Edmund C. Stoner and the discovery of the
maximum mass of white dwarfs
Michael
Nauenberg
Department of Physics
University of California, Santa Cruz,
CA 95064
The
existence of a mass limit for white dwarfs is usually attributed solely to the
late astrophysicist Subrahmanyan
Chandrasekhar, and this limit is named after him[1]. But as is often the case, the
history of this discovery is more nuanced. In this note I will show that the
existence of a maximum mass was first established by Edmund C. Stoner, a
physicist who began experimental research
under the supervision of Rutherford at the Cavendish in Cambridge, but
later switched to theoretical work. Rutherford recommended Stoner to a position at the Physics department of the
University of Leeds where he spent his entire career[2]. According to G. Cantor, he was
Òprobably the leading Cavendish-trained theoretical physicist of
the 1920's ''[3], although he learned theory mostly on his own, and became known for his work on magnetism[4].
Unfortunately, Stoner suffered
from diabetes and poor health which restricted his travels, and this may account for the fact that he
did not receive wider recognition for his achievements.
In
1924 Stoner wrote a paper on the
distribution of electrons among atomic levels[5]. In the preface of the fourth edition of his classic book,
ÒAtomic Structure and Spectral LinesÓ,
Arnold Sommerfeld gave special mention to Ò einen grossen Fortschritt [a
great advancement]Ó brought about
by StonerÕs analysis, which then
came to the attention of Wolfgang Pauli, and played and important role
in his formulation of the
exclusion principle in quantum physics [6]
. Therefore, it is not surprising that
StonerÕs interest in white dwarfs
was aroused by Ralph H. Fowler's suggestion[7] [8] that this exclusion
principle could be applied to solve a major puzzle, the origin of the
extreme high density of white dwarfs [9]
[10],
which could not be explained by classical physics. Eddington expressed this puzzle as follows:
``
I do not see how a star which has once got into this compressed state is ever
going to go out of it... The star will need energy in order to coolÉIt would seem that the star will be in an awkward
predicament when its supply of subatomic energy fails. Imagine a body
continually losing heat but with insufficient energy to grow cold ! '' [11].
At the
time, the conventional wisdom was
that the source of internal pressure
which maintained all stars
in equilibrium against
gravitational collapse was the internal pressure of the matter composing the star which had
been heated into a gas presumably, according to Eddington, by Òsubatomic energyÕÕ. But when this supply of energy is
exhausted and the star cools, Fowler
proposed that a new equilibrium
would ensue, even at zero temperature, due to the ÒdegeneracyÓ pressure of the electrons caused by the
exclusion principle in quantum mechanics. Fowler, however, did not attempt to
determine the equilibrium properties of such a star which he regarded as
Òstrictly analogous to one giant molecule in the ground state''. Apparently he
was unaware that at the time,
Llewellyn H. Thomas had developed a mathematical method to solve this problem in atomic physics[12]. Subsequently, Stoner applied the minimum energy principle to
obtain the equilibrium properties of such dense stars[13] in a constant density approximation,
by substituting for the internal
energy density Fowler's
non-relativistic equation of state
for a degenerate electron gas [14].
In particular, he found that the density increases with the square of the mass
of the star[15]. In such a gas
the mean momentum of an electron
is proportional to the cube root of the density , and Wilhem Anderson, a
Privatdozent at Tartu University, Estonia, who had read StonerÕs paper, noticed that for the mass of
a white dwarf comparable to
or higher than the mass of
the Sun, the density calculated
from StonerÕs non-relativistic
mass-density relation implied that the electrons become relativistic [16]. Hence, Anderson concluded that in this regime, this
relation gave Ògršblich
falschen Resultaten [gross false results]Ó for the properties of a white dwarf. He attemped to extend the equation of
state of a degenerate electron gas
to the relativistic domain, but he gave an incorrect formulation which,
fortuitously, indicated that StonerÕs minimum energy principle implied a
maximum value for the white dwarf mass. Alerted by AndersonÕs paper, Stoner
then derived the correct relativistic
equation of state, and re-calculated, in a constant density approximation, the
properties of white dwarfs for arbitrary densities[17] [18].
Thus, he obtained, now on solid theoretical grounds, the surprising result that when the density approaches
infinity, the mass of the star reaches a maximum value.
