Hydrostatic equilibrium andstellar structure in f(R)-gravity	  Salvatore Capozziello		

Università di Napoli “Federico II”
and 	INFN Sez. di Napoli		

In collaboration with	M. De Laurentis, S.D. Odintsov, A. Stabile	

PRD 83, 064004 (2011)	

PRD 85, 044022 (2012)	

KMI, Nagoya, August 3, 2013	Outlines	

Hydrostatic equilibrium of stellar structures	

The Newtonian limit of f(R)-gravity	

Stellar hydrostatic equilibrium in f(R)-gravity	

Solution of the standard and modiﬁed Lanè-Emden equations	Discussion and conclusions	

Next steps	Setting the problem	• Several open questions in modern Astrophysics ask for newparadigms.	

• No evidence of Dark Energy and Dark Matter at fundamental
level (LHC, astroparticle physics, ground based experiments…).	

• Such problems could be framed extending GR at infrared scales	• GR does not work at ultraviolet scales (no quantum gravity theory
up to now).	

• f(R)-gravity as minimal extension but other modiﬁcations arepossible.	

• Several stellar structures cannot be addressed by the standard
theory of stellar evolution (magnetars, variable stars, etc..)	

• Big issue: Is it possible to revise stellar theory in view of extendedgravity?	

Hydrostatic equilibrium of stellar structures	

The condition of hydrostatic equilibrium in
Newtonian dynamics is	

² p is the pressure,	² Φ is the gravitational potential,	² ρ is the density	

The Poisson equation	

Since we are taking into account only static and stationary
situations, here we consider only time independent solutions	

In general, the temperature τ appears and the density ρ satisﬁes an	

equation of state of the form 	

R. Kippenhahn and A. Weigert, Stellar Structures and Evolution (Springer-Verlag, Berlin, 1990).	

Hydrostatic equilibrium of stellar structures	

We assume that there exists a polytropic relation
between p and ρ of the form	

K is the polytropic constant that can be obtained as a combination of
fundamental constants	

The constant ϒ is the polytropic exponent.	

Note that Φ > 0 is in the interior of the model, since we deﬁne thegravitational potential as -Φ	

Inserting the polytropic equation of state, we obtain	Hydrostatic equilibrium of stellar structures	For γ≠ 1, the above equation can be integrated
giving	we have chosen the integration constant to give Φ = 0 at surface ( ρ = 0)	  Is the polytropic index	

Inserting the above relation into the Poisson equation, we obtain adiﬀerential equation for the gravitational potential	

Hydrostatic equilibrium of stellar structures	

Let us deﬁne now the dimensionless variables:	

where the subscript c refers to the center of the star and the relation between ρand Φ	

At the center (r = 0), we have z = 0, Φ = Φc, , ρ = ρc and therefore w = 1	Then we obtain the standard Lané-Emden equation describing	

the hydrostatic equilibrium of stellar structures in the Newtonian 	theory	

The Newtonian limit of f(R) - gravity	Let us start with a general class of Extended Theoriesof Gravity (ETG) given by the action	

varying the action with respect to the metric we obtainthe ﬁeld equations	

S. Capozziello, M. De Laurentis Phys. Rep. 509, 167-321 (2011)	

S. Capozziello , M. Francaviglia, Gen. Relativ. Gravit. 40, 357 (2007)	

The Newtonian limit of f(R) - gravity	

In order to achieve the Newtonian limit of the theory
the metric tensor has to be approximated as follows:	

The Ricci scalar formally becomes	

The n-th derivative of Ricci function can be developedas	

here R.n denotes a quantity of order O(n)	

S. Capozziello, A. Stabile, and A. Troisi, Phys. Rev. D 76, 104019 (2007)	

The Newtonian limit of f(R) - gravity	

Field equations at O (2)-order, that is at the
Newtonian level, are	

Δ is the Laplacian in the ﬂat space Rtt = Δ Φ and, for thesake of simplicity, we set f’ (R)_0 = 1 	

We recall that the energy-momentum tensor for aperfect ﬂuid is		  p is the pressure and is the energy density	The Newtonian limit of f(R) - gravity	

Being the pressure contribution negligible in the ﬁeldequations in the Newtonian approximation, we have	

modiﬁed Poisson equation	

ρ is now the mass density	

For f’’(R) = 0 we have the standard Poisson equation	

This means that as soon as the second derivative of f (R) is
diﬀerent from zero, deviations from the Newtonian limit of GRemerge	

Stellar hydrostatic equilibrium in f(R) - gravity	

From the Bianchi identities we have	

If the dependence on the temperature is negligible, thisrelation can be introduced into ﬁeld equations, whichbecomes a system of three equations for p, Φ and	R.(2) and can be solved without the other structure	

equations.	

