\section{Introduction}
Over the last decade a few experimental/observational contexts have been
identified in which remarkably some ultrasmall effects
introduced at the Planck scale,
of the type that can be of interest for
the study of the quantum-gravity problem, could be tested
(see, {\it e.g.},
Refs.~\cite{grbgac,kifune,biller,gacNature1999,gacPiranPRD,jaconature,PiranNeutriNat,gacPolonpapLRR}).
This had a significant impact on the perspective adopted by many of those
studying the quantum-gravity problem, especially in light of the fact that
up to the mid 1990s it was instead assumed as a given that
the smallness of the characteristic distance scale of quantum gravity,
the ``Planck length" $L_P \sim 10^{-35}m$, would render
these effects forever inaccessible.
While this is certainly exciting, it should be stressed that
the very few examples of  tests of Planck-scale physics that have been identified so far
provide us only with opportunities to explore the Planck-scale regime in a rather
limited way.
In particular,  most of these tests involve (more or less directly) the laws of propagation
of free particles in vacuum, and therefore provide only a rather narrow window
on the quantum-gravity realm.

The recent paper in Ref.~\cite{Camacho}, by Camacho, could be rather relevant
for these concerns, since it focuses on a context that has very limited overlap with
the properties of free particles in vacuum: Camacho considers the implications
of possible Planck-scale modifications of the energy-momentum ``dispersion" relation
for the analysis of the Chandrasekhar model~\cite{Chandrasekhar} of white dwarfs.
And the results of the preliminary analysis reported by Camacho in Ref.~\cite{Camacho}
were rather exciting, since they provided evidence of a rather striking amplification
of the Planck-scale effects in the limit in which the mass of the white dwarf approaches
the Chandrasekhar limit.

Due to the preliminary nature of Camacho's investigation it remained unclear whether this
amplification could be turned into a resource from a phenomenology perspective,
and also the nature (conceptual origin) of the mechanism producing such
a large amplification was not explored. But clearly the encouragement those preliminary
results provided for the prospect of testing Planck-scale effects in macroscopic systems,
should provide sufficient motivation for further investigation.

A first follow-up study was reported in Ref.~\cite{sethNANE}, providing a first few
more in-depth indications, showing in particular that the amplification of Planck-scale effects
reported by Camacho is unlikely to provide the basis for a valuable phenomenological programme,
at least not in the context of the Chandrasekhar model and of its Newtonian description
of gravitational effects.

We are here mostly interested in characterizing this amplification mechanism
from a conceptual perspective, possibly in ways that could inspire the search
of other options for an associated phenomenological programme (the Chandrasekhar-model
option, as observed in Ref.~\cite{sethNANE} and further deduced here, is indeed not viable).
And from a broader perspective we are also interested in this amplification mechanism
in relation to its implications for the intuition that guides quantum-gravity research:
this intuition nearly exclusively refers, as the regime where quantum-gravity effects
are nonnegligible, to the regime of ultra-short (Planckian) length scales, but we were hoping
(and indeed we found, as here reported) that a possible alternative/complementary perspective
might arise from investigation of Camacho's amplification mechanism.
From this perspective besides further analysis of the Newtonian case (which we also give here)
 it is valuable to contemplate a possible role for general-relativistic
effects, and this aspect, not considered in Refs.~\cite{Camacho,sethNANE},
is indeed here preliminarily investigated.

We are also interested in these issues because of their relevance for another aspect of
the recent quantum-gravity literature, which concerns the possibility
of ``Planck-scale deformations"
of spacetime symmetries~\cite{gacdsr1a,gacdsr1b,dsrO1,dsrO2,dsrnature,dsrO3,dsrO4}).
Both Ref.~\cite{Camacho} and Ref.~\cite{sethNANE} introduced Planck-scale modifications of
the dispersion relation
in such a way that the underlying spacetime symmetries would be broken. Some of the new
aspects we here introduce in the analysis are suitable for the description of modified
dispersion relations in a deformed-symmetry scenario. And in this respect we report
preliminary evidence that this type of analysis can be used to discriminate
between the two possibilities.