Two years
after the appearance of the first paper by Stoner on the Ò limiting density of
white dwarfsÓ [19] Chandrasekhar published a paper [20] with a similar title Òarriving at the order of magnitude of
the density of white stars from different considerationsÓ, which was
communicated by Fowler to the Philosophical Magazine . Since the non-relativistic pressure - density
relation for a degenerate electron gas is a power law with exponent 5/3, Chandrasekhar
realized - having read EddingtonÕs book ÒThe Internal Constitution of the
StarsÓ [21]
, which he had obtained as an
essay prize - that the solution of the hydrostatic equation for gravitational
equilibrium appropriate to a low mass white dwarf was the Lande-Emde polytropic
solution with index n=3/2. This
solution leads to the same mass – density relation previously found by
Stoner in the uniform density approximation, but with a proportionality coefficient
smaller by a factor about two . Meanwhile, Stoner, in collaboration with Frank Tyler, had calculated the minimum energy of a white dwarf assuming a density distribution
corresponding to the n=3/2
polytrope [22]
obtaining the same result as Chandrasekhar , and somewhat earlier Edward A. Milne also
had carried out this calculation [23]. In his paper, Chandrasekhar ignored Òrelativistic-mass correctionsÓ,
because he did not yet know how to incorporate them,
while Stoner already had shown, as an example, that
for the white dwarf companion of Sirius these corrections gave a density almost an order of magnitude larger
than the non-relativistic calculation. In Òsome historical
notes[24], Chandrasekhar recollects that he had
found that the degenerate electrons become relativistic [25]
for white dwarfs with masses which are comparable or larger than the mass of
the Sun. His calculation in the extreme relativistic limit appeared separately
in a very short paper
(two pages long) on Ò the maximum mass of ideal white dwarfs [26]. Again Chandrasekhar was able to obtain his result with great
ease, because the relevant solution of the differential equation for
gravitational equilibrium for the extreme relativistic equation of state of a
degenerate electron, which has an exponent 4/3, corresponds to the n=3 Lane-Emde
polytropic solution, which also
appears in Eddington's book 11
. It turns out that in this case the mass is independent of the
central or mean density of the star. Chandrasekhar acknowledged that his result
was in surprising Òagreement'' with Stoner's result , but he also claimed,
without giving any proof, that it was a maximum mass for a white
dwarf. Later, in an interview with
Spencer Weart [27], Chandrasekhar acknowledged that
ÒÉat first I didnÕt understand what
this limit meant and I didnÕt know how it would end [28],
and how it related to the 3/2 low mass polytropes. But all that I did when I was in England and wrote my second
paper on itÓ.
But a proof that the critical mass is a
maximum mass already had been given
in the uniform density approximation by Stoner, who also had shown analytically
that the mass of a white dwarf is a monotonically increasing
function of the density which is finite at infinite density, while it took
Chandrasekhar several additional months before he found a rough argument to show that at the critical mass the
density becomes infinite[29]. His awareness of StonerÕs analysis,
however, was left unmentioned, although it is clear that
it must have given him some
confidence in the validity of his result.
At about this time,
the physicist Lev D. Landau, who had recently finished his studies in
Leningrad, was visiting the ETH in
Zurich where Rudolf Peierls was PauliÕs Assistent [30].
During this visit, Landau worked with Peierls on relativistic quantum field theory, and he developed the quantum theory of
diamagnetism associated with a degenerate electron gas in a metal. Thus, it is
not surprising that he should also consider the role of quantum degeneracy of
an electron gas, including the
implications of special relativity, for the properties of stars. He was motivated by the work on stellar
structure of Milne, whom he
criticized for Òmaking physical assumptions only for the sake of mathematical
convenienceÓ[31]. Like Stoner, Landau recognized that
the equilibrium state of dense stars is
a minimum of the energy. By
applying this principle to the extreme relativistic equation of state, he found
that the total energy 
, where 
is the mean
density and 
is a constant
which depends on the mass of the star. Hence, depending on the sign of 
, the star would
either Òexpand or collapse to a pointÓ to attain the minimum value of 
. The criterion
separating these two regimes corresponds to 
, which in the uniform density approximation leads to StonerÕs solution for the
critical mass. Instead, Landau
solved this problem by considering the equation for the chemical
potential , which in this case corresponds to the maximum energy or
Fermi energy of an electron in a
degenerate electron gas. The resulting differential equation is analogous to the Thomas-Fermi equation [32],
which in the relativistic regime is equal to the Lane-Emde n=3 polytropic equation. Thus, starting from the same principles enunciated earlier
by Stoner, but solving the resulting equations without making StonerÕs uniform density approximation, Landau
obtained the exact value for the critical mass [33], but with a different numerical value
than that given by Chandrasekhar[34]. Since there is some confusion in the literature concerning the dates [35] associated with LandauÕs work, it should be pointed point out that
Landau submitted his paper on dense stars for publication five months before the appearance of ChandrasekharÕs
first paper [36] on this
subject, and therefore it is
very unlikely that he would have
been aware of ChandrasekharÕs work [37]
Stoner's fully
relativistic analytic solution, in the uniform density approximation [38],
for the mass-radius dependence of the dense stars is shown graphically in Fig.
