Corresponding author ]

# The size of the Sun

M.A. Fardin  [<EMAIL_ADDRESS>Université de Paris, CNRS, Institut
Jacques Monod, F-75013 Paris, France  The Academy of Bradylogists, 75013
Paris, France  M. Hautefeuille  Institut de Biologie Paris Seine, Sorbonne
Université, 7 quai Saint Bernard, 75005 Paris, France

###### Abstract

Why does the Sun have a radius around 696000 km? We will see in this article
that dimensional arguments can be used to understand the size of the Sun and
of a few other things along the way. These arguments are not new and can be
found scattered in textbooks. They are presented here in a succinct way in
order to better confront the kinematic and mechanical viewpoints on size. We
derive and compare a number of expressions for the size of the Sun and relate
large and small scales. We hope that such presentation will be useful to
students, instructors and researchers alike.

In The Character of Physical Law Richard Feynman stated that “every
theoretical physicist who is any good knows six or seven different theoretical
representations for exactly the same physics. He knows that they are all
equivalent, and that nobody is ever going to be able to decide which one is
right at that level, but he keeps them in his head, hoping that they will give
him different ideas for guessing.” Feynman (2017) Following Feynman’s advice,
we provide several different expressions for the radius of the Sun. More
generally, this will lead us to expressions for the size of other stars and
astronomical bodies like planets and satellites, and eventually to a
discussion of size in a broader sense.

Many excellent textbooks review the multiple sciences involved in the
understanding of stars, their mechanics, thermodynamics, acoustics, the
intricacies of radiative and convective processes, the interplay of atomic and
nuclear physics, or the crossover between quantum and relativistic phenomena.
To assist writing this article, we principally used the book Stellar Structure
and Evolution Kippenhahn et al. (1990). These many approaches cover widely
varying scales and viewpoints, and they can be followed quite rigorously in
many cases, yielding very good agreement with observations. The present
article proposes a walk through some of the most thought-provoking formulas
offered by this field in order to show how the different viewpoints are
related, painting a kaleidoscopic answer to the question: ‘what is the size of
the Sun?’

Volumes have been written on the physics of stars because each argument that
we will highlight can actually be pushed to a high degree of precision through
adequate consideration of the geometric and dynamical subtleties underlying
them. To enable a wide-ranging exploration in the short span of this article,
we will have to neglect these subtleties; we will rely mostly on dimensional
arguments. Formulas that will be presented have to be understood as
approximate, neglecting small numerical factors of order $1$. For instance, we
will say that the volume of a sphere of radius $R$ is $R^{3}$, omitting the
factor $4\pi/3$. Similarly, we will neglect careful integration of spatially
varying fields like pressure and density and only rely on approximate bulk
averages. At no point shall we venture to state exact results, so we have used
the sign ‘=’ to stand for approximate equality, where others usually use
‘$\propto$’ or ‘$\simeq$’. We have reserved these last two symbols to
respectively stand for scaling relations and approximate numerical results.
Thus, when we will say that $A\propto B$, we mean that $A$ is proportional to
$B$, and when we say that $A\simeq au_{a}$, we mean that the value of $A$ in
units $u_{a}$ is approximately the number $a$.

## Hydrostatic equilibrium

The most common way to approach the size of the Sun and similar stars is to
consider that it derives from an equilibrium between two competing factors.
The first factor is gravity, which tends to compress the Sun. It can be
expressed as a force per unit volume $\Psi$ (dimensions
$\mathcal{M}\mathcal{L}^{-2}\mathcal{T}^{-2}$). The second factor is the
pressure inside the sun, resisting further compression. It can be expressed as
a force per unit area $\Sigma$
($\mathcal{M}\mathcal{L}^{-1}\mathcal{T}^{-2}$). From these two quantities,
dimensions can combine to produce a length:

$R=\frac{\Sigma}{\Psi}$ (1)

This length $R$ is the radius of the Sun, understood as a ratio of pressure
and force density. This approach is generally called that of ‘hydrostatic
equilibrium’, because it basically has the same form as the relation between
the pressure inside a fluid and its own weight Landau and Lifshitz (2013).
This argument of hydrostatic equilibrium first emerged in the late 19th
century in studies by Lane, Kelvin and Helmholtz Powell (1988); Arny (1990).
The equilibrium can also be written as $\Psi=\Sigma/R$, which relates the
force density to a pressure gradient, or as $\Psi R=\Sigma$, which relates the
gravitational inward pressure on the left to the outward pressure on the
right. The radius can also be seen as the result of the balance of forces
$\Psi R^{3}=\Sigma R^{2}$. Of course, neither pressure nor force density are
homogeneous inside the Sun, so the quantities $\Psi$ and $\Sigma$ should be
understood as giving average orders of magnitude.

To evaluate the relevance of Eq. 1, we can express the force density as the
product between the average density of the Sun and the acceleration of gravity
on its surface: $\Psi=\rho g$. Using the experimental values $\rho\simeq
10^{3}$ kg/m3 and $g\simeq 3~{}10^{2}$ m/s2, this gives $\Psi\simeq
3~{}10^{5}$ N/m3 Stix (2012). Since the observed radius of the Sun is $R\simeq
7~{}10^{8}$ m, for Eq. 1 to be valid would imply that the average pressure of
the Sun is $\Sigma\simeq 2~{}10^{14}$ Pa. In practice, the pressure of the Sun
varies from about $10^{-2}$ Pa in its corona Gómez et al. (2019), to
$3~{}10^{16}$ Pa in the core, so $\Sigma\simeq 2~{}10^{14}$ Pa must be
understood as a bulk average Stix (2012).

