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PHYSICS ESSAYS 32, 3 (2019)
Eddington’s mass-luminosity relation and the laws of thermodynamics
Stephen J. Crothers1,a) and Pierre-Marie Robitaille2,b)
1PO Box 1546, Sunshine Plaza 4558, Queensland, Australia 2Department of Radiology and Chemical Physics Program, The Ohio State University, Columbus, Ohio 43210, USA
(Received 10 June 2019; accepted 27 July 2019; published online 13 September 2019)
Abstract: Ever since its formulation by A. S. Eddington, the mass-luminosity relation has been viewed as a triumph for theoretical astronomy and astrophysics. The idea that the luminosity of the stars could be controlled solely by their mass was indeed a revolutionary concept. The proof involved two central aspects: (1) the belief that stars could be treated as ideal gases in hydrostatic equilibrium, and (2) that the opacity of Capella could be used as a reference mark applicable to other stars. Yet, when the mass-luminosity relation was advanced, no thought was given to the need for thermodynamic balance. Within thermodynamic expressions, not only must the dimensions (hence units) be consistent on each side of the equals sign, but the extensive nature of the properties must also balance. Namely, thermodynamic expressions must be balanced by properties which are extensive to the same degree. In this regard, mass is an extensive thermodynamic property and can be represented by a homogenous function of degree 1. Conversely, the luminosity of a star is neither extensive nor intensive, but rather can be represented by a homogenous function of degree 2/3. Consequently, the mass-luminosity expression is thermodynamically unbalanced and stands in violation of the laws of thermodynamics. VC 2019 Physics Essays Publication. [http://dx.doi.org/10.4006/0836-1398-32.3.353]
Resume: Depuis sa formulation par A. S. Eddington, la relation masse-luminosite a ete consideree comme un triomphe pour l’astronomie theorique et l’astrophysique. L’idee que la luminosite des etoiles puisse ^etre contr^olee uniquement par leur masse etait en effet un concept revolutionnaire. La preuve impliquait deux aspects centraux: 1) la conviction que les etoiles pouvaient ^etre traitees comme des gaz ideaux en equilibre hydrostatique, et 2) que l’opacite de Capella pouvait ^etre utilisee comme un repe`re applicable a d’autres etoiles. Pourtant, lorsque la relation masse-luminosite a ete mise au point, la necessite d’un equilibre thermodynamique n’a pas ete prise en compte. Dans les expressions thermodynamiques, non seulement les dimensions (donc les unites) doivent ^etre coherentes de chaque c^ote du signe egal, mais la nature extensive des proprietes doit egalement ^etre equilibree. A savoir, les expressions thermodynamiques doivent ^etre equilibree par des proprietes extensives au m^eme degre. A cet egard, la masse est une propriete thermodynamique extensive et peut ^etre representee par une fonction homoge`ne de degre 1. Inversement, la luminosite d’une etoile n’est ni extensive ni intensive, mais peut plut^ot ^etre representee par une fonction homoge`ne de degre 2/3. En consequence, l’expression masse-luminosite est thermodynamiquement desequilibree et constitue une violation des lois de la thermodynamique.
Key words: Mass-Luminosity Relation; Astronomy; Astrophysics; Thermodynamics.

I. INTRODUCTION
Pondering the nature of the stars in 1911, Halm was the ﬁrst to advance that “intrinsic brightness and mass are in direct relationship.”1 Ejnar Hertzsprung eventually echoed Halm,2 also noting that the brightness of a star could depend on its mass.3,4 However, it was Arthur Stanley Eddington5–7 who ﬁrst derived a mathematical relationship between the mass and luminosity of the stars based on ideal gases in hydrostatic equilibrium. There was only one additional requirement; namely, that the resulting line must pass through mass and luminosity values for the star Capella.5–7
a)sjcrothers@plasmaresources.com b)robitaille.1@osu.edu

Eddington required that this star’s internal opacity be shared by all others on the main sequence.5–7 The apparent success
of the derivation seemed to offer proof that Eddington’s entire approach was valid—the stars could be treated as ideal
gases. But in fact, he initially believed that only the giants could be considered in such fashion.7
Since that time, Eddington’s ideas have come to guide
progress in stellar structure and evolution, his massluminosity relationship2 playing a central role in astronomy and astrophysics. Yet, despite the wide modern acceptance
of Eddington’s mass-luminosity relationship, James Jeans
strongly disputed the idea. His objections and ensuing battle with Eddington were based on a fundamentally different approach to how science should be developed.4 Jeans argued
that “…there is no general relation between the masses and