1. His result is compared with ten
numerical calculations, shown by circles, which Chandrasekhar
obtained five years
later by integrating numerically the differential equations of gravitational
equilibrium with Stoner's relativistic pressure-density equation of state[39].
Fig. 1 The dark line is a plot of the
scaled radius, 
vs. scaled mass, 
of Stoner's 1930 analytic solution in the uniform density approximation. The circles are the
solutions published in 1935 by Chandrasekhar, who numerically integrated the equations of gravitational
equilibrium using Stoner's pressure-density relativistic equation of state. The
mass is given in units of the critical mass 
,
and the radius in units of a length 
for which 
in the non-relativistic limit, 
.
The dashed line is the non-relativistic solution 
.
This remarkable agreement is
surprising, because Stoner's result was based on the uniform density
approximation, while ChandrasekharÕs was obtained by integrating the equations
of gravitational equilibrium. The main difference is in the scales of mass and
of length, e.g. Chandrasekhar's critical mass 
is 20 % smaller that Stoner's. Before
1935, following ideas of Milne [40]
, Chandrasekhar had developed only a crude composite model for a white dwarf [41]
in which the non-relativistic approximation was assume to be valid for
increasing mass until the central pressure became equal to the pressure given
by the extreme relativistic equation at the same density. For a larger mass, he
applied this relativistic equation to a central region of the star, and the
non-relativistic equation for an external region of the star bounded by a
surface defined when these two equations gave the same pressure at equal
densities.
Stoner was encouraged by Arthur S.
Eddington, regarded as Òthe most distinguished astrophysicist of his
timeÓ [42], to pursue the implication of his relativistic equation of
state on the maximum density and temperature of white dwarfs, and he
communicated Stoner's two papers on this subject to the Monthly Notices of the
Royal Astronomical Society[43][44] [45].
Eddington's 1932 correspondence with Stoner (see Appendix and Fig. 2) deepens
further the mystery why several years later, in a well known public attack [46][47]on Chandrasekhar's similar work on
white dwarfs [48], Eddington unexpectedly rejected the
relativistic equation of state, and the profound implications of the existence
of a white dwarf mass limit for the fate of stars with masses exceeding this limit[49]. Apparently Eddington had found that
relativistic degeneracy was
incompatible with his fundamental theory, and later confessed to
Chandrasekhar that he would have to abandon this theory if relativitivistic
degeneracy was valid[50]
. Eddington's criticisms [51]
were entirely unfounded[52] but his enormous prestige led to the
acceptance of his views by many in
the astronomical community, and to an early rejection of Chandrasekhar's work.
After Eddington questioned the validity of the relativistic equation of state
for a degenerate electron gas, Chandrasekhar went for support to several of the
great pioneers of the modern quantum theory, including Dirac who was in
Cambridge, and to Bohr and
Rosenfeld who he had met during a visit at Bohr's Institute in
Copenhagen. They assured him of the validity of the relativistic equation of
state [53],
and advised him to ignore Eddington's objections [54],
but Chandrasekhar continued relentlessly to pursue this matter, writing a paper
with Christian M0ller on
relativistic degeneracy[55],
and persuading Rudolf Peierls to give another proof [56]
of its validity. During this controversy, however, Chandrasekhar apparently did
not mention Stoner and his earlier derivation of this equation, which is
neither referenced in his paper
with M0ller nor in the
paper by Peierls. In an appendix to the first paper [57]
in which he applied StonerÕs equation,
Chandrasekhar claimed to offer a Òsimpler derivationÓ of it, but it turned out to be essentially the same one given by Stoner. Here Chandrasekhar gave an
acknowlegdment to Stoner with the remark that Ò this equation has been derived by Stoner (among others)Ó,
but the ÒothersÓ remain unidentified,
and in reality they donÕt
exist. He also mentioned Ò that Stoner had previously made some calculations concerning the (p,
)
relations for a degenerate gasÓ,
neglecting to give reference to a paper by Stoner [58]
where a derivation of this pressure-density relation and his numerical tables appeared. For several more years Stoner continued
to work on the equation of state for finite temperatures, publishing extensive
tables of Fermi-Dirac functions [59]
which later turned out to be also very useful for improved calculations of the properties of white
dwarfs [60]. During his controversy with Eddington,
Chandrasekhar also did not mentioned LandauÕs independent derivation in 1931 of the critical mass
of dense stars, although by then he had met Landau during his 1934 visit to Russia where he had presented his
work. However. Nevertheless,
in his Òhistorical notesÓ [61],
Chandrasekhar complained Òthe
tendency in some current literatureÓ to give Landau priority in this discovery,
and he never gave reference to LandauÕs work.