## Sound speed and gravitational acceleration

The hydrostatic equilibrium as expressed in Eq. 1 is the archetype of a
mechanical expression for a length scale. By this, we mean that the radius
$R$, with dimension $\mathcal{L}$, is expressed as a ratio between two ‘mass-
carrying quantities’, i.e. quantities with mass in their units. In the case
where the force density can be expressed as a weight density $\Psi=\rho g$,
the hydrostatic equilibrium can also be turned into a purely kinematic
expression:

$R=\frac{v_{s}^{2}}{g}$ (2)

where $v_{s}=(\Sigma/\rho)^{\frac{1}{2}}$ is the ‘speed of sound’ inside the
Sun, i.e. the speed of mechanical waves Landau and Lifshitz (2013). In this
expression of the radius, the terms of the ratio on the right-hand side do not
have any mass in their units. The dimensions of the speed $v_{s}$ are
$\mathcal{L}\mathcal{T}^{-1}$, and the dimensions of the acceleration $g$ are
$\mathcal{L}\mathcal{T}^{-2}$. Overall, the ratio $v_{s}^{2}/g$ has the
dimensions of a distance. Here again, the terms of the equation can be moved
around so as to gain new insight. For instance, one can notice that $R/v_{s}$
is the time required for sound waves to travel across the Sun. According to
Eq. 2, this time scale should be equal to $v_{s}/g$. We will shortly see how
to interpret this time scale.

If the radius and accelerations are known, Eq. 2 can be used to estimate the
average sound speed inside the Sun, as $v_{s}=(Rg)^{\frac{1}{2}}\simeq
4~{}10^{5}$ m/s. This value is indeed the right order of magnitude of the bulk
average Stix (2012); Christensen-Dalsgaard (2002).

## The gravitational constants

We mentioned that at the surface of the Sun the gravitational acceleration is
$g\simeq 275$ m/s2, yet on Earth it is $g\simeq 9.8$ m/s2. This difference is
due to the fact that behind the acceleration $g$ hides the mass and the radius
$R$ itself. Indeed, the weight $F_{i}=m_{i}g$ of an object of mass $m_{i}$ at
the surface of the Sun can be expressed more generally from Newton’s formula
as $F_{i}=GMm_{i}/R^{2}$, where $G\simeq 6~{}10^{-11}$ kg-1 m3 s-2 is the
universal gravitational constant, and where $M\simeq 2~{}10^{30}$ kg is the
mass of the Sun. Then, we can replace $g$ by $GM/R^{2}$ and express the
density of the Sun as $\rho=M/R^{3}$. By inserting these expressions for $g$
and $\rho$ into Eq. 2 and solving for $R$ we get:

$R=\Big{(}\frac{GM^{2}}{\Sigma}\Big{)}^{\frac{1}{4}}$ (3)

If instead we express the mass in terms of the density, we get:

$R=\Big{(}\frac{\Sigma}{G\rho^{2}}\Big{)}^{\frac{1}{2}}$ (4)

These two equations are mechanical in the sense defined above: they involve
quantities with mass in their dimensions. However, in contrast to Eq. 1 these
equations involve three rather than two quantities.

## Sound/escape speed and free-fall time

To a gravitational field with acceleration $g=GM/R^{2}$ one can also associate
an ‘escape speed’ $v_{e}=(GM/R)^{\frac{1}{2}}$, which gives the typical speed
necessary to escape the attraction of a body of mass $M$ and radius $R$.
Coincidentally, this speed is also the orbital speed at a distance $R$. Either
way, $v_{e}$ is the scale of speed set by gravity. One can see that the escape
speed can be expressed as $v_{e}=(gR)^{\frac{1}{2}}$, which is also the
expression of the sound speed deduced from Eq. 2. Hence, the radius of the sun
can be understood as corresponding to the identity between the sound speed and
the escape speed, $v_{e}=v_{s}$:

$\Big{(}\frac{GM}{R}\Big{)}^{\frac{1}{2}}=\Big{(}\frac{\Sigma}{\rho}\Big{)}^{\frac{1}{2}}$
(5)

This approach is usually associated with Jeans, who derived the condition for
a gas cloud to collapse into a star to be $v_{e}\gtrsim v_{s}$ Jeans (1902).

Now that we know that for a gravity-bound body at hydrostatic equilibrium the
sound speed is equal to the escape speed, we can give an interpretation to the
time scale $v_{s}/g$. Indeed, since $v_{s}=v_{e}=(gR)^{\frac{1}{2}}$, then we
can define a time scale
$\tau=v_{s}/g=(R/g)^{\frac{1}{2}}=(G\rho)^{-\frac{1}{2}}$. This time scale is
often called the ‘free-fall time’ and corresponds to the time that a body
would take to collapse under its own gravitational attraction Kippenhahn et
al. (1990). For the Sun, $\tau\simeq 1$ h. This time scale does not depend on
the absolute size of the object but on its density. Since the Sun and a human
being have similar densities, they would collapse after a similar time of one
hour.