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luminosities of stars….”8 Milne summarized the situation in these words: “No words are needed to praise Eddington’s achievement in calculating the state of equilibrium of a given mass of gas, and in calculating the rate of radiation from its surface. What was wrong was Eddington’s failure to realize exactly his achievements: he had found a condition for a star to be gaseous throughout; by comparison with the star, Capella, he had evaluated the opacity in the boundary layers; and he had made it appear unlikely that the stars in nature were gaseous throughout. His claims were the contrary; he claimed to have calculated the luminosity of the existing stars; he claimed to show that they were gaseous throughout; and he claimed to have evaluated the internal opacity of the stars. Jeans deserves great credit for being the ﬁrst critic to be sceptical about these claims of Eddington’s theory, in spite of the attractive plausibility with which the theory was expounded. I think that even today there is much misconception amongst astronomers about the status of Eddington’s theory.”9
Various characteristics of stars are currently derived from the mass-luminosity relation, such as stellar lifetimes and stellar masses.10,11 However, careful examination of the mass-luminosity equation reveals that it constitutes a violation of the laws of thermodynamics.

II. EDDINGTON’S MASS-LUMINOSITY EQUATION AND THERMODYNAMICS
The mass-luminosity relation is an empirical plot correlating stellar luminosity with mass for main sequence stars. In 1924, Eddington, in advocating the theory of gaseous stars, derived this expression, ﬁnding to his surprise5,6 that this expression could be made to ﬁt the empirical plot by forcing the line derived to pass through the star Capella. This was accomplished by inferring that all stars shared Capella’s opacity.5–7
However, the resulting expression did not ﬁt the entire main sequence. Consequently, today, Eddington’s work has been modiﬁed using certain power relations that are restricted to corresponding portions of the main sequence. Setting L for luminosity and M for mass, astronomy maintains that for the lower main sequence, L / M1:6 and for the upper main sequence L / M5:4.12 In general,13 L / Mn, 1 < n < 6, with the value of n chosen ad hoc to ﬁt the desired portion of the main sequence mass-luminosity plot. For Sun-like stars, n ¼ 4, and for Eddington’s overarching curve, n ¼ 3.
Still, Eddington’s approach to this problem is worth reviewing. First, one can begin by denoting the total pressure, P, within a gaseous star. Then

P ¼ pG þ pR;

(1)

where pG is gas pressure and pR is radiation pressure.6 Next, the Stefan-Boltzmann law relating luminosity of a thermal emitter to its temperature and surface area can be invoked

L ¼ erAT4;

(2)

where e is emissivity (0 e 1, a unitless property of the emitter), r is the Stefan-Boltzmann constant, A is the surface

area, and T is the temperature. For a blackbody e ¼ 1, in which case,

L ¼ rAT4:

(3)

Note that in this expression, temperature is an intensive thermodynamic property.14 However, area and luminosity are neither intensive nor extensive.14 But the quotient of
luminosity with area (L/A) is intensive. Equations (2) and (3)
are therefore thermodynamically balanced, since T (hence also T4) is intensive. By combining Eq. (3) with gravitational
acceleration,

GM

g ¼ r2 ;

(4)

employing the hydrostatic equation

dP ¼ Àgq;

(5)

dr

and the equation of radiative equilibrium

dpR ¼ À kqH ;

(6)

dr

c

in spherical symmetry, A ¼ 4pr2, with the radiation power per unit area H ¼ L/A, Eddington6 arrives at the mass-
luminosity relation

L ¼ 4pcGMð1 À bÞ ;

(7)

k0

where c is the speed of light in vacuum, k0 is a constant stellar opacity, G is the constant of gravitation, and M is the stellar mass. The b term is a constant pure number related to total pressure P, gas pressure pG, and radiation pressure pR, by

pR ¼ ð1 À bÞP;

(8)

pG ¼ bP:

Equation (7) is not restricted to “a perfect gas.”6 Even so, it is already clear that Eq. (7) stands in violation of the laws of thermodynamics because luminosity L is neither intensive nor extensive, whereas mass M is extensive. Luminosity is a homogeneous function of degree 2/3 whereas mass is a homogeneous function of degree 1.14 Since the stellar opacity is intensive and all the other terms on the right side of Eq. (7) are constants, Eq. (7) is not thermodynamically balanced.
To relate Eq. (7) to the gaseous theory of stars, Eddington6 invoked, outside its proper setting, the equation for an ideal gas (which is thermodynamically balanced), in the form

<

pG ¼ l qT;

(9)

where pG is gas pressure, < is the universal gas constant, l is molecular weight in terms of the hydrogen atom, q is den-
sity, and T is absolute temperature. The equation for radiation pressure is6

Physics Essays 32, 3 (2019)

355

pR

¼

1 3

aT4;

(10)

where a is a constant (the Stefan-Boltzmann constant). Combining Eqs. (8)–(10) Eddington6 obtained

aT4

P ¼ 3ð1 À bÞ :

(11)

Elimination of T then yields

"

#1

P¼

3<4ð1 À bÞ 3 4

al4b4

q3;

(12)

which is a polytropic form. From this Eddington6 deduced thatc)

1

À

b

¼

0:00309M2l4b4 MH2

;

(13)

where MH is the mass of the Sun. Thus, for any given gaseous star of mass M and molecular weight l, the related value for b can be found from the corresponding solution to Eq. (13), then substituted into Eq. (7) for luminosity.
Putting Eq. (13) into Eq. (7) gives

0:01236pcGl4b4M3

L¼

k0MH2

:

(14)

This equation is not thermodynamically balanced either,
which can be easily seen by dividing through by stellar surface area A ¼ 4pr2 to yield

L A

¼

0:00309cGl4b4M3 k0MH2 r2

¼

erT4:

(15)

Luminosity per unit area is intensive, by the StefanBoltzmann law, since temperature is intensive, but M3=MH2 r2 is neither intensive nor extensive because mass is extensive (a homogeneous function of degree 1) and radius is not extensive (a homogeneous function of degree 1/3).14 The right side of Eq. (15) equates temperature, which is intensive, to a combination of terms that is not intensive. Hence, Eqs. (7) and (14) are invalid.
The thermodynamic inconsistency arises by the incorporation of luminosity with gravitational acceleration via Eqs. (5) and (6). The assumption that luminosity, a surface phenomenon, can be combined with the acceleration due to gravity, a bulk phenomenon, although the bulk is required to facilitate both cases, is false, as it leads directly to insurmountable violations of the laws of thermodynamics by producing nonintensive temperature. In similar fashion, the assumption that gravitational potential energy can be combined with the kinetic energy of an ideal gas to determine temperature stands in violation of the kinetic theory of gases and the laws of thermodynamics, producing perpetual

c)The details of the derivation of Eq. (13) are unimportant for the argument herein.

motion machines of both the ﬁrst and second kind,15,16 which are forbidden by the laws of thermodynamics. A gas cannot compress itself (i.e., “gravitationally collapse”) to do work on itself and raise its own temperature. It is for this reason that the temperature equations for stars are thermodynamically unbalanced,15–17 and therefore inadmissible. In quite similar fashion, the equations advanced for black hole temperature and black hole entropy violate the laws of thermodynamics.15–26

III. GENERALIZATION OF EDDINGTON’S MASS-LUMINOSITY EQUATION

In hydrostatic equilibrium, the maximum luminosity a star can have is given by the Eddington Limit27

4pGc

LEd ¼ j M;

(16)

where j is the Rosseland mean opacity. Mass M is extensive

but luminosity LEd is not. The left side is homogeneous degree 2/3 but the right side is homogeneous degree 1. The

Eddington Limit is therefore invalid. Denoting the luminos-

ity of the Sun by LH and its mass by MH, Eq. (16) can be written as


LEd ¼

4pGcMH j LH

M LH MH :

(17)

The generalized mass-luminosity relation is28–30

L

¼

 M n aLH MH

1 < n < 6;

(18)

wherein “… n is about 4 for Sun-like stars, 3 for more massive stars and 2.5 for dim red main sequence stars.”28 The

constant a is adjustable ad libitum according to the mass M in order to obtain a desired ﬁt between plot and curve.30 The

case of n ¼ 1, where radiation pressure is said to dominate, is

the Eddington Limit. However, the case n ¼ 1 is ruled out by

astronomy, not by violations of the laws of thermodynamics,

but on the basis that such stars are unstable [hence the lower

inequality for Eq. (18)]. Comparing Eqs. (17) and (18),


a ¼ 4pGcMH :

