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I suggest that stars introduce mass and density scales that lead to “naturalness” in the Universe. Namely, two ratios of order unity. 1) The combination of the stellar mass scale, _{*}(_{
p
}, _{
e
}, _{Pl}, and the Chandrasekhar mass leads to a ratio of order unity that reads _{
p
} is the proton mass. 2) A system with a dynamical time equals to the nuclear life times of stars, _{nuc*}, has a density of _{
λ*} = _{Λ}/_{D*} ≈ 10^{−7}−10^{5}. Although the range is large, it is critically much smaller than the 123 orders of magnitude usually referred to when _{Λ} is compered to the Planck density. In the pure fundamental particles domain there is no naturalness; either naturalness does not exist or there is a need for a new physics or new particles. The “Astrophysical Naturalness” offers a third possibility: stars introduce the combinations of, or relations among, known fundamental quantities that lead to naturalness.

The naturalness topic is nicely summarized by Natalie Wolchover in an article from May 2013 in Quanta Magazine.^{1}
^{2}

My answer to the first point is that in astrophysics dimensional analysis does work when stars are considered as fundamental entities. This answers the second question as well. If the nuclear lifetime of stars is taken to be a dynamical time of the Universe, then naturalness emerges from the observed cosmological constant. No fine tuning is required.

Many relations among microscopic quantities and their relations with macroscopic quantities are discussed by

This essay does not discover anything new, but rather suggests to include stars as “fundamental entities” when considering naturalness in our Universe. As naturalness was discussed in talks and popular articles, I use them as references. I also limit the discussion to two commonly discussed quantities in relation to naturalness, the Planck mass and the cosmological constant (dark energy). Many other relations and coincidences can be found in

The Planck mass that starts the discussion on naturalness is defined as

It is many orders of magnitude above the mass of the Higgs boson and all other fundamental particles. If we constrain ourselves to the particle world, no naturalness exists (e.g.

Consider the Chandrasekhar mass limit _{Ch}. This is the maximum mass where a degenerate electron gas can support a body against gravity. The electrons are relativistic at this mass limit, and the expression reads_{
p
} = 1.673 × 10^{−24} g is the proton mass, and _{1} ≃ 3.1 is composed of pure numbers (no physical constants), and _{1}(^{2} ≃ 0.8 for white dwarfs in nature where _{BCh} (e.g.,

The mass of stars, namely, gravitationally bound objects that sustain hydrogen nuclear burning, is determined by the requirement that hydrogen burns to helium. From below it is limited by brown dwarfs, where the star cannot compress and heat enough to ignite hydrogen. The minimum mass for a star is _{∗} > 0.08_{⊙}. The maximum stellar mass of hundreds solar masses is not well determined, but radiation pressure limits the upper mass (e.g.,

In the logarithmic scale the range of this ratio is approximately −2 to 1.4, much-much smaller than the 17 orders of magnitude difference between the mass of the Higgs boson and the Planck mass. Moreover, if the ratio is with the Planck mass rather than _{BCh}, then the ratio is closer to unity, as it reads

It is important to emphasise that the mass of stars is determined by the requirement that hydrogen experience thermonuclear burning to helium. The Chandrasekhar mass is determined from the pressure that a degenerate electrons gas can hold against gravity. Nothing demands them to be equal. But they are. Namely, the ratio of the Chandrasekhar mass, that is composed of the Planck and the proton masses, to stellar mass is of order one.

Of course, the properties of stars are determined by the properties of the four fundamental forces, as all of them are involved in the nuclear burning and stellar structure, and the properties of the particles involved. The question is what combination of the fundamental constants of the forces and of the particles’ properties gives two quantities whose ratio is ≈1? The answer here is that stars form this combination as^{2} more than to

In other words, much as the proton “forms” a combination from the properties of the quarks and the electric and color forces to give a mass, the proton mass _{
p
}, so do stars. But stars build a much more complicated combination, and with many more of the fundamental constants and forces, and the output of this relation is not quantized, but it is rather a continuous function.

I note that _{Pl∗} ∼ 1 is expected. However, they had to use numbers from more complicated calculations than just order of magnitude estimates. They specifically use the nuclear burning temperature of hydrogen, _{
H
}, and take a factor of ^{−2} in the expression _{
H
} = _{
e
}
^{2}. _{
B
}

There is also the demand that the baryonic density in the Universe be high enough for stars to form in the first place (e.g.,

The naturalness has several implications. One of them is that regular stars can lead to white dwarfs with a mass close to and above the Chandrasekhar mass. White dwarfs with that mass or above, and iron cores of massive stars with that mass, explode eventually as a supernova. White dwarfs explode as thermonuclear supernovae where carbon and oxygen burn to nickel; cores of massive stars explode as core-collapse supernovae where a neutron star is formed. The typical kinetic energy of the ejected gas in supernovae,

The radius of an idealized white dwarf supported by a degenerate non-relativistic electrons gas is given by_{WD} is the white dwarf mass and the constant _{2} ≈ 1 is composed of pure numbers. For other forms of this expression for the white dwarf radius see _{BCh} in

Due to the factor (^{5/3} the real radius is smaller by a factor of

This is the typical kinetic energy of the mass ejected in supernova explosions of either massive stars (core collapse supernovae) or of white dwarfs (Type Ia supernovae). Simply the explosion energy is of the order of the binding energy of an electron-degenerate star.

Accurate calculations give lower binding energy values to exploding white dwarfs and collapsing cores by a factor of several. This is because the internal energy has a positive value. The explosion kinetic energy is then several times the binding energy of the degenerate core. But this does not change the argument.

The factor

When a core of a massive star collapses to a neutron star it releases a total energy of

The usual approach to search for naturalness is to compare the observed density of the dark energy _{Λ} = 7 × 10^{−30} g cm^{−3} with the Planck density _{Pl}(_{Λ} = 10^{123}. We in astrophysics are not accustomed to such astronomical numbers. This “unnatural” ratio is referred to as the cosmological constant problem (e.g.,

As we saw in previous sections, stars introduce a (complicated) combination of the fundamental quantities to give a mass ratio of order unity, that is, an astrophysical mass naturalness (

Stars spend most of their nuclear lives burning hydrogen to helium. The nuclear life time of stars depends mainly on the initial mass of the star, with _{nuc∗}(0.1_{⊙}) ≈ 10^{13} year, _{nuc∗}(_{BCh}) ≈ 10^{9} year, and _{nuc∗}(_{⊙}) ≈ 10^{7} year.

I ask now the following question. What system will have a dynamical time scale _{
D*} that is equal to the nuclear life-time of stars _{nuc}? The answer is a system that has an average density of

This density comes from the nuclear life time of stars that depends on many fundamental parameters. Namely,

The point here is that stars introduce the basic relation among these fundamental quantities.

The second natural number defined in this essay is therefore

Although the range is large, it is critically much smaller than the 123 orders of magnitude usually referred to when _{Λ} is compered to “natural density.” Moreover, a ratio of unity sits just near the center of this range.

I conclude that stars introduce a nuclear time scale, whose associated dynamical time scale leads to a density about equal to the dark energy density. Again, the nuclear time scale of stars is determined by a complicated relation of fundamental quantities, constants and particle properties. Stars combine the fundamental quantities to lead to a naturalness.

It is important to emphasise that the approach here is different than the question “Why did the cosmological constant (dark energy) become significant only recently?” (e.g.,

In the present approach the age of the universe has no importance at all. The same argument presented here holds as soon as hydrogen becomes the main element in the universe; the first minute of the universe, at an age of 10^{−16} times the present Universe age. The same argument will be true when the universe be 10^{16} times its present age (as long as the dark energy density stays constant; see

It is true that if the cosmological constant (dark energy) had been much larger, stars would not have formed (see, e.g.,

The astrophysical naturalness approach disfavours any time-variation of the fundamental constants of nature. In principle, the different constants can vary as to maintain the ratios (4), (5), and (12) around unity. However, the typical stellar mass _{*} (_{D*} (

This holds as well to the value of the cosmological constant

Overall, the astrophysical naturalness approach, that holds that stars, despite being very complicated, serve as a basic entity in our Universe, makes the Universe simpler in both introducing naturalness and in arguing that fundamental constants, including the cosmological constant (dark energy), do not vary with time.

The naturalness question I studied here can be posed as follows: “What is the combination of the fundamental constants and particle properties that leads to a ratio of two values that is of order unity?” In the present essay I showed that stars introduce these combinations that give what might be termed “Astrophysical Naturalness.”

Stars introduce the stellar mass given in _{D*} given by a very complicated relation (

Nathan Seiberg summarizes his talk by a diagram that leaves two basic options, 1) abandon naturalness, or 2) go beyond known physics/particles to find naturalness. Here I take a third option which is basically to add stars as a basic entity in our Universe, much as the proton is a composite particle. This brings out naturalness in a beautiful way, at least in the eyes of an astrophysicist.

In

The arguments presented here are not the anthropic principle, e.g., as presented by

An overall summary is that the astrophysical naturalness approach makes the Universe simpler in both introducing naturalness and in arguing that fundamental constants do not vary with time.

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I thank Adi Nusser for useful discussions, and two referees for very useful comments.