Later on, in his 1939
book [62]
on stellar structure where he
reproduced his work on
white dwarfs, Chandrasekhar mentioned that the Òequation for the internal energy of an electron gasÓ [63]
was derived by E. C. Stoner, but again he neglected to refer to StonerÕs
explicitly derivation of the pressure-density relation, and his numerical tables for such a gas, although in 1934 he had to reproduce these tables with
higher accuracy, because these
tables were essential for his numerical integrations of the
differential equations for gravitational equilibrium [64]. He stated that Ò the existence of this
limiting mass was first isolated
by Chandrasekhar , though its existence had been made apparent from earlier
considerations by Anderson and Stoner ÉÓ [65].
One is left wondering, however, what he meant by this assertion [66]
, because it was Stoner and not Chandrasekhar who first Òisolated'' the
limiting mass by giving explicitly the
dependence of this mass on natural constants [67].
In some ÒBiographical Notes" [68]
in his book, Chandrasekhar gives a reference to two of StonerÕs five papers on the
properties of white dwarfs [69], but merely comments that in these
papers ÒStoner makes some further
applications of Fowler's ideas'' [70], not giving the reader any idea of the
important concepts and results regarding the properties of white dwarfs
contained in these seminal papers. By such obfuscation, Chandrasekhar gave rise to the current neglect of
Stoner's work.
In Kamesh Wali's
excellent biography of Chandrasekhar [71],
Stoner, is not mentioned even once, and his name also does not appear in
Spencer Weart's transcript [72]
of his lengthy interview with Chandrasekhar in 1977. More recently, in his book
ÒThe Empire of Stars'' Arthur Miller remarks that Òit was indeed extraordinary that a nineteen-year-old Indian
youth [Chandrasekhar] had managed to make a discovery that had eluded the great
minds of European astrophysics'' [73]
. Although Miller briefly refers
to Anderson and to Stoner, he claimed that they Òhad never examined the
ramifications'' [74] of the
relativistic equation of state. But
as we have shown here, with respect to Stoner MillerÕs claim is incorrect. Miller also did not mention here that Landau discovered the limiting mass when he was only
twenty three years old.
According to Chandrasekhar's account of his discovery, which he
repeated on numerous occasions [75], both Fowler and Milne were at first not
interested in this result , and five years later Eddington publicly ridiculed
him for engaging in Òstellar buffoonery" [76]
. This episode has become one of the best known legends in astronomy, and has
been told to generations of students in this field. They have been given,
however, only a partial historical account, because StonerÕs important role has
always been passed over in silence.
Actually, the early reception of the discovery of the limiting mass also
appears to have been more nuanced. When Chandrasekhar arrived in Cambridge and
mentioned his discovery to Fowler, in effect Fowler responded that he had been
scooped by Stoner [77].
Likewise, from references in a paper by Milne [78],
it is clear that Milne also was
aware of Stoner' s work, because he applied it to his own theory of
stellar interiors, without, however, examining the implications of relativity.
Therefore Fowler and Milne's supposed lack of interest in Chandrasekhar's
account of the limiting mass may partly have been due to the fact that they did
not considered it to be a novel discovery. Moreover, early on both Milne and
Eddington encouraged Chandrasekhar to do further research on the white dwarf
problem, while at the same time, Eddington also encouraged Stoner to work on
this problem. Surprisingly, Eddington even offered to collaborate with Stoner
(see Appendix II) , who was in Leeds, rather than with Chandrasekhar, who was
at his own institute in Cambridge. Evidently, Eddington recognized that Stoner could apply the fully relativistic equation of
state for a degenerate electron gas at arbitrary densities,
while at the time Chandrasekhar
could consider only the non-relativistic (low density) and extreme relativistic
(infinity density) limits. This prevented Chandrasekhar from carrying out a
complete analysis of the properties of white dwarfs [79]until
five years after Stoner had done a comparable analyis in the uniform density
approximation.