\section{Chandrasekhar model with modified dispersion relation}
It is convenient for us to start by summarizing briefly the argument
proposed by Camacho in Ref.~\cite{Camacho}, so that the perspective from which
we are planing to contribute will be clearer.
The analysis of Ref.~\cite{Camacho}
adopts a class of candidate modifications of the dispersion relation,
parametrized\footnote{Camacho considered the more general case
of a quantum-gravity correction of the form $\eta \vec{p}^2 ({E}/{E_p})^\alpha$
with the additional parameter $\alpha$ giving the power of the leading-order
quantum-gravity correction. For simplicity we here focus on the most optimistic scenario
in which $\alpha =1$. Since we are investigating whether there are any chances of
significant effects it makes sense to focus on the case in which the effects
would be strongest.} by a parameter $\eta$  in terms
of the Planck scale $E_p$ (we use units such as $c=\hbar=1$)
\begin{equation}
E^2-m^2 = \vec{p}^2 + \eta \vec{p}^2 \left(\frac{E}{E_p}\right) \label{MDR},
\end{equation}
which has been much studied in the literature (see, {\it e.g.},
Refs.~\cite{aemn1,plb1997,grbgac,gambinipullin,urrutiaPRL,gacdsr1a,gacdsr1b}).
And Ref.~\cite{Camacho}
observes that such a modification of the dispersion relation would primarily affect
the analysis of the Chandrasekhar model for white dwarfs
by producing a correspondingly modified expression
for the zero point energy of the system:
\begin{eqnarray}
E_0=\frac{2V}{h^3}\int_0^{p_F}  dp ~ 4\pi p^2 \sqrt{p^2 + m_e^2}
\left[1+\frac{\eta}{2}\frac{p^2}{p^2
+m_e^2}\left(\frac{\sqrt{p^2+ m_e^2}}{E_p}\right)\right] \label{Ezerocorr1} ~,
\end{eqnarray}
 where $m_e$ is the electron mass and $p_F$ is the Fermi momentum.

 From Eq.~(\ref{Ezerocorr1}) one finds
\begin{eqnarray}
E_0=\frac{m_e^4 V}{\pi^2}\left(f(x_F)+\frac{\eta}{2}\left(\frac{m_e
}{E_p}\right)g(x_F)\right)\label{Ezerocorr}
\end{eqnarray}
in terms of the dimensionless variable $x_F \equiv \frac{p_F}{m_e}$ and with
\begin{eqnarray}
f(x_F)&=&\frac{x_F}{4}(x_F^2+1)^{3/2}-\frac{x_F}{8}\sqrt{x_F^2+1}
-\frac{1}{8}\ln\left(x_F+\sqrt{x_F^2+1}\right) ~,\label{effe}\\
g(x_F)&=& \int_0^{x_F}x^4 dx = \frac{1}{5} x_F^5 ~.
\end{eqnarray}

The result (\ref{Ezerocorr}) was then used by Camacho~\cite{Camacho}
to study the implications of the quantum-gravity effects
for the pressure due to Pauli repulsion
in the ultrarelativistic $(x_F>>1)$ limit:
\begin{equation}
P_0 = -\frac{\partial E_0}{\partial V} \simeq \left(\frac{m_e^4}{12\pi^2}\right)\left(x_F^4-x_F^2
+ \frac{4}{5}\eta\frac{m_e}{E_p}x_F^5\right) ~.\label{pizerorelcorr}
\end{equation}

This is rather significant, at least conceptually, since within
 the Chandrasekhar model \cite{Chandrasekhar},
 the stability of white dwarfs is due to the equilibrium between
gravitational pressure and Pauli repulsion.
Therefore the result of Eq.~(\ref{pizerorelcorr})
should balance the gravitational pressure, which in the Chandarsekhar model is
described using Newtonian gravity:
\begin{equation}
P_0=\frac{G}{4\pi}\left(\frac{8 m_P}{9\pi}\right)^2 m_e^4\frac{\bar{M}^2}{\bar{R}^4}\label{Pgrav},
\end{equation}
where we introduced the notation $m_e$ for electron mass, $m_P$ for the proton mass, and
\begin{equation}
\bar{M}=\frac{9\pi }{8 m_P}M ~, \qquad \bar{R}= m_e R  ~.
\end{equation}
Camacho's quantum-gravity modified pressure-balance equation
therefore takes the form
\begin{eqnarray}
K\frac{\bar{M}^{4/3}}{\bar{R}^4}-K\frac{\bar{M}^{2/3}}{\bar{R}^2}
+\frac{4}{5}\eta\frac{m_e}{E_p}K\frac{\bar{M}^{5/3}}{\bar{R}^5}
=K'\frac{\bar{M}^2}{\bar{R}^4}~, \label{equaz}
\end{eqnarray}
in which $K=\left(\frac{m_e^4}{12\pi^2}\right)$
and $K'=\frac{G}{4\pi}\left(\frac{8m_P}{9\pi}\right)^2 m_e^4$.

Solving Eq.~(\ref{equaz}) for the radius one finds
\begin{equation}
\bar{R}=\bar{M}^{\frac{1}{3}}\sqrt{1-\frac{K'}{K}\bar{M}^{\frac{2}{3}}}
-\frac{4}{5}\eta\frac{m_e}{E_p}\frac{\bar{M}^{\frac{1}{3}}}{2(1
-\frac{K'}{K}\bar{M}^{\frac{2}{3}})} ~,\label{dodici}
\end{equation}
which is most intelligibly characterized in terms of a correction to
the Chandrasekhar result:
\begin{equation}
R = R_{Chan}\left(1-\frac{2}{5}\eta\frac{m_e}{E_p}\frac{1}{(1
-(\frac{M}{M_0})^{\frac{2}{3}})^{\frac{3}{2}}}\right) ~,\label{Camachos}
\end{equation}
where
\begin{equation}
R_{Chan}=\frac{3}{2}\frac{\pi^{\frac{1}{3}}}{m_e}\left(\frac{M}{m_p}\right)^{\frac{1}{3}}\sqrt{1-\left(\frac{M}{M_0}\right)^{\frac{2}{3}}}
\end{equation}
is the Chandrasekhar radius and $M_0=(\frac{9\pi}{8 m_p})^2(\frac{1}{3\pi G})^{\frac{3}{2}}$
is called Chandrasekhar limit and represents the maximum stable mass for a white dwarf in the
Chandrasekhar model.

