "0187 "0185 9"55×104 1"4×108 The amount of radiation per gram (E/) is a quantity of great interest : log E'=log E--log M, =log E + log (M,/M)--log M,, = -0"4 Mbol. + 2"22 +log (M,/M).

  Equilibrium of' Dense Stavs.

  relahon p=tcp~. '[he dlstrlbutmns in polytropic stars for a number of values of ~/(or its equi~,alent n', defined by V=l+l/n') have been worked out by Emden. The results are quoted by Eddington *. Table IV. gives the ratio of the • A. S. Eddington, ' The Internal Constitution of the Stars,' p. 89 (C. U. P. 1926).

  Equilibrium of _Dense Stars.

  957 maximum (central)to the mean density. Uniform density corresponds to n' = O.

  Equilibrium of Dense Stars.959Froin this it is possible to draw the following conclusions for Van Maanen's Star :-M/Ms > "4, p > 650,000, E' < "0008.

~

  * ft. It. Jeans, ~The UnL'erse Arouad Us,' p. 3]0. Downloaded By: [Davis, Edward] At: 18:18 4 January 2011 http://mc.manuscriptcentral.com/pm-Eq~dlibri~on of ])ense Stars. 961 T/~e Fermi Statistics (~f Electron Gas at Lower .Densities.

  J. H. Jeans, ~ Astronomy and Cosmogony,' p. 141. Phil. Mag. S. 7. Vol. 9. No. 60. M)~j 1930. 3 R . C. S~oner o,~ the the table. For the sun * the central density is about 140, and the temperature can hardly be less than 40 x 106. This gives p = ,~kT(l + 0"48 x 10 -~) ..... (38)

  963o[ stability, they are not nearly large enough. It would seem, therefore, either that the theory as to the conditions for stability requires modification, or that there are factors involved which are at present unrecognized.

)

  Note that this expression gives a value for the mean density of Sirius B in fairly good agreement with the modern value but not in good agreement with the 1930's value. Note also that there is no suggestion of a limiting mass with Fowler's non-relativistic equation (equation 2).

Following the appearance of Stoner's paper, Wilhelm Anderson, a German physicist working at Tartu University in Estonia, noticed that for masses beyond about a solar mass the electron energies become relativistic

[10]

. So Fowler's equation of state, which is derived using Newtonian Mechanics, will not be valid for the more massive stars. Anderson derived a relativistic equation of state and, despite his derivation being not strictly correct, deduced from it that there is a limiting mass for white dwarfs of 0.68 M solar ; with the modern value 2 = e µ this becomes 1.1 M solar . Stoner responded to Anderson in a second paper [11] (reproduced as a facsimile following this commentary) in which he pointed out the error in Anderson's derivation of the relativistic equation of state and derived the correct equation -Stoner's equation is now known as the Stoner-Anderson equation. In the extreme relativistic limit this equation of state tends to the simple form

  solar obtained by numerical integration of the equations of stellar structure. It is interesting to note that Stoner did not write out the expression above for M L explicitly, so

	missed the significant observation that M L is proportional to the Planck mass	  	G hc	  	1	/	2	cubed.
	The Planck mass occurs whenever gravity, quantum mechanics and relativity play an
	essential role; for example see [12].
	We can understand the reason for the existence of a limiting mass using the constant
	energy approximation as follows. With the non-relativistic equation of state [equation (2)],
	as the star shrinks the kinetic energy E K increases faster than the potential energy E G so
	there is always an equilibrium radius. But with the limiting form of the equation of state
	[equation (5)], E K becomes proportional to N 4/3 /R, where R is the radius of the star and N the F r number of electrons. As the gravitational energy E G is proportional to N 2 /R, both E K and E G o increase at the same rate as the star contracts. So if K G E E > the star will collapse and if
	P
	e
	e r
	R
	e
	v i e
	w
	O n l
	y

2 = e µ this gives M L = 1.56 M solar , which should be compared with the more accurate value of 1.44 M

  Fowler (1926) 'On Dense Matter' Monthly Notices of the Royal Astronomical Society 37 114-122. 4. M Zeilik and S A Gregory (1998) Introductory Astronomy and Astrophysics 4th Ed. Saunders Publishing. 5. F W Herschel (1785) Philosophical Transactions of the Royal Society of London 75 40-126. 6. W S Adams (1925) Proceedings of the National Academy of Sciences 7. M Barstow et al. (2005) 'HST Spectroscopy of the Balmer Lines in Sirius B' Monthly Notices of the Royal Astronomical Society 362 1134. 8 A S Eddington (1926) 'The Internal constitution of the Stars' First edition Dover 1959 pp.The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

	o r P e e r o r P e e r 21. M S Longair (2006) 'The Cosmic Century' (footnotes to Chapter 4), Cambridge University F P r i e e v e e R r r o r P e e 3. R H F LXXKV[I. The Eq~dllbri,,,n of l)e~se Stars. By ED~U.~D Press. C. STONEa, Ph.D. (Ca,rib.), Reader in Physics, University of Leeds ~. F o Introduction.
	R R R w
	e v i e w e v i e w e O v i n l e w y
	O n l O n l O n l
	y y y

9. E C Stoner (1929) 'The limiting density in White Dwarf Stars' Philosophical Magazine 7,7: 41 63-70. 10. W Anderson (1929) 'Uber die Grenzdichte der Materie und der Energie' Zeitschrift fur Physik 54 851-856. 11. E C Stoner (1930) 'The Equilibrium of Dense Stars' Philosophical Magazine, 9:60,944-963.