There is no evidence in
his writings that Chandrasekhar understood the relationship between his
mathematical approach which was
based on EddigntonÕs hydrostatic equation for gravitational equilibrium [80], and Stoner's minimum energy principle [81],
although already in 1931 this relationship had been elucidated by LandauÕs
independent work [82] [83]
In 1983 Chandrasekhar was awarded the Nobel prize, but in his acceptance
speech, which mainly consisted of a historical review of his work on white dwarfs,
he did not include a single reference to either Stoner or Landau. This general neglect of Stoner's
seminal work on white dwarfs helps explain why, with a few notable exceptions [84]
[85]
[86], Stoner's contributions and his priority
in the discovery of the maximum mass of white dwarfs have been forgotten now.
Appendix
Eddington's Feb. 28, 1932 letter to Stoner
In light of
Eddington's famous controversy with Chandrasekhar at a 1935 meeting of the Royal Astronomical
Society in which Eddignton quipped, without giving any reference to Stoner, that the relativistic
equation of state for a degenerate electron gas
Ò...is based on a combination of
relativity mechanics and non-relativity quantum theory, and I do not regard the
offspring of such a union as born in lawful wedlock ...Ó [87]
it is
remarkable that three years earlier Eddington had been in communication with
Stoner about this equation of state, encouraging Stoner in his work, and even suggesting that they collaborate on an
investigation of the effect of this equation on stellar structure. In a letter
to Stoner on Feb. 28, 1932 ( see Fig. 2), Eddington wrote:
ÒI have been thinking that a combination
of your work and mine would make quite definite the state of the question as to
upper limits to the temperature and density of a star of given mass. This is
very important, e.g. in regard to theories of subatomic energy and does not
seem to be as well understood by astronomers as it might be...Ó
Then he added that
ÒI suggest that it would be very useful
to tabulate 
[Stoner's relativistic equation for the
pressure 
as a function of the density 
]
or 
,
others who have written on the subject seem to consider only the two extremes
of ordinary [
]
and relativistic degeneracy [
],
whereas we are actually most concerned with intermediary conditions Ò
By ÒothersÓ Eddington evidently was referring here
to the work of Milne [88]
and of Chandrasekhar [89]
who, at the time, had been taking
into account such Òintermediary condiitionsÓ by a crude interpolation
scheme between two density regimes where either the non-relativistic or the
extreme relativistic pressure-density relations were assumed to be applicable[90].
Eddington continued:
Ò
While the critical mass may have some interest of its own, it does not affect
the more fundamental questions. It is useless to suggest a theory of subatomic
energy involving temperatures of 
degrees which might be possible for
Sirius but could not possibly apply to Krueger 60.
We
have been fairly generous in upper limits, so that (especially if there is
abundance of hydrogen) the critical mass is probably much greater than the
sun'sÓ
Evidently, at the time
Eddington's primary interest was the applications of Stoner's relativistic
equation of state to find limits on the temperatures required for the
production of subatomic energy in stars. The passage of his letter quoted here
reveals that in 1932 Eddington did not have objections to Stoner's relativistic
equation of state for a degenerate electron gas, which together with Stoner's
minimum energy principle implied the existence of a critical mass. Moreover, he
understood that the magnitude of this critical mass depended on the inverse
square of the molecular weight 
, which had generally been assumed to be
equal to 2.5. Hence, one can understand his remarks that for a hydrogen star,
the critical mass would Òprobably be much greater than the sun's'', because in
this case 
,
and the critical mass would be about nine times larger than the mass of the
sun.
Stoner followed Eddington's
suggestions by publishing additional numerical tables of his relativistic
equation of state [91],
and by calculating the maximum density and temperature of dense stars in the
the uniform density approximation for arbitrary densities and for the polytropic
density distribution in the non-relativistic and extreme relativist limits [92].