The remarkable aspect of Camacho's result (\ref{Camachos}) is the huge amplification of the
quantum-gravity effect for $M \rightarrow M_0$: the quantum-gravity correction has the inevitable
factor with Planck-scale suppression $\frac{m_e}{E_p}$ (and this is of order $\sim 10^{-22}$)
but in this instance it also involves
a factor $(1-(\frac{M}{M_0})^{\frac{2}{3}})^{-\frac{3}{2}}$ which formally diverges
as  $M \rightarrow M_0$.
For some value of $M$ that is very close but finitely smaller than $M_0$ the correction
is of $O(1)$.
This result is of clear significance from a conceptual perspective, and we shall provide
further elements to underline this significance here.
From the conceptual perspective it is however interesting to investigate whether it is
possible that
this unexpected feature is a peculiarity of scenarios with broken Lorentz symmetry,
or could be also a feature of scenarios with deformation of Lorentz symmetry in the sense
of ``doubly-special relativity"~\cite{gacdsr1a,gacdsr1b}. Both in Camacho's original paper~\cite{Camacho}
and in a follow up study~\cite{sethNANE} there was no evidence that this feature might
be prevented in symmetry-deformation scenarios, but we shall here show that this is a possibility.

Most importantly, one would hope that this unexpected and striking feature could be exploited
for experimental searches of quantum-gravity effects, in the spirit of the recent literature
on ``Quantum-Gravity Phenomenology"~\cite{grbgac,kifune,biller,gacNature1999,gacPiranPRD,jaconature,PiranNeutriNat,gacPolonpapLRR}).
From this perspective Camacho~\cite{Camacho} focused his investigations on
two white dwarfs, specifically G156-64 and Wolf 485 A, arguing that
a comparison of the properties of these two white dwarfs could provide
valuable information on the correction term in his formula (here reported in Eq.~(\ref{Camachos}).
One of our objectives is a more careful
analysis of the possibility of using data on white dwarfs
like G156-64 and Wolf 485 A as opportunities to place meaningful constraints on the model.
And our first observation on this point
is that these white dwarfs cannot be described in the ultrarelativistic
regime, on which Camacho focused, since for them the condition $x_F>>1$ is not satisfied.
This will lead us to generalize Camacho's analysis to include proper handling of the nonrelativistic
regime, which (besides the specifics of white dwarfs
like G156-64 and Wolf 485 A) could also be of rather general interest.

We also plan to consider a somewhat broader class of features
both for the Planck-scale effects and in the characterization of the stars.
The only study that so far has extended Camacho's original observation
was reported in Ref.~\cite{sethNANE} and focused mainly on
improving the picture advocated by Camacho in the direction of obtaining
a more
robustly quantitative analysis, including in particular the effects of
a non-homogeneous mass-density
distribution within the star,
but still focusing exclusively on Planck-scale effects in the dispersion relation
and on a Newtonian description of gravitational effects.

We will offer also a preliminary investigation of general-relativistic corrections
(Section~VI), and, perhaps most importantly, we combine Planck-scale effects
for the dispersion relation with Planck-scale effects in the law of composition
of momenta. This type of combination of Planck-scale effects is relevant
for establishing whether the mechanism under study is characteristic of
quantum-gravity pictures with a breakdown of spacetime symmetries
or could also be found in pictures with ``deformation" of symmetries.

\section{Deformed integration measure}
Both Ref.~\cite{Camacho} and Ref.~\cite{sethNANE}
considered modifications of the Chandrasekhar model implied by quantum-gravity scenarios
with Planck-scale modifications of the dispersion relation. And in both papers it was
assumed that the results obtained would be applicable to any proposal with modified dispersion
relation, indepedently of other structures present in the model, and in particular
that they would be applicable equally well to scenarios with breakdown of Lorentz
symmetry and scenarios with deformed Lorentz symmetry.