Let us suppose that matter still satisﬁes a polytropicequation	

Stellar hydrostatic equilibrium in f(R) - gravity	

we obtain an integral-diﬀerential equation for thegravitational potential, that is	is the Green function	

Stellar hydrostatic equilibrium in f(R) - gravity	

Adopting again the dimensionless variables	

is a characteristic length linked 	

to stellar radius ξ	The Lanè-Emden in f(R)-gravity becomes	Solutions of the standard andmodiﬁed Lanè-Emden equations	The task is now to solve the modiﬁed 	

Lane´-Emden equation and compare	

its solutions to those of the standard Newtonian theory	

Only for three values of n, the classical solutions have analyticalexpression	Solutions of the standard andmodiﬁed Lanè-Emden equations	The surface of the polytrope of	index n is deﬁned by the value z = z(n). whereρ = 0 and thus w = 0	For n = 0 and n = 1 the surface is reached for a ﬁnite value of z = z(n)	

The case n = 5 yields a model of inﬁnite radius	

It can be shown that for n<5 the radius of polytropic models is ﬁnite; for n>5
they have inﬁnite radius	One ﬁnds z(0)GR = √ 6 , z(1)GR = π, z(5)GR = ∞ 	

A general property of the solutions is that z(n) grows monotonically with thepolytropic index n	Solutions of the standard andmodiﬁed Lanè-Emden equations	

Apart from the three cases where analytic 	

solutions are known, the classical 	Lane´-Emden has to be solved numerically, considered with the expression for
the neighborhood of the center	

at lowest orders, a classiﬁcation of solutions by the index n, that is	The case ϒ=5/3 and n= 3/2 is the nonrelativistic limit while the case γ=4/3
and n = 3 is the relativistic limit of a completely degenerate gas.	

Solutions of the standard andmodiﬁed Lanè-Emden equations	

For the modiﬁed Lane´-Emden, we have anexact solution for n = 0, in fact	

where the boundary conditions w(0) = 1 and w’(0) = 0 are satisﬁed	

A comment on the GR limit (that is f(R.) à R) of above solution is necessary.	

In fact, when we perform the limit m à ∞ we do not recover exactly w(0)GR (z).	

The diﬀerence is in the deﬁnition of quantity ξ0	

In GR it is	Solutions of the standard andmodiﬁed Lanè-Emden equations	The point z(0)f(R) is calculated by imposing	w(0)f(R)(z(0)f(R) ) =0 and by considering theTaylor expansion	

we obtain	

Since the stellar radius ξ is given by deﬁnition ξ =ξ0 z(0)f(R) we obtain	

By solving numerically the constraint, we ﬁnd the modiﬁed expression of the radius	

If m à ∞ we have the standard expression valid for the Newtonian limit of GR	Solutions of the standard andmodiﬁed Lanè-Emden equations	In the f(R)-gravity case, for n=0, the radius issmaller than in GR	

In the case n= 1 we obtain	

If we perturb this equations we have	The coeﬃcient e-mξ < 1 is the parameter with respect to which we perturb	

And then	Solutions of the standard andmodiﬁed Lanè-Emden equations	And the solutions is easily found to be	Also in this case, for mà∞, we do not recover exactly w(1)GR(z)	

The reason is the same of the previous n = 0 case	Analytical solutions for other values of n are not available	Solutions of the standard andmodiﬁed Lanè-Emden equations	Finally we report the gravitational	

potential proﬁle generated by a sphericallysymmetric source of uniform mass with radius ξ	We can impose a mass density of the form		  	  Θ is the Heaviside function and M is the mass	

By solving ﬁeld equations inside the star and considering the boundary conditionsw.(o) = 1 and w’(0)=0, we get	In the limit mà