The radius $R$ can be understood as the distance traveled by sound over a time
equal to $\tau$. Generally, we have

${R=v_{s}\tau=v_{e}\tau}$ (6)

## Schwarzschild radius

Gravitation associates a special radius to any object with a given mass. This
is the Schwarzschild radius Schwarzschild (1916), which corresponds to the
size of a black hole with that mass. For the Sun, this radius is
$R_{s}=GM/c^{2}\simeq 1$ km, where $c\simeq 3~{}10^{8}$ m/s is the speed of
light. We can use Eq. 3 to express the size of the Sun in terms of its
Schwarzschild radius as:

$R=R_{s}\frac{c^{2}}{(v_{e}^{3}v_{s})^{\frac{1}{2}}}=R_{s}\Big{(}\frac{c}{v_{e}}\Big{)}^{2}=R_{s}\Big{(}\frac{c}{v_{s}}\Big{)}^{2}$
(7)

In this identity, the dimension of length is provided by $R_{s}$, and the rest
of the equation is a dimensionless ratio built out of the speeds $v_{s}$,
$v_{e}$ and $c$. In the limit where $v_{e}=v_{s}=c$, we have $R=R_{s}$, i.e.
if the escape/sound speed is equal to the speed of light, then the object is a
black hole.

## Solar internal energy

So far, we expanded on Eq. 1 by specifying the content of the force density
$\Psi=\rho g=\rho^{2}GR=GM^{2}/R^{5}$ when the inward force comes from
gravity. In contrast, we barely investigated the pressure $\Sigma$; we just
related it to the sound speed. Density is quite naturally connected to a ratio
between mass and volume, as in $\rho=M/R^{3}$. In a similar way, the
dimensions of pressure suggest that it can be expressed as a density of energy
since
$\mathcal{M}\mathcal{L}^{-1}\mathcal{T}^{-2}$=$\mathcal{M}\mathcal{L}^{2}\mathcal{T}^{-2}/\mathcal{L}^{3}$.
Thus, one can define a solar energy as $E=\Sigma R^{3}$ and express the radius
of the Sun as:

$R=\frac{E}{F}=\frac{GM^{2}}{E}$ (8)

where $F=\Psi R^{3}=GM^{2}/R^{2}$ is the scale of the self gravitational
force. The second expression will later be useful. The first equation can also
be rearranged to express the solar energy as $E=Mv_{e}^{2}=MgR\simeq 10^{42}$
J. As a comparison, Type Ia supernovae release an energy on the order of
$10^{44}$ J Branch and Tammann (1992).

If one uses the density $\rho=M/R^{3}$, the energy $E$, and the free-fall time
$\tau=(G\rho)^{-\frac{1}{2}}$, then the radius of the Sun can be written as

$R=\Big{(}\frac{E}{\rho}\Big{)}^{\frac{1}{5}}\tau^{\frac{2}{5}}$ (9)

This expression is reminiscent of the formula for the spreading of an
explosion blast in the Taylor-Sedov regime Taylor (1950); Sedov (1993), which
includes supernovae. Here, the free-fall time $\tau$ replaces the time since
the beginning of the explosion. The size of the Sun is similar to that of an
explosion frozen at a time $\tau$ after ignition.

## Equation of state and polytrope

Note that Eq. 3 should not be used to infer that
$R\propto\Sigma^{-\frac{1}{4}}$, nor should Eq. 4 be used to infer that
$R\propto\Sigma^{\frac{1}{2}}$. Indeed, both mass and density can depend on
pressure. The actual scaling between size and pressure will depend on the
‘equation of state’ of the body; i.e. on the specification of an additional
relation between $\Sigma$ and $\rho$ Landau and Lifshitz (2013). We will see
later that in some contexts this relation can be specified from microscopic
considerations, but it can also be set phenomenologically by prescribing that
the equation of state be that of a ‘polytrope’ Horedt (2004):

$\Sigma=K\rho^{\gamma}=\Sigma_{r}\Big{(}\frac{\rho}{\rho_{r}}\Big{)}^{\gamma}$
(10)

where in the first equation the dimensions of the proportionality factor $K$
are $\mathcal{M}^{1-\gamma}\mathcal{L}^{3\gamma-1}\mathcal{T}^{-2}$. The
dimensionless exponent is usually written as $\gamma=(n+1)/n$, where $n$ is
called the ‘polytrope index’. In the rightmost equation we define the
proportionality factor in terms of a reference density and pressure as
$K=\Sigma_{r}/\rho_{r}^{\gamma}$. The polytrope is expected to be valid in the
vicinity of the reference values.

Using Eq. 4 in conjunction with the polytrope equation, we can express the
radius $R$ as:

$R=\Big{(}\frac{K}{G}\Big{)}^{\frac{1}{2}}\rho^{\frac{\gamma-2}{2}}=\frac{\Sigma_{r}^{\frac{1}{2}}}{\rho
G^{\frac{1}{2}}}\Big{(}\frac{\rho}{\rho_{r}}\Big{)}^{\frac{\gamma}{2}}$ (11)

This approach is usually associated with Lane and Emden Lane (1870); Emden
(1907). We will now see that values of $K$ and $\gamma$ can sometimes be
computed from microscopic models.