The proposal~\cite{gacdsr1a,gacdsr1b} of deformed Lorentz symmetry, in the DSR sense (``doubly-special
relativity"), has generated rather significant interest and a relatively large literature
(see, {\it e.g.}, Refs.~\cite{gacdsr1a,gacdsr1b,dsrO1,dsrO2,dsrnature,dsrO3,dsrO4}).
Already with the first results~\cite{gacdsr1a,gacdsr1b} of this research programme
it was established that modified dispersion relation of the type (\ref{MDR}) can be introduced
in a deformed-symmetry scenario but only if accompanied by a modification
of the law of composition of momentum.
The studies reported in
Ref.~\cite{Camacho} and Ref.~\cite{sethNANE} focus on the fact that there is a clear
explicit role for the dispersion relation in the Chandrasekhar analysis,
while there is no explicit role for the law of composition of momentum.
However, we want to here stress that, as shown already in Refs.~\cite{gacmajid,kowaMEASURE},
the law of composition of momenta affects the rules of integration over energy momentum
space, and these are crucially relevant for the Chandrasekhar analysis.

The considerations reported in Refs.~\cite{gacmajid,kowaMEASURE},
while showing robustly that integration over energy-momentum is affected,
are insufficient to fully specify the new rules of integration.
Still the arguments reported in Refs.~\cite{gacmajid,kowaMEASURE}
imply that the net effect should be describeable in terms of
a deformed measure of integration of the type
\begin{equation}
d^4p\rightarrow e^{\theta E/E_p}d^4p,\label{misuradeformata}
\end{equation}
with $\theta$ a parameter that will need to be adjusted on the basis of  future
better insight on the mechanism, but should most likely be
an integer multiple of $\eta$, with one of the values $\eta,-\eta,-3\eta$,
according to Ref.~\cite{gacmajid},  or the value  $3 \eta$,
according to Ref.~\cite{kowaMEASURE}.

In our Planck-scale modification of the Chandrasekhar model
we shall take into account both of modifications of the dispersion relation and
of modifications of the integration measure.

In closing this section let us give a simple illustrative derivation
of deformed integration measure that follows from a modified law of
composition of momenta.
For this illustrative example we focus
on the following
modified law of
composition of momenta
\begin{equation}
p \dot{+} q=\{E(\vec{p})+E(\vec{q}),\vec{p}+e^{-\eta E(\vec{p})/E_p}\vec{q}\} ~,
\label{psum}
\end{equation}
which is in use in the
literature on the $\kappa$-Minkowski noncommutative
spacetime~\cite{kappaM1,kappaM2,gacmajid,kappaM3}.
We look for the implications of this modified law of composition of momenta
for the function $F'(p)$ which is the integrand of $F(p)$: $F=\int F' dp$. And let us observe
that the law of composition of momenta (\ref{psum}) suggests that, for each spatial component
of momentum,
\begin{eqnarray}
F'(p)=\frac{F(p\dot{+}\Delta p)-F(p)}{\Delta p}=\frac{F(p+\Delta p e^{-\eta E/E_p})
-F(p)}{\Delta p}\simeq
 \frac{\partial F(p)}{\partial p}e^{-\eta E/E_p} ~.\label{rapinc}
\end{eqnarray}

This in turn suggests that in one spatial dimension one should have
\begin{eqnarray}
F=\int F' dp = \int F' dp = \int \frac{\partial F(p)}{\partial p}e^{-\eta E/E_p} dp~,\label{intONEDIM}
\end{eqnarray}
amounting effectively to a change of integration measure with
respect to the standard case: $dp \rightarrow e^{-\eta E/E_p}dp$. In the case here of interest,
with 3 spatial and 1 time dimension, one ends up with
\begin{equation}
d^4p \rightarrow e^{-3\eta E/E_p}d^4p ~,
\end{equation}
which is indeed compatible with (\ref{misuradeformata})
for $\theta = -3 \eta$.

\section{Chandrasekhar model with modified dispersion relation and modified momentum composition}
In light of the observations reported in the previous section we are interested
in an analysis of somewhat broader scope than the ones reported in Refs.~\cite{Camacho,sethNANE},
contemplating the implications for the
Chandrasekhar model of both a modification of type (\ref{MDR})
of the dispersion relation and of a modification of the law of composition of momenta
such that (on the basis of the observations reported in the preceeding section,
and approximating $e^{\theta E/E_P}\simeq (1+\theta {E}/{E_p})$)
\begin{equation}
\left(1+\theta\frac{E}{E_p}\right)d^4p ~.\label{misuradeforme}
\end{equation}
This deformation of the integration measure, which results from a corresponding deformation
of the law of composition of momenta, affects the Chandrasekhar description of white dwarfs already
at the level of the relationship between the Fermi momentum $p_F$,
the number  $N$ of fermions in the system and  the volume $V$ of the star:
\begin{equation}
N=\frac{2V}{(2\pi)^3}4\pi\int_0^{p_F}p^2 \left(1+\theta\frac{E}{E_p}\right)dp ~.
\label{defpieffe}
\end{equation}
In particular, in the ultrarelativistic limit this leads to
\begin{equation}
p_F=\left(\frac{3}{4}\frac{N}{V}(2\pi)^2\right)^{\frac{1}{3}}\left(1
-\frac{1}{4}\frac{\theta}{E_p}\left(\frac{3}{4}\frac{N}{V}(2\pi)^2\right)^{\frac{1}{3}}
\right)\label{pfultrarel}~.
\end{equation}
Here the Planck-scale correction is such that for positive $\theta$
it decreases (with respect to the case without Planck-scale corrections)
the value of the Fermi momentum at a given density, or equivalently one can describe the
effect as going in the direction of allowing,
for given value of the Fermi momentum, to
place more particles in a given volume.
The opposite holds for negative $\theta$, which is a case with
increased value of the Fermi momentum at a given density, or lower density
at given Fermi momentum.