In the last of his five papers on white dwarfs , Stoner took into account the
effect of radiation pressure on the equilibrium state of white dwarfs. In the
introduction he reviewed his previous work:
ÒThe question of limiting densities in
connection with white dwarf stars has already been discussed in a series of
papers. In the first of these ( reference 10)- the relativity effect being
considered in the second (reference 14) - the case of a sphere of uniform
density was considered. The results may be considered as giving rough upper
limits for the mean density. In the third paper (reference 16) the
effect of non-uniform (polytropic) density distribution was discussed, some of
the conclusions being similar to those reached by Chandrasekhar (reference 15)
at about the same time.Ó
Stoner had applied an inequality, which
had been published earlier by Eddington[93], to obtain the maximum possible value
of the density and the temperature of a star under the assumption that the
central pressure was the sum of the pressure due to a degenerate electron gas
and the pressure of radiation [94],
finding that
Ò... the maximum values [of density and
temperature] can be fixed by these considerations provided that the star has a
mass below a critical value Ò,
namely, the
mass limit which Stoner had obtained previously in the absence of radiation.
Fig. 2 Feb. 28, 1932 letter from Eddington to Stoner
encouraging Stoner to apply his relativistic equation of state to obtain upper
limits to the density and temperature of dense stars of a given mass. In
EddingtonÕs figure the dashed
curves are plots of pressure vs. density to the power 
curves for
different star masses 
, 
, which he obtained under the assumption that the ratio of
radiation and gas pressure inside a star is constant 11.
The solid curve is a sketch of StonerÕs
relativistic pressure-density relation for a degenerate gas. ( Courtesy
of the Trinity College library in Cambridge, England, which holds the copyright
to this letter, and the University of Leeds library, where this letter is
located under a
POSTSCRIPT
After the completion of this article, I had the opportunity in Nov. 2007 to examine some of StonerÕs correspondence at the University of Leeds Brotherton library, where I found some additional letters which bear directly on the subject of my article, and support its conclusions. In particular, there are two letters from Milne to Stoner, and the draft of three letters from Stoner to Milne trying to clarify the relation between their different approaches to the theory of white dwarfs. MilneÕs approach to white dwarfs was based on the prevalent astrophysical method for stellar structure - the solution of the hydrostatic differential equation for gravitational equilibrium which later was applied also by Chandrasekhar. The Milne -Stoner correspondence illustrates the difficulty that astrophysicists had in understanding StonerÕs alternative derivation of the properties of white dwarfs, which was based on his novel application of the minimum energy principle. In addition, the Stoner papers contain two previously unknown letters from Chandrasekhar to Stoner, in one of which he acknowledges that StonerÕs papers Òare of great value to meÓ ( see Fig. 3). Unfortunately a copy of StonerÕs response to these letters does not appear either among his papers at the University of Leeds, or among ChandrasekharÕs papers at the Regenstein library of the University of Chicago
Milne published a long article in the November 1930 issue of the Philosopical Magazine entitled Ò The Analyses of Stellar StructureÓ (see ref. ??) in which he discussed the configuration of the core of very low luminosity stars. Following previous work of Fowler , Anderson and Stoner, Milne assumed that this core consisted of a gas in a Òdegenerate state (ref. ?? p. 30) , and in reference to StonerÕs two papers he remarked in a footnote that
Ò StonerÕs investigation of white dwarfs assumes that the only source of energy is gravitational contraction. As soon as the possibility of an internal supply of subatomic energy is admitted, his Òmaximum densityÓ condition is invalidatedÓ
This footnote elicited a response from Stoner who drafted a letter to Milne, dated
January 23 , 1931, attempting to clarify his derivation of a maximum density condition for a white dwarf:
Ò May I comment
on the relation between your results and mine. It is of course
true, as you state in the footnote (p.30) [ ref. ??] that an internal supply of subatomic energy
invalidates my equilibrium condition, but it should be equivalent to
your result for L=0 (
)Ò
Here L is the
luminosity and 
is the ratio of the degenerate non-relativistic
electron gas pressure to the total internal pressure which includes the pressure due to electromagnetic radiation. Under the assumption that
this ratio is a constant
throughout the star, Milne had
applied the Emde-Lande solution to
the equation of hydrostatic equilibrium to obtained a relation between the central density and the mass of the
white dwarf. When radiation is
neglected, 
, and Stoner pointed out that after taking
into account the non-uniform density distribution for the non-relativistic pressure density
relation 
,
MilneÕs relation was the same
as his own.
But a dispute between Stoner and Milne ensued after Milne replied to him on January 24, 1931. This dispute illustrates the problem that astrophysicists had in understanding StonerÕs approach:
Ò It was very kind of you to write in such a friendly way, and I appreciate both that and your sending me your papers. I studied your papers on white dwarfs [refs. ??} with great care, and only made my reference to them in my last paper after re-reading them.