## Microscopic density and pressure

Whereas mass and energy are typically understood as ‘extensive’ properties and
depend on the size of the system, density and pressure are ‘intensive’ and are
expected to be independent of the size of the test volume. In fact, this
independence of size only holds down to microscopic scales. Let us call $r$,
$m$ and $\varepsilon$ the size, mass and energy of the smallest scale where
$m/r^{3}=\rho$ and where $\varepsilon/r^{3}=\Sigma$. We call this scale the
microscopic scale. For sizes smaller than $r$, density and/or pressure
significantly vary from the macroscopic values. Using Eq. 4 we can write:

$R^{2}=\frac{\varepsilon}{Gm^{2}}r^{3}=\frac{r^{3}}{d}$ (12)

We know from Eq. 8 that $d=Gm^{2}/\varepsilon$ is the size of a body of mass
$m$ at hydrostatic equilibrium under its own gravity and pressure
$\varepsilon/r^{3}$. However, this size $d$ must be different from $r$.
Indeed, by construction we have $r<R$ and $d<R$, and Eq. 12 states that
$R^{2}d=r^{3}$, thus $d<r<R$.

The identity between the microscopic and macroscopic pressures,
$\varepsilon/r^{3}=E/R^{3}$ is sometimes formulated as the ‘Virial theorem’
Clausius (1870):

$\varepsilon=Gm\rho R^{2}=Gm\frac{M}{R}$ (13)

This equation relates the potential energy between the macroscopic mass $M$
and the microscopic mass $m$ to the microscopic energy $\varepsilon$. The
energy $\varepsilon$ is usually understood as a kinetic energy involving the
sound speed $(\varepsilon/m)^{\frac{1}{2}}=(\Sigma/\rho)^{\frac{1}{2}}$.

## Microscopic equilibrium

In Eq. 12 both lengths $d$ and $r$ are associated with the same energy
$\varepsilon$ and mass $m$, but not in the same way. How can one express the
size $r$? Can this be done in analogy with what we derived for the size of the
Sun? Expressions such as $R=\Sigma/\Psi$ or $R=E/F$ are too vague to ever be
false. They owe their generality to their vagueness. If one specifies the mass
$M$, or the force $F$, or the energy $E$–or their densities ($\rho$, $\Psi$,
$\Sigma$), then one can express more detailed results. This is what we did for
the Sun, where the force comes from gravity. For the size $r$ we can similarly
state without too much risk that:

$r=\frac{\varepsilon}{f}$ (14)

where $f$ is a yet unspecified force. The only thing we know is that $f\neq
Gm^{2}/r^{2}$; i.e. the dominant force at the microscopic scale is not self-
gravity. Indeed, otherwise we would have $r=\varepsilon r^{2}/Gm^{2}=d=R$
according to Eq. 12. Density and pressure would be far from intensive. If
density and pressure are constant down to a scale of size $r<R$, then this
scale must be governed by forces beyond gravity.

## Electromagnetic equilibrium

The dominant force $f$ at scale $r$ can take many forms depending on contexts,
but one example has proven to be very fruitful. This is the case where
Newton’s law of gravitation (15) is replaced by Coulomb’s law of
electrostatics (16):

$\displaystyle F$ $\displaystyle=\frac{GM^{2}}{R^{2}}$ (15) $\displaystyle f$
$\displaystyle=\frac{k(ne)^{2}}{r^{2}}$ (16)

For the gravitational force we assumed self-gravitation of a mass $M$ and size
$R$, as in the case of the Sun. For the electrostatic force we assumed two
opposite charges $ne$, where $e\simeq 1.6~{}10^{-19}$ C is the elementary
charge and $n$ is an integer. The two charges are separated by a distance $r$.
The Coulomb constant $k\simeq 9~{}10^{9}$ kg m3 s-2 C-2 and the elementary
charge can be combined into a single quantity $S_{0}=ke^{2}\simeq
2~{}10^{-28}$ kg m3 s-2 (no more units of charge). We then define
$S=n^{2}S_{0}$. The number $n$ is usually small due to screening between
positive and negative charges.

Using $f=S/r^{2}$ and $r=\varepsilon/f$ we can obtain the electromagnetic
equivalent of the hydrostatic equilibrium:

$r=\frac{S}{\varepsilon}$ (17)

In the same way that the energy $E$ and pressure $\Sigma$ were left
unspecified in the gravitational case, here the nature of the energy
$\varepsilon$ is unknown. For the gravitational case, we used the values of
$M$ and $R$ to deduce the values of $E$ or $\Sigma$. Here, values of $r$ and
$S$ would be enough to deduce a value of the energy $\varepsilon$ or pressure
$\varepsilon/r^{3}$.

## Electromagnetic kinematics

The substitution of the electromagnetic force in place of the gravitational
force changes the definition of the kinematic quantities described so far. The
formulas derived for the hydrostatic equilibrium of a body under its own
gravity can be extended to the electromagnetic case by using the following
substitutions:

$\displaystyle g=\frac{GM}{R^{2}}$ $\displaystyle\rightarrow\frac{S}{mr^{2}}$
(18) $\displaystyle v_{s}^{2}=\frac{\Sigma}{\rho}$
$\displaystyle\rightarrow\frac{\varepsilon}{m}$ (19) $\displaystyle
v_{e}^{2}=\frac{GM}{R}$ $\displaystyle\rightarrow\frac{S}{mr}$ (20)
$\displaystyle\tau^{2}=\frac{1}{G\rho}$
$\displaystyle\rightarrow\frac{mr^{3}}{S}$ (21) $\displaystyle
R_{s}=\frac{GM}{c^{2}}$ $\displaystyle\rightarrow\frac{S}{mc^{2}}$ (22)

For instance, in analogy with Eq. 2 one can say that $r=v_{s}^{2}/g$, only if
the sound speed is redefined as $v_{s}\equiv(\varepsilon/m)^{\frac{1}{2}}$,
and if the ‘electromagnetic acceleration’ is redefined as $g\equiv
S/(mr^{2})$.