For the determination of zero-point energy both the modification of the dispersion relation
and the modification of the integration measure play a role:
\begin{eqnarray}
E_0&=&\frac{2V}{h^3}\int_0^{p_F}  dp\left(1+\theta\frac{\sqrt{p^2
+m_e^2}}{E_p}\right) 4\pi p^2\sqrt{p^2+m_e^2} \left(1+\frac{\eta}{2}\frac{p^2}{p^2+m_e^2}\left(\frac{\sqrt{p^2
+m_e^2}}{E_p} \right)\right) ~. \label{Ezerocorr1nostra}
\end{eqnarray}

The Planck-scale features we are introducing in the Chandrasekhar model are all encoded
in the Fermi momentum and in the zero-point energy.
We are going to first consider the ultrarelativistic behaviour, and then, in the next section,
offer some observations on the nonrelativistic limit.


Concerning the equations of state one
straightforwardly finds\footnote{Some of the results in this section
are given in terms of  the mass $m_f$
of some fermions.
Within the context of the Chandrasekhar model, on which we focus in this section, $m_f$
should be taken as the electron mass, but in Section VI we shall refer to these formulas again,
and within the perspective of Section VI $m_f$ could still be the mass of electrons, but could also
be the mass of neutrons.}:
\begin{equation}
\epsilon(x_F)=\frac{E_0}{V} = \frac{3}{2}K\left(x_F(1+2x_F^2)\sqrt{1+x_F^2}
-\ln(x_F+\sqrt{1+x_F^2})+\frac{1}{5}\frac{m_f}{E_p}((\theta
+\frac{\eta}{2})x_F^5+\theta x_F^3)\right)\label{equazstatoA} ~,
\end{equation}
\begin{equation}
 P_0(x_F)=- \frac{\partial E_0}{\partial V} = \frac{1}{2}K\left( x_F(2x_F^2-3)\sqrt{1+x_F^2}
+\ln(x_F+\sqrt{1+x_F^2})+\frac{16}{5}\frac{m_f}{E_p}(\theta
+\frac{\eta}{2})x_F^5\right) ~,\label{equazstatoB}
\end{equation}
where $K=\frac{m_f^4}{12\pi^2}$, $\epsilon$
is the energy density, $P_0$ is the pressure of the fermionic gas, and,
also taking into account (\ref{pfultrarel}),
\begin{eqnarray}
x_F&\equiv&\frac{p_F}{m_e}=\frac{\bar{M}^{\frac{1}{3}}}{\bar{R}}\left(1
-\frac{\theta}{4}\frac{m_e}{E_p}\frac{\bar{M}^{\frac{1}{3}}}{\bar{R}}\right)
~.\label{xf}
\end{eqnarray}

These observations allow us to derive
a form of the pressure-balance equation
which takes into account
 both the modification of the dispersion relation and the modification
of  the law of composition of momenta
\begin{eqnarray}
&K\frac{\bar{M}^{4/3}}{\bar{R}^4}\left(1
-\theta\frac{m_e}{E_p}\frac{\bar{M}^{\frac{1}{3}}}{\bar{R}}\right)
-K\frac{\bar{M}^{2/3}}{\bar{R}^2}\left(1
-\frac{\theta}{2}\frac{m_e}{E_p}\frac{\bar{M}^{\frac{1}{3}}}{\bar{R}}\right)+&\nonumber\\
&+\frac{8}{5}(\theta+\frac{\eta}{2})\frac{m_e}{E_p}K\frac{\bar{M}^{5/3}}{\bar{R}^5}
=K'\frac{\bar{M}^2}{\bar{R}^4} ~, &\label{equazdeform}
\end{eqnarray}
where on the left-hand side we used Eq.~(\ref{equazstatoB}),
while the right-hand side describes the pressure due to the Newtonian potential,
as done in Eq.~(\ref{Pgrav}).