12. D Garfinkle (2009) 'The Planck Mass and the Chandrasekhar limit' American Journal of Physics 77 683-687.

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TABLE I .

 I

	Corresponding values of x, F(x), log10 n, and loglo M.
		n=5"876 X 10~9X 3,
		= 1"751 × 1034 IF(x) y/~.
	x.	F(x).	loglo n.	logl~ 31.
	"1	'0200	26"7690	31"6947
	-2	"0394	27'6720	32"1364
	'3	"0581	28'2003	"3895
	'4	"0756	"5753	"5609
	'6	-1071	29"1036	"7879
	"8	"1330	"4783	"9291
	1"0	"1537	"7690	33'0233
	1'5	"1887	80-2973	"1569
	2	"2085	"(;720	"2220
	3	"2280	31-2003	-2801
	4	"2366	"5753	'3042
	5	'2410	"8660	"3162
	6	"2436	32"1036	'3233
	8	"2463	"4783	"3303
	10	"2476	"7690	"3339
	20	"2494	33'6720	"3385
	40	"2498	34'5753	"3396
	60	"2499	35"1036	"3399
	100	"2500	"7690	"3401

TABLE II .

 II Observational Data.

		Spectral 'l'ype.	Mv[s.	Mbol.	M/Ma"
	~irms B ..................	A7	11"3	11'2	"85
	% Eridani B ............	A0	11"2	10'8	"44
	Procyon B .............	--	16	---	"37
	Van 3h~,~nen's star ......	ZF	14'3	14"3	--
	J. H. Jeans, 'Astronomy and Cosmogony,' p. 63 (Camb. Univ. Press~
	1928).

TABLE IV .

 IV Ratio of 1V~aximum to Mean Density in Polytropes. suggested by the values in TableIV. These results, however, are somewhat uncert'tin, and for the present the calculations are left in the form which is strictly appropriate for the case of the homogeneous sphere.* Eddingtou, lee. cir. p. 87. Van Maanen's Star is of great interest, as it is the smallest star known, being about the same size as the earth. From the data in Table

	958	Dr. E. C. Stoner on the
		Van Jlaanen's Star and _Procyon B.
	............ ~' . ........... po/D ......... calculation, however, is based on the assumption that n is oo 5/3 3/2 7/5 4/3 0 3/2 5/2 3 t 6"00 11'4 24"i 54"4 uniform, aud different averaging processes wouhl have to be carried out in deriving expressions for the gravitational and kinetic energies in terms either of no (the central electron concentration) or the mean n. Some rough calculations indicate that the values derived for the limiting concen-trations approximate to those ibr no; but a much more elaborate investigation would be necessary if the variation o~ density were to be properly taken into account. The indications are that the values obtained for the limiting density approach those for the limiting central density, and that the limiting mass may be :l~g~]iy c .han~tad ~:nh.a~! Sirius B still falls within the range f h c t kinetic equilibrium can occur ; and that the limiting mean densities may be smaller than those calculated approximately o r P e e r R e v i e w O n l in the ratios F y

TABLE V .

 V Compatib]e Values for Van Maanen's Star. TableV. are given compatible values of MIMe, p, and E r, together with the calculated P0 for each value of the mass. Incidentally, this table gives the values of P0 for a range of masses.

		Te=7000.	R='00955.
	M/M e.	E'.	1 O--SO.	10--% 0 talc.
	"2	"0016	"33	"19
	• 4	"0008	"65	1"14
	'6	"0005	"98	3"7
	• 8	"0004	1"30	13"1
	10	"0003	1"63	104
	]n

TABLE VI .

 VI

		Compatible Values for 1)rocyon B.
		M/M~= "37.	P0 ealc. ='84 x 106.
	Te.	M]bol,	ltd.	p.	:E'.
	3000	14:	'056	2"4 × 108	'00 t
	4000	15"5	'018	1" 1 × 10 ~	"0003

  Stoner on the o~ Eridani B. I~ those ratios are approximately correct, o~ Eridani B may be said to approach the limiti,~g condensed state fairly closely, while Sirius B is still far removed from it. If the observational dat't (TableII.) and the essentials el the equilibrimn theory are correct, Van Maanon's Star is certainly the most dense that is known, though it is impossible to say how nearly it approaches the condensed limit.If the observational data are correct, again, there is for stars of this type no simple relation between the mass and luminosity--as is shown by the absolute bolometric magMtude of o~ Eridani B being smaller than that el Sirius B, although its mass is smaller. The radiation per gram, moreover, does not decrease steadily with the mass of the star, as had been suggested *.

	F o r P e e 960 Dr. E. C. TABLn VII.
	r Observed and Deduced Data, for White Dw;,r[s.
	M/M.d. "85 "44 ll,200 T,. 8,000 R I1. 0312 "0 lS7 e 40,000 ,). 95,00q Van ]~:L~nen's Star . :>'4 o 2 E,-idani B ......... 7,003 • 01)95 ),650,000 >I'IX106 ~,,, ('ale. 23X10 s 1"4 X 10 ~	)'.. 1'1× l(/:*~ 0065 E'. 1"6 X 10 ~L -0185 6"4,<100"~) <2"0008
	v
	i e
	w
	O
	n l .) suggest that the maximum mean density may be something approaching 50 times smaller than P0 f0r Sirius B, and 6 times smaller for y

Sirius :B ...............

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Acknowledgements

should like to thank Edward Davis, Richard Jameson, Malcolm Longair, Peter Maksym and Derek Raine for reading the manuscript and offering helpful suggestions.