## Stoney mass and Planck mass

In the electromagnetic expressions, we have used the radius $r$, mass $m$ and
energy $\varepsilon$, the same symbols we used in Eq. 12 to describe the
smallest scale at which density and pressure are equal to their macroscopic
values. How can one be sure that electromagnetic forces dominate over gravity
at the scale $r$? Using Eq. 12 and Eq. 17 we can write the size of the Sun in
terms of the size obtained from electromagnetic equilibrium as:

$R=\frac{(SG^{-1})^{\frac{1}{2}}}{m}r$ (23)

This expression is legitimate if electromagnetism is the relevant force at the
scale $r$. The strengths of the gravitational and electromagnetic forces can
be expressed respectively as $Gm^{2}$ and $S$ (both with dimensions
$\mathcal{M}\mathcal{L}^{3}\mathcal{T}^{-2}$). When $m=M$, the electromagnetic
force is completely negligible since $GM^{2}\gg S$ ($10^{50}\gg 10^{-28}$ kg
m3 s-2, assuming $S=S_{0}$). One can then derive the mass $m_{S}$ that would
correspond to the case where the electromagnetic and gravitational forces have
similar strengths, i.e. when $R=r$:

$m_{S}=\Big{(}\frac{S}{G}\Big{)}^{\frac{1}{2}}$ (24)

In cases where the strength of the electromagnetic forces is well
characterized by the value $S_{0}$ obtained for a pair of elementary charges,
then $m_{S}\simeq 10^{-9}$ kg. This mass is called the Stoney mass Tomilin
(1999). Thus, Eq. 23 can be rewritten as:

$R=\frac{m_{S}}{m}r$ (25)

Similarly, we can write $d=(m/m_{S})^{2}r$. Note that if a material can
locally deviate substantially from neutrality then the value of $S$ and
$m_{S}$ must be increased accordingly.

One can also define the fine structure constant as $\alpha=S_{0}/\hbar c\simeq
1/137$, with Planck’s constant $\hbar\simeq 10^{-34}$kg m2 s-1 and reach:

$R=\alpha^{\frac{1}{2}}\frac{m_{P}}{m}r$ (26)

where the Planck mass is $m_{P}=m_{S}/\alpha^{\frac{1}{2}}=(\hbar
c/G)^{\frac{1}{2}}$ Tomilin (1999).

So far, we have used electromagnetism to derive a relation between the
macroscopic $R$ and the microscopic $r$, but we have not specified any value
for $r$ and $m$. From Eq. 17 we know that specifying $\varepsilon$ will lead
to $r$, but $m$ seems relatively free, as long as $m\ll m_{S}$. We know from
Eq. 25 that if the mass $m$ associated with the size $r$ is much smaller than
$m_{S}$ then electromagnetism will be dominant over gravity. However other
forces may dominate over electromagnetism, in which case Eq. 25 or 26 loose
their validity.

## Atomic units

A particularly interesting choice of microscopic size and mass considers the
standard atomic units based on the hydrogen model with one nucleon (proton) of
mass $m_{0}$ and one electron of mass $m_{e}$. This model is particularly
useful for stars since they are mostly composed of hydrogen. In this context,
we have $S=S_{0}$, $m=m_{0}\simeq 1.7~{}10^{-27}$ kg, and the Hartree energy
and Bohr radius Tomilin (1999):

$\displaystyle\varepsilon_{0}$
$\displaystyle=m_{e}\Big{(}\frac{S_{0}}{\hbar}\Big{)}^{2}=\frac{\hbar^{2}}{m_{e}r_{0}^{2}}\simeq
4~{}10^{-18}~{}\text{J}$ (27) $\displaystyle r_{0}$
$\displaystyle=\frac{S_{0}}{\varepsilon_{0}}=\frac{\hbar}{m_{e}\alpha c}\simeq
5~{}10^{-11}~{}\text{m}$ (28)

where we used the electron mass $m_{e}\simeq 9~{}10^{-31}$ kg. Note that the
ratio $S_{0}/\hbar$ has the dimensions of a speed, and is often expressed as
$\alpha c$. With these units, pressure and density have values comparable to
$\Sigma$ and $\rho$:

$\displaystyle\Sigma_{0}=\frac{\varepsilon_{0}}{r_{0}^{3}}$
$\displaystyle\simeq 3~{}10^{13}~{}\text{Pa}\approx\Sigma$ (29)
$\displaystyle\rho_{0}=\frac{m_{0}}{r_{0}^{3}}$ $\displaystyle\simeq
10^{4}~{}\text{kg/m}^{3}\approx\rho$ (30)

With these atomic units we can obtain the following radius:

$R_{0}=\frac{\hbar^{2}}{(GS_{0})^{\frac{1}{2}}}\frac{1}{m_{0}m_{e}}=\frac{m_{P}^{2}}{\alpha^{\frac{1}{2}}m_{0}m_{e}}\ell_{P}\simeq
5~{}10^{7}~{}\text{m}$ (31)

In the last equation we express the reference size in terms of the Planck
length $\ell_{P}=(\hbar G/c^{3})^{\frac{1}{2}}\simeq 6~{}10^{-34}$ m, which is
expected to be the smallest possible size. The radius is associated with the
following mass:

$M_{0}=\frac{m_{S}^{3}}{m_{0}^{2}}=\alpha^{\frac{3}{2}}\frac{m_{P}^{3}}{m_{0}^{2}}\simeq
2~{}10^{27}~{}\text{kg}$ (32)

These expressions only involve fundamental constants. We will say that $R_{0}$
and $M_{0}$ are the reference size and mass. These types of expressions for
the size and mass of stars are usually associated with the astrophysicist
Chandrasekhar Chandrasekhar and Milne (1931), who used these reference values
in the derivation of the maximum mass of stable white dwarfs (now called the
Chandrasekhar limit). The depth of these formulas has been noticed on several
occasions Weisskopf (1975); Carr and Rees (1979); Garfinkle (2009); Burrows
and Ostriker (2014). Note that Eq. 32 can also be expressed, referring to Eq.
24, as an equation expressing the relative strengths of the electromagnetic
and gravitational interactions:

$(GM_{0}^{2})(Gm_{0}^{2})^{2}=S_{0}^{3}$ (33)

From the reference size and mass, one can also derive reference values for
speed (sound/escape), acceleration and time (free-fall):

$\displaystyle v_{0}$
$\displaystyle=\Big{(}\frac{GM_{0}}{R_{0}}\Big{)}^{\frac{1}{2}}=\frac{\alpha}{\beta^{\frac{1}{2}}}c\simeq
5~{}10^{4}~{}\text{m/s}$ (34) $\displaystyle g_{0}$
$\displaystyle=\frac{v_{0}^{2}}{R_{0}}=\frac{\alpha^{4}}{\beta^{2}}\frac{F_{P}}{M_{0}}=\frac{\alpha^{\frac{5}{2}}}{\beta^{2}}\frac{F_{P}}{m_{P}}\Big{(}\frac{m_{0}}{m_{P}}\Big{)}^{2}\simeq
50~{}\text{m/s}^{2}$ (35) $\displaystyle\tau_{0}$
$\displaystyle=\frac{R_{0}}{v_{0}}=\Big{(}\frac{\beta}{\alpha}\Big{)}^{\frac{3}{2}}\Big{(}\frac{m_{P}}{m_{0}}\Big{)}^{2}\tau_{P}\simeq
10^{3}~{}\text{s}$ (36)

where $\beta=m_{0}/m_{e}\simeq 1836$ is the dimensionless ratio between the
mass of a nucleon (proton or neutron) and the mass of an electron. The
constant $\tau_{P}=(G\hbar/c^{5})^{\frac{1}{2}}$ is the Planck time, and
$F_{p}=c^{4}/G$ is the Planck force Tomilin (1999).

## From natural satellites to stars

Given how we crudely neglected numerical factors, $R_{0}$ is not such a bad
approximation for the radius of the Sun. However, there are stars with sizes
smaller or larger than the Sun, for which the mass $m$ and energy
$\varepsilon$ must differ from the atomic units. There can be atoms larger
than Hydrogen, with $m>m_{0}$, and there could be atomic energies different
from Hartree’s formula. Deviations from the reference value computed in terms
of atomic units can be revealed quite strikingly by plotting the density
$M/R^{3}$ versus the radius $R$ for an array of astronomical bodies from
natural satellites to stars Zombeck (2006), as shown in Fig. 1. As can be seen
the atomic units provide the size of the crossover between stars and planets,
but stars, planets and smaller bodies can significantly deviate from the
reference point. Moreover, stars display a different trend than smaller
bodies.

For stars, the density usually decreases as the radius increases. Whereas a
star like the Sun has a density close to that of water, stars a hundred times
larger can have a density smaller than that of air. This trend can be
understood roughly by invoking the Virial theorem in Eq. 13, which states that
$\rho=\varepsilon/GmR^{2}$. Thus, if the microscopic energy per unit mass
$\varepsilon/m$ is constant then $\rho\propto R^{-2}$. In Fig. 1, the dotted
line provides this scaling in the case where $\varepsilon=\varepsilon_{0}$,
assuming $m=m_{0}$. As is obvious, this choice of atomic energy strongly
underestimate the density. The dotted dashed line gives a better fit for an
energy $\varepsilon=100\varepsilon_{0}$. We shall explain such difference
shortly.

For planets and smaller bodies, the density is roughly constant, close to the
reference atomic value $\rho_{0}$. For the density to remain constant for
bodies of different sizes, the Virial theorem imposes that the kinetic energy
per unit mass should follow $\varepsilon/m\propto R^{2}$. This can also be
stated as $\varepsilon/r^{3}\propto R^{2}$, or as $\Sigma\propto R^{2}$. The
smaller the size the smaller the internal pressure $\Sigma$.

Figure 1: The density as a function of radius for a few astronomical objects,
from moons and planets to stars Zombeck (2006). The vertical dashed line is
$R_{0}$ defined in Eq. 31. The horizontal dashed line is the density
$\rho_{0}$ obtained from Eq. 30. The dotted and dotted-dashed lines represent
the predictions of the Virial theorem (Eq. 13), respectively with
$\varepsilon=\varepsilon_{0}$ (Hartree energy), $\varepsilon=k_{B}T$ (thermal
energy), with a constant temperature $T=10^{7}$ K. Note that the density is
here defined as $M/R^{3}$ instead of $M/(4\pi/3)R^{3}$, which gives values
slightly larger than standard results.

The mass $m$, energy $\varepsilon$, and size $r$ of the microscopic scale can
be fairly independent from one another. The microscopic parameters can produce
different types of pressure $\varepsilon/r^{3}$ and density $m/r^{3}$, beyond
the values obtained in terms of atomic units. In the microscopic realm, the
relationship between $\varepsilon$, $m$ and $r$ is the equivalent of the
macroscopic equation of state. In general, we can express the size $R$ in Eq.
13 from the reference values as:

$R=\Big{(}\frac{\varepsilon}{\varepsilon_{0}}\Big{)}^{\frac{1}{2}}\Big{(}\frac{m}{m_{0}}\Big{)}^{-1}\Big{(}\frac{r}{r_{0}}\Big{)}^{\frac{3}{2}}R_{0}=\Big{(}\frac{\Sigma}{\Sigma_{0}}\Big{)}^{\frac{1}{2}}\frac{\rho_{0}}{\rho}R_{0}$
(37)

For stars or for planets, the particular relationships between the microscopic
parameters can then be used to constrain this equation further.

## The Eddington model

To give but one example of the ways in which the size of astronomical bodies
can deviate from the reference value, let us consider the Eddington stellar
model Eddington (1988). This model is sometimes called ‘the standard stellar
model’, and it is usually described in textbooks concerned with stars
Kippenhahn et al. (1990). Discussing this model also gives us the opportunity
to underline the fact that our approach so far has been a bit stereotypical of
an astrophysical viewpoint, in contrast to the astronomical perspective.
Typical astronomical observations of stars rarely yield their size or mass,
the most common observables being rather their luminosity and spectral
characteristics. These observables are then transformed to mechanical
quantities like mass, pressure, density, etc. The link for such a
transformation is temperature.

In the Eddington model and others of its kind, the equation of state relating
pressure and density uses the star temperature $T$. In practice, temperature
varies inside stars from the core to the envelope, and $T$ must be understood
as a bulk average. Using Boltzmann’s constant $k_{B}\simeq 1.38~{}10^{-23}$ m2
kg s-2 K-1, an energy can be translated into a temperature using
$\varepsilon_{T}=k_{B}T$. In this framework Hartree’s atomic energy would
correspond to a temperature $T_{0}\simeq 3~{}10^{5}$ K. Thus, for the stars in
Fig. 1, the energy $\varepsilon=100\varepsilon_{0}$ corresponds to a
temperature $T\simeq 3~{}10^{7}$ K, which is the right order of magnitude for
the temperature in stars like the Sun Stix (2012).

Eddington’s model proposes that the pressure inside a star be given as a fixed
combination of a gas pressure and a radiation pressure, where the gas pressure
follows the ideal gas law $\Sigma_{1}=(\rho/m)\varepsilon_{T}$, and where the
radiation pressure follows Stefan-Boltzmann law
$\Sigma_{2}=\varepsilon_{T}^{4}/(c\hbar)^{3}$. If $\theta\in[0,1]$ is the
proportion of thermodynamic pressure then $1-\theta$ is the proportion of
radiation pressure, such that $\theta\Sigma=\Sigma_{1}$ and
$(1-\theta)\Sigma=\Sigma_{2}$, where $\Sigma$ is the total pressure. One can
then replace the thermal energy $\varepsilon$ by $m\theta\Sigma/\rho$ to
express the radiation pressure as:

$(1-\theta)\Sigma=\frac{\Sigma^{4}}{(c\hbar)^{3}}\Big{(}\frac{m\theta}{\rho}\Big{)}^{4}$
(38)

This equation can be solved for the total pressure $\Sigma$. By noticing that
$c\hbar=S_{0}/\alpha$ the total pressure can then be expressed as:

$\Sigma=\frac{1}{\alpha}\Big{(}\frac{1-\theta}{\theta^{4}}\Big{)}^{\frac{1}{3}}\Big{(}\frac{m_{0}}{m}\Big{)}^{\frac{4}{3}}\Big{(}\frac{\rho}{\rho_{0}}\Big{)}^{\frac{4}{3}}\Sigma_{0}$
(39)

If one assumes that $m=m_{0}$ and that $\theta$ is constant, then this
equation provides a fully specified polytrope with $\gamma=\frac{4}{3}$ (and
references $\rho_{r}=\rho_{0}$ and
$\Sigma_{r}=((1-\theta)/\theta^{4})^{\frac{1}{3}}\Sigma_{0}$). With this
equation of state, the radius and mass of the star become:

$\displaystyle R$
$\displaystyle=\frac{1}{\alpha^{\frac{1}{2}}}\Big{(}\frac{1-\theta}{\theta^{4}}\Big{)}^{\frac{1}{6}}\Big{(}\frac{\rho}{\rho_{0}}\Big{)}^{-\frac{1}{3}}R_{0}$
(40) $\displaystyle M$
$\displaystyle=\frac{1}{\alpha^{\frac{3}{2}}}\Big{(}\frac{1-\theta}{\theta^{4}}\Big{)}^{\frac{1}{2}}M_{0}$
(41)

Stars following this equation of state can have different radii but they all
share the same mass. If one assumes that the Sun follows such equation of
state, it would imply that radiation only account for 20$\%$ of the pressure.
Note that this equation of state implies $\rho\propto R^{-3}$. The data in
Fig. 1 are closer to $\rho\propto R^{-2}$, which can be interpreted by
changing the proportion of radiation and thermodynamic pressures as the size
increases. Arguments aimed at deciphering the value of $\theta$ for different
populations of stars usually rely on a decomposition of the size of a star
into layers (core, envelopes, etc.) with different amounts of radiation and
convection. Schematically, greater accuracy on the size of stars requires
their radius to be decomposed additively, for instance as
$R=R_{core}+R_{env}$, where the sizes of the core and envelop are then
decomposed multiplicatively as in the many ways we followed in this article
Kippenhahn et al. (1990). This degree of precision goes beyond our scope but
is inescapable if one wishes to recover the properties of the wide variety of
stars in the cosmos.