From Eq.~(\ref{equazdeform})
one straightforwardly obtains (also see analogous derivation in Section~II)
a relationship between radius and mass of the white dwarf, which is most
insightfully described in terms of a correction to the analogous relationship
that holds in the standard Chandrasekhar model:
\begin{eqnarray}
R=R_{Chan}\left(1-\left(\frac{13~\theta
+ 4~\eta}{10} \right)\frac{m_e}{E_p}\frac{1}{\left(1
-(\frac{M}{M_0})^{\frac{2}{3}}\right)^{\frac{3}{2}}}\right).\label{Camachos2}
\end{eqnarray}
This result, which characterizes our variant of the Chandrasekhar model
in the ultrarelativistic regime,
exactly reproduces Camacho's formula (\ref{Camachos})
upon taking $(13/4~\theta + \eta) \rightarrow \eta$.
The introduction of a deformed measure of integration, required in frameworks
with a modified law of composition of momenta, has not produced stronger
or different effects, but it is noteworthy that the effects due to the deformed measure
of integration, with parameter $\theta$, are exactly of the same order of magnitude
of the ones induced by the modification of the dispersion relation.
This implies that in particular frameworks with only a modified law of composition of
momenta (but no modification of the dispersion relation) would still predict
exactly the same features. And on the other hand in frameworks with both
a modified dispersion relation and a modified law of composition of momenta
one cannot a priori exclude a cancellation of the main Planck-scale effect
(the cancellation would occur if $\theta = - 4/13~\eta$).

In closing this section we observe that
by casting the result in the form (\ref{Camachos2})
(which we here adopted for allowing quick comparison
to the results of Ref.~\cite{Camacho})
some of the implications might be overlooked.
The form of Eq.~(\ref{Camachos2}) underlines that the effects
become significant when the mass of the white dwarf gets very close to
the Chandrasekhar mass, and this is a characterization that
provides little intuition for the type of features that should be sought in order
to find an amplification of Planck-scale effects in contexts that
are different from the one of the study of white dwarfs.
We notice however that Eq.~(\ref{Camachos2}) can be rewritten
equivalently as follows
\begin{eqnarray}
R=R_{Chan}\left(1-\frac{3}{2}\left(\frac{13\theta+4\eta}{10}\right)\frac{\pi^2}{m_p
m_e^2 E_p}\rho_{Chan}\right) \, ,   \label{Camachos3}
\end{eqnarray}
where $\rho_{Chan}=M/(4\pi R_{Chan}^3/3)$, and this provides the intuition that the effects are amplified
in presence of ultra-high densities ($ \rho_{Chan}\sim 2m_p m_e^2 E_p/(3\pi^2)$).
There is a certain ``quantum-gravity folklore" assuming that strong effects
could only erise for systems of Planck-length size, but this result
(in spite of its ``mere academic" nature) provides encouragement for a new intuition
according to which one could look for systems of extremely high density
(perhaps very-unusually high density) but not necessarily Planck-length size.

\section{Nonrelativistic limit}
The Planck-scale effects that can be introduced in the ultrarelativistic
limit of the Chandrasekhar model are conceptually very significant, because
they provide a rather remarkable example of amplification of Planck-scale effects,
and particularly because this could encountered in the description of a
macroscopic system (in admittedly ``ideal" conditions of ultra-high densities).
However, while it is indeed intriguing conceptually,
in the specific context of the Chandrasekhar model these results
should at present be viewed as of merely academic interest, since the
effects of the Planck-scale corrections become significant only for very small
white-dwarf radius, and for real stars of such small radius and
high densities the Chandrasekhar model is completely inapplicable,
particularly  as a result
of the fact that it ignores the implications of weak interactions
(which are definitely not negligible in those regimes~\cite{weakIMPORTANT}).

As mentioned, as a possible opportunity of
 realistic application of Planck-scale effects for the Chandrasekhar model
 Camacho~\cite{Camacho} considered two white dwarfs,
 G156-64 and Wolf 485 A,
but, as we already stressed above, the ultrarelativistic limit
on which Camacho focused is not relevant for these white dwarfs
since~\cite{NaneStrane} $x_F\simeq 0.68$ for G156-64
and $x_F\simeq 0.50$ for Wolf 485 A.
And actually
the majority of
white dwarf stars are in the non relativistic $x_F\lesssim 1$
regime
(the biggest $x_F$ is the one of Syrius B which is $x_F\simeq 1.08$).

These observations motivate us to further extend the analysis proposed by
Camacho (to which we already added
the feature of a possible modification of the law of composition of momenta)
by considering also the nonrelativistic regime.
For this we must first establish
the relationship between $x_F$, $\bar{M}$
and $\bar{R}$ in the nonrelativistic  limit.
From Eq.~(\ref{defpieffe}), assuming indeed $x_F \ll 1$,
one finds
\begin{equation}
p_F=\left(\frac{3}{4}\frac{N}{V}(2\pi)^2\right)^{\frac{1}{3}}\left(1
-\frac{1}{5}\frac{\theta}{E_p}\frac{(\frac{3}{4}\frac{N}{V}(2\pi)^2)^{\frac{2}{3}}}{2
m}\right),\label{pfnonrel}
\end{equation}
which leads to
\begin{equation}
x_F=\frac{\bar{M}^{\frac{1}{3}}}{\bar{R}}\left(1
-\frac{1}{10}\frac{m_e}{E_p}\frac{\bar{M}^{\frac{2}{3}}}{\bar{R}^2}\right)~.
\end{equation}
This must be taken into account in the analysis of the
 zero-point energy, which
 is given by
\begin{equation}
P_0 =-\frac{\partial E_0}{\partial V} =\left(\frac{m_e^4}{15\pi^2}\right)\left(x_F^5
+(\eta+\theta)\frac{m_e}{E_p}x_F^5\right)~,\label{pizerononrelcorr}
\end{equation}
as easily found assuming $x_F \ll 1$ in Eq.~(\ref{Ezerocorr}).