I noticed of
course that dimensionally our limiting
formulas [for the mass-density relation of a non-relativistic degenerate
white dwarfs] are identical. Any
physical principle would I suppose yield 
proportional to 
.
I had not notice the very remarkable agreement between your and my numerical constants. But I tried in vain to reconcile the principle behind your formula with that
behind mine [my italics]Ó.
On p. 66 of your Jan. 1929 Phil. Mag. Paper you say ÒÉ
decrease in the volume occupied by the electrons necessitates an increase in
the average momentum and if the gravitational energy is insufficient for this,
further contraction will be impossibleÓ. Therefore, according to your principle
if another source of energy is present which could provide the energy necessary
to accomplish further contraction should be possible, i. e., if L
0,
greater densities that your 
should be possible. My analysis shows,
however, (on its hypothesis) that when L 
0
the density much increase with decreasing L but cannot exceed a value which, as
you say, coincides with your 
.
It is a rather delicate but fundamental difference. Your principle does not
forbid the existence of densities greater than you 
when L 
0,
although it forbids the reaching of this stage when L=0 by a process of
contraction; but again even when L=0 it does not forbid a 
>
your 
when L=0 – configurations of density greater than your 
could exist exist when L=0 although the dwarfs could not obtain them by actual contraction. A
configuration actually constructed with density 
could survive in the state at L=0 as you
principle goes Ó
Milne continued at some length criticizing StonerÕs approach, and then went on to complain that
Ò Astrophysics suffers from a plethora of assumptions as it is . To import an additional principle [ StonerÕs minimum energy principle] when the principles of equilibrium alone should suffice to disclose all possible configurations of equilibrium seems to me to be complicating the subject in a retrograde wayÓ.
But MilneÕs arguments were incorrect, as Stoner explained to him at some length in his two response letters. Like his contemporaries and even modern astrophysicists [95], Milne revealed that he did not understand the connection between StonerÕs minimum energy principle, and the equation of hydrostatic gravitational equilibrium which astrophysics generally have applied to solve problems of stellar equilibrium. On January 26, 1931 Stoner replied that
Ò ÉIf another source of energy is present it might appear that my condition is incorrect.
But the presence of another source of energy (at least of the types considered) necessitates to involve T>0. The gas is no longer at absolute zero and the pressure is greater by the addition of radiation pressureÉany situation giving rise to T>0 and the density will, as you find, be lessÓ
Milne remained unconvinced, and replied on Jan. 28, 1931,
Ò Dear Stoner,
I am very interested in getting to the bottom of our respective poin to view. So I will venture just to analyze your argument a stage further, as it appears to meÓ
Then Milne argues again that according to his application of StonerÕs minimum energy principle, an additional amount of energy added to a white dwarf leads to a further contraction and, therefore, to a higher density while his own method leads to the opposite conclusion. He concludes that
Ò My whole point; that other process might evade your conclusion and that therefore the conditions of your equilibrium method establishes something beyond what your principle does.Ó
In a Òrough draftÓ of a letter to Milne dated Feb 2 1931, Stoner responds Ò I will consider the case you mention – proton electron neutralization energy used up- to increase the kinetic energy Ò, and shows by and application of his minimum energy principle that the Ò star would expand and not contractÓ, concluding that
Ò so it seems to me that the apparent difficult you mentioned is overcomedÓ [96] Thus, Stoner ended his arguments with Milne, remarking that
Ò I do hope
I have made it clear that É
what matters is not that my way of getting at it is better than yours
– it certainly isnÕt - but
simply that the physical principles used were equivalent for the particular case; that the fact [that
the] same result was obtain for the 
distribution was not fortuitous.Ó
Remarkably, throughout this fascinating exchange, Milne never mentioned StonerÕs application of special relativity to obtain the fully relativistic equation of state for a degenerate electron gas, the important modification that this equation implied on the maximum density of a white dwarf, and StonerÕs discovery hat such stars had a limiting mass. Indeed, relativistic effects were also not mentioned in MilneÕs 1930 article.