(19)

j LH

Thus, a is altered by adjustment of j. Furthermore, a is not just an adjustable constant; it has thermodynamic character but is neither intensive nor extensive since j is intensive. Consequently, Eq. (18) is thermodynamically unbalanced and therefore invalid because it equates luminosity (homogeneous degree 2/3) on the left, to a combination of terms on the right having homogeneous degree 1/3, easily seen by substituting Eq. (19) into Eq. (18),

L

¼

 4pGcMH j LH

 M
LH MH

n

1 < n < 6:

(20)

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If R is the radius of the star of mass M, Eq. (20) can be

written

L

¼


4pGcMH 3LHRM

M

n

j LH

3RM MH

1 < n < 6:

(21)

Dividing through by the stellar surface area A ¼ 4pR2 gives

L

¼


4pGcMH LHRq

M

n

¼

erT4

A

j LH

3M MH

1 < n < 6;

(22)

where q is the mean stellar density. This can be simpliﬁed as

follows:

L

¼

   4pGc Rq

M

nÀ1

¼

erT4

A

j

3 MH

1 < n < 6:

(23)

The mean stellar density q is intensive but the radius R is not. The right side equates intensive temperature to a combination of terms that is not intensive. Hence, Eq. (21) is invalid for all values of n. Consequently Eq. (18) is also invalid for all values of n. Setting n ¼ 1 in Eq. (23) and multiplying through by area A recovers the Eddington Limit.
Even more extreme ad hoc adjustments to Eddington’s theoretical mass-luminosity equation have been proposed; for example, that due to Cuntz and Wang31 for nearby late-K and M dwarf stars on data sampled by Mann et al. 32 as calibration for distant late-type stars

 M nðMÞ

L ¼ LH MH

;

nðMÞ ¼ À141:7M4 þ 232:4M3 À 129:1M2

(24)

þ 33:29M þ 0:215;

0:20MH < M < 0:75MH;

in stark revelation of the mere curve ﬁtting nature of the whole exercise, with complete disregard for the laws of thermodynamics a priori. Mann et al.32 ﬁrst advanced the following empirical equations for radius, luminosity, and mass, respectively, (in solar units), against effective temperature Teff:
RÃ ¼ À16:883 þ 1:18 Á 10À2Teff À 2:709 Á 10À6Te2ff þ 2:105 Á 10À10Te3ff;
LÃ ¼ À0:781 þ 7:40 Á 10À4Teff À 2:49 Á 10À7Te2ff þ 2:95 Á 10À11Te3ff ;
MÃ ¼ À22:297 þ 1:544 Á 10À2Teff À 3:488 Á 10À6Te2ff þ 2:650 Á 10À10Te3ff: (25)

Using the expressions for LÃ and MÃ in Eqs. (25), Cuntz and Wang31 then derived Eqs. (24).
IV. CONCLUSIONS
The mass-luminosity relation violates the laws of thermodynamics in the very same way as the equations for temperatures of gaseous stars and black holes,15–17 and for black hole entropy.15–26
Perplexed by oddities in the theory of gaseous stars, Eddington6 remarked
“We must conclude either that we have been misled altogether in the theory of the massluminosity relation or that in dense stars like the sun the material behaves as a perfect gas.”
Eddington did not consider the third possibility: Astronomy has been misled in the theory of the mass-luminosity relation, as the laws of thermodynamics reveal, and the Sun and stars are not gaseous (perfect or otherwise).33 Ultimately, the luminosity of a star depends upon its surface area and its vibrational lattice structure.33 Only condensed matter has a lattice structure.
Confronted with numerous violations of the laws of thermodynamics, astronomy and cosmology seek to move the laws of thermodynamics34,35 in order to allow gaseous stars and black holes. They justify their actions through the presumed existence of these objects from mere hypothesis. Yet, the laws of thermodynamics are revealing that the theories of gaseous stars and black holes are invalid. The laws of thermodynamics exist, ascertained from experimental physics, and cannot be disregarded or arbitrary altered in order to permit theories that violate them; for otherwise any and all violations of thermodynamics can be circumvented by simply changing the laws of thermodynamics at will. Astronomy and cosmology have built themselves upon thermodynamically invalid foundations and must therefore be entirely reassessed.33,36–38
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