We therefore find, combining this result (\ref{pizerononrelcorr})
with Eq.~(\ref{Pgrav}), that the Planck-scale-modified
 pressure-balance equation in the nonrelativistic limit takes the form
\begin{eqnarray}
\frac{4}{5}K\frac{\bar{M}^{\frac{5}{3}}}{\bar{R}^5}\left(1
+(\theta+\eta)\frac{m_e}{E_p}\right)=K'\frac{\bar{M}^2}{\bar{R}^4},
\end{eqnarray}
which can be equivalently described as the following relationship
between mass and radius of the star:
\begin{eqnarray}
\bar{R}=\frac{4}{5}\frac{K}{K'}\bar{M}^{-\frac{1}{3}}\left(1+(\theta+\eta)\frac{m_e}{E_p}\right)~.
\end{eqnarray}

 This result excludes the possibility of any meaningful phenomenological studies
of these effects in the nonrelativistic,  $x_F \ll 1$, regime,
since we are finding that the correction to the mass-radius formula is purely governed
by the ratio $m_e/E_p$ ($\sim 10^{-22}$).


\section{Deformed Tolman-Oppenheimer-Volkoff equations}
Our main conclusion is that the picture proposed by Camacho
is probably not relevant for the phenomenology of white dwarfs
(which appeared to be camacho's original motivation), but might
carry a rather significant message from the conceptual perspective,
since it shows, within a sort of ``toy model", that there is some chance
of finding macroscopic manifestations of Planck-scale effects.
It appears likely that future searches of more realistic descriptions
of macroscopic Planck-scale effects would have to take into account
general relativistic effects, rather than be confined (like Camacho's and
Chandrasekhar's analyses) to the Newtonian limit.
As a first contribution in this direction in this section we offer
a preliminary characterization of how the class of Planck-scale effects
we are considering would affect
the Tolman-Oppenheimer-Volkoff (TOV) equations~\cite{TOV1,TOV2}, which
provide a way for introducing general-relativsitic
effects in the Chandrasekhar picture.

The starting point for our considerations
are the TOV equations\footnote{In this section we also set $G=1$}:
\begin{eqnarray}
\begin{cases}
\frac{dm}{dr}=4\pi\epsilon(P)r^2\\
\frac{dP}{dr}=-\frac{P+\epsilon(P)}{r(r-2m)}(4\pi Pr^3+m(r))
~.\end{cases}
\end{eqnarray}
Here $r$ is the distance from the center of the star, and the equations
simply give the general-relativistic description of the radial pressure gradient,
under the assumption of spherically symmetric mass distribution (so that $M = \int_0^R m(r) dr=\int_0^R
4 \pi r^2 \epsilon(r) dr$).

These TOV equations are solvable upon providing an equation of state, giving $\epsilon(P)$.
we shall of course be interested in the outcome of these TOV equations for the case
in which the equation of state is obtained from (\ref{equazstatoA})-(\ref{equazstatoB}),
so that it is an equation of state that takes into account our Planck-scale effects.

On the basis of the observations reported in the previous sections we expect
the Planck-scale effects to be significant only in the ultrarelativistic regime,
and we therefore focus on solving\footnote{For $x_F \gg 1$ solving the TOV equations
is of little interest from a strict astrophysics perspective because of stability
concerns~\cite{Landau}.
From our perspective the analysis of the regime $x_F \gg 1$ of the TOV equations
is still meaningful, since we are here only interested in
estimating the possible magnitude of the effects.} the TOV equations for $x_F \gg 1$.
For this reason we will solve TOV equations in the
ultrarelativistic regime, in which it is convenient
to make use of the variable
\begin{equation}
t=4\ln\left(x_F+\sqrt{1+x_F^2}\right)~,
\end{equation}
also taking into account that, as stressed in Section IV,
our Planck-scale effects of modification of the law of composition of momenta
also introduce Planck-scale corrections, described in Eq.~(\ref{xf}),
for the formula that gives $x_F$ (whereas $x_F$ would not be affected by any Planck-scale
effects in scenarios that exclusively modify the dispersion relation).