## White dwarfs, neutron stars and black holes

Object |  Pressure |  Radius
---|---|---
White dwarf
(classical) |  $\hbar^{2}/m_{e}r^{5}$
$\Sigma_{0}(r_{0}/r)^{5}$
$\Sigma_{0}(\rho/\rho_{0})^{\frac{5}{3}}$ |
$R_{0}(\rho_{0}/\rho)^{\frac{1}{6}}$
White dwarf
(relativistic) |  $\hbar c/r^{4}$
$\Sigma_{0}\alpha^{-1}(r_{0}/r)^{4}$
$\Sigma_{0}\alpha^{-1}(\rho/\rho_{0})^{\frac{4}{3}}$ |
$R_{0}\alpha^{-\frac{1}{2}}(\rho_{0}/\rho)^{\frac{1}{3}}$
Neutron star
(ideal) |  $\Sigma_{0}\alpha^{3}(\varepsilon_{n}/\varepsilon_{0})^{4}$
$\Sigma_{0}\alpha^{-1}(\rho_{n}/\rho_{0})^{\frac{4}{3}}$ |
$R_{0}\alpha^{\frac{1}{2}}\beta^{-1}\simeq 2$km
Black hole
(of mass $M_{0}$) |  $c^{8}/G^{3}M_{0}^{2}$
$\Sigma_{0}\alpha^{-4}(\rho_{n}/\rho_{0})^{\frac{4}{3}}$
$\Sigma_{0}(\rho_{b}/\rho_{0})^{\frac{4}{3}}$ |
$R_{0}\alpha^{2}\beta^{-1}\simeq 1$m
Table 1: Pressure and associated radius for different types of astronomical
objects beyond the main sequence of stars. For a given object, the various
expressions for the pressure are equivalent, assuming that the mass of the
microscopic scale is $m=m_{0}$. The ideal neutron star considers a star with a
nuclear density $\rho_{n}=m_{0}/r_{n}^{3}\simeq 10^{18}$ kg/m3, where
$r_{n}=\hbar/m_{0}c$ is the Compton wavelength of the nucleon, and a nuclear
energy $\varepsilon_{n}=m_{0}c^{2}$. For the black hole of mass $M_{0}$, the
density is $\rho_{b}=c^{6}/G^{3}M_{0}^{2}$. We recall that
$\beta=m_{0}/m_{e}\simeq 1836$ and $\alpha=S_{0}/\hbar c\simeq 1/137$.

All expressions of $R$ from Eq. 3 to Eq. 9 are essentially equivalent, they
just present the same relations under different disguises. All these formulas
correspond to an object bound by its own gravity, which is resisted by an
unspecified internal pressure $\Sigma$, which can be expressed as an energy
$E$ if need be. These expressions are expected to be valid for all objects in
Fig. 1. Actually, these equations are expected to be valid for some objects
beyond the range of the figure, in particular for stars outside the ‘main
sequence’.

The stars represented in Fig. 1 all lie on the ‘main sequence’, where the
internal pressure is some mixture of thermodynamic or radiation pressures,
well described by the Eddington model. However, some stars are governed by
different types of internal pressure, which can lead to masses and radii lying
outside the range of Fig. 1. Table 1 gives a few classical examples, left for
the reader to ponder.

## Conclusion

A diversity of ‘ideas for guessing’ led us to more than a dozen formulas
providing equivalent approximations of the size of the Sun. The size of the
Sun can be seen as the result of a balance of forces or pressures, or as the
consequence of an equation relating sound and escape speed, or as a frozen
explosion. The size of the Sun can be expressed in terms of the size of its
equivalent black hole, or in terms of the size of its microscopic
constituents. We saw that some of these formulas also apply to other objects
bound by gravity including different types of stars, as well as planets, and
to some extent to microscopic objects bound by forces beyond gravity.

The gigantic gap between the microscopic realm of atoms and the astronomical
realm of planets and stars is quite daunting. In this paper, we provided a few
simple arguments that can help bridge this divide, demonstrating the strong
links between the quantum microcosm and the much larger scales dominated by
gravity.

The path we took to connect all these different formulas is one out of many.
Any written story has a beginning and an end, so we had to start somewhere,
with the hydrostatic equilibrium, and finish somewhere else, with the
Eddington model and a table of additional cases based on different kinds of
pressure. The beginning was in no way a means to an end, nor the end our goal.
We invite the reader to explore their own path across the fascinating
landscape of stellar physics, which provides an exciting laboratory for
thought experiments about what we collectively mean by the ‘size’ of
something.

Acknowledgments: This paper was inspired by discussions with Nina Gnadig,
Philip Avigan, Paul Cahen, Cyprien Gay and Vivek Sharma, and was fostered by
the joyful atmosphere of the Ladoux-Mège lab.

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