In his correspondence with Milne, Stoner made an interesting comment which revealed his attitude towards his astrophysical research. In the draft of his January 26, 1931 letter to Milne, he concluded
Ò Of course the question is a small one , but I would not like you to regard my brief [crossed out] mild excursion into theoretical astrophysics with too much scornÉ I find that with a certain amount of teaching to do, a self imposed duty of keeping abreast in magnetism, I can [not] devote nearly as much time to A. P [astrophysics] as I would like – and the white dwarf business must be regarded rather as a `digressionÕ Ò
On March 31, 1931 Chandrasekhar wrote to Stoner requesting copies of his two first papers on white dwarfs affirming that these papers were of Ògreat valueÓ to him (see Fig. 3). Moreover, only two weeks
later Chandrasekhar acknowledged a subsequent paper by Stoner and Tyler (see ref. ??) , notifying Stoner that Ò I also considered the density of white dwarfs from the polytropic point of view. My results are in complete agreement [my italics] with yoursÉÓ But in his published papers Chandrasekhar never referred to this paper which was thus consign to oblivion. In this letter, Chandrasekhar also gave Stoner
a preview of his paper on Ò how the consideration of the relativistic effect modifies
the analysisÓ which later appeared in the Monthly Notices (see ref. ??). Unfortunately, a copy of StonerÕs response to these letters, does not appear either among his papers, or in the archives of the University of Chicago which contains the correspondence of Chandrasekhar. This is strange because the Stoner archives reveal that Stoner was meticulous in responding to letters send to him, and the second letter from Chandrasekhar would have been of particular interest to him.
On August 19, 1929, Chandrasekhar had written a letter to his father saying that
Ò As far my paper I had nearly completed, writing it out, a paper by a German, Wilhelm Anderson, appeared discussing the same problem. Even mathematically this treatment was identically to mine. So the satisfaction is that I was able to do it independently. I do not intend sending it for publication.Ó [97]. Earlier that year, Anderson had published several articles in the Zeitschrift fur Physik , but Chandrasekhar did not mention which
one he was referring to. Given his interest in relativity and quantum mechanics, and his
work on Fermi-Dirac statistics, however, it is most likely that it was AndersonÕs paper in the June 1929 issue of Zeit. f. Physik (ref. ??) on the effect of special relativity on
these statistics that caught his attention. Furthermore, a few months later
Chandrasekhar came across FowlerÕs
1926 paper arguing that
Fermi-Dirac statistic gave
rise to the internal pressure of a
degenerate electrons gas which supported a white dwarf
against gravitational collapse. In this paper, Anderson argued
that ordinary matter and radiation were different ÒphasesÓ of the same
fundamental substance, and showed
that, as a consequence of special relativity, Ò when the pressure
increases the difference
between a gas consisting of ordinary matter and of light quanta (black-body radiation) always becomes lessÓ. AndersonÕs derivation showed that in the extreme
relativistic limit, the pressure of a
degenerate gas of atoms was proportional to 
,
but his derivation was flawed.
Therefore, it is surprising that Chandrasekhar wrote that AndersonÕs faulty treatment was
equivalent to Òhis ownÓ which he had obtained ÒindependentlyÓ. Further evidence that this paper was the one Chandrasekhar had in mind when writing to his father is that two
months letter another paper by Anderson appeared where he
mentioned that a paper by Stoner had just appeared (ref.
??) on the mass
dependence of the maximum density
of stars which ignored the effect of special relativity. Already then Anderson
commented that Ò for stars of masses comparable to the mass of the Sun, StonerÕs
[non-relativistic] formula leads to Ògross false resultsÓ (see Introduction). Anderson also promised to discuss StonerÕs article in a later
publication, and his critique was
published in the issue of Zeit. f. Physik
which appeared on August 12 ,1929.
Hence, this would not have been available to Chandrasekhar at the time he wrote to his father, but undoubdetly he found have read it later on.
This account reveals that already by the Fall of 1929, Chandrasekhar had become aware through AndersonÕs work, that the non-relativistic pressure density relation for a degenerate electron gas had to be modified to include the effects of special relativity, just like Stoner had acknowledge earlier. There cannot be much doubt that Chandrasekhar, who regularly read the Zeit. f. Physics , could not have missed AndersonÕs articles which appeared well over a year before he left on his fateful voyage to England, and become also aware of StonerÕs early work on white dwarfs and his failure to take into account the effects of special relativity. Hence, ChandrasekharÕs famous story ( ref. ??) that it was during his August 1930 voyage from India to England when he first considered these effects cannot be considered to be valid. Moreover, after he arrived on August 19, 1930, he had to wait another month in a half before he met with Fowler who had been on vacation. During this intervening period it is hard to believe that the would still not have looked up StonerÕs paper, but in his historical notes (see ref. ) that is what he claimed when he said that these papers were first called to his attention by Fowler.
Acknowledgements
I would like to thank
Werner Israel for useful comments and for information about W. Anderson, and Malcolm MacGregor for helpful
editorial comments.