In the ultrarelativistic regime ($x_F \gg 1$ and, as a result, also $t \gg 1$)
the TOV equations, if one indeed assumes that the Planck-scale-corrected $\epsilon(P)$
is obtained from (\ref{equazstatoA})-(\ref{equazstatoB}), take the form\footnote{For compactness,
in this section we work in units such that $K=2/(3\pi)$, which are not unusual in studies
of TOV equations. For a neutron gas
 this means that the unit of lenght has been fixed to
 be $l=1.36 \cdotp 10^4 m$, while the unit of mass
 is $\mu=1.83\cdotp 10^{31} Kg$.}
\begin{eqnarray}
\begin{cases}
\frac{dm}{dr}=\frac{1}{2}r^2e^t+\frac{1}{2}r^2(\frac{3}{2}\theta+\eta)\frac{m_f}{E_p}e^{\frac{5}{4}t}\\
\\
\frac{dt}{dr}=-\frac{4}{r(r-2m(r))}\left(\frac{r^3}{6}e^t+m(r)\right)
-\frac{r^3}{r(r-2m(r))}(\frac{23}{9}\theta+\frac{7}{4} \eta)\frac{m_f}{E_p}e^{\frac{5}{4}t}
\end{cases} \label{sistema3}
\end{eqnarray}

From these combined equations one can straightforwardly obtain
the explicit form of the $r$-dependence of $m$ and of (the exponential of) $t$
at leading order in $m_f/E_p$:
\begin{eqnarray}
e^{t(r)} &=&\frac{3}{7}\frac{1}{r^2}
+\frac{m_f}{E_p}\left(\frac{3}{7}\right)^{\frac{1}{4}}\left(\frac{884}{441}\theta
+\frac{271}{196}\eta\right)r^{-\frac{5}{2}} ~, \label{soluzione1}\\
m(r)&=&\frac{3}{14}r
+\frac{m_f}{E_p}\left(\frac{3}{7}\right)^{\frac{1}{4}}\left(\frac{2335}{882}\theta
+\frac{335}{196}\eta\right)r^{\frac{1}{2}} ~.\label{soluzione2}
\end{eqnarray}
These results can in turn be used
to establish the relation between mass and energy density, which is given by
\begin{equation}
m(\epsilon)\simeq \left(\frac{3}{14}\right)^{\frac{3}{2}}\epsilon_0^{-\frac{1}{2}}\left(1
+\frac{m_f}{E_p}f(\theta,\eta)\epsilon_0^{\frac{1}{4}}\right)~,\label{emmediro2}
\end{equation}
where $\epsilon_0$ is the central density of the star
 and  $f(\theta,\eta)=\frac{7}{3}\left(\frac{2777}{441}\theta+\frac{1611}{392}\eta\right)
+2^{\frac{1}{4}}(\frac{3}{2}\theta+\eta).$

It is noteworthy that our result (\ref{emmediro2}) shows that the Planck-scale effects
can be significant for ultra-high densities ($\epsilon_0 \sim (E_p/m_f)^4$),
without necessarily requiring the abstraction of a star of Planck-length size.
We are therefore encountering once again the characteristics already discussed
for our analysis of the modified Chandrasekhar model.

\section{Closing remarks}
The fact, established over the last decade, that there are some (however rare)
windows on the Planck-scale realm represents a, previously unexpected, significant
opportunity for quantum-gravity research. Now that we have the first
few examples~\cite{gacPolonpapLRR}
of phenomenological
analyses establishing access to effects introduced  genuinely at the Planck scale
we have in some sense the luxury of starting to worry about the fact that
these first few examples only probe certain specific aspects of the Planck-scale realm,
and look for ways to achieve a broader ability to investigate this realm.
The study we here reported, taking off from related previous analysese reported in
Refs.~\cite{Camacho,sethNANE}, starts to set the stage for new ways to probe
the Planck-scale realm in the context of macroscopic systems in astrophysics.

We unfortunately found that it is unlikely that, as
originally hoped by Camacho~\cite{Camacho}, near-term observations
of white dwarfs might play a role in the phenomenology of certain Planck-scale effects.
Yet the result of our analysis provides encouragement for the idea that
there is indeed a new regime of phenomena were tiny Planck-scale effects could
be observably large, and it is the regime of extremely high densities.
The densities that appear to be required for observably large Planck-scale effects
are ultra-high, and it might require a dedicated multi-stage research programme
to identify the most promising applications that might allow to
unveil such effects. And such a programme will need to face the challenges posed
by the still only partial ordinary-physics understanding of some of the
most interesting macroscopic systems in astrophysics, particularly when gravitational
collapse (and associated high densities) is involved. But the payoff that could expected
appears to be well worth the effort, since such a novel window on the Planck-scale
realm could have particularly significant impact on our ability to investigate the
quantum-gravity problem.

Also for what concerns the more academic/conceptual side of the issues here discussed
these studies should motivate further investigation, particularly for what concerns the
discrimination between scenarios with broken spacetime symmetries
and scenarios with deformed spacetime symmetries in the DSR sense.
As discussed in Section III through refinements of the analysis here reported
it should be possible to gain further insight on the type of observables that
can be used to discriminate between these two alternative scenarios for the fate of spacetime
symmetries at the Planck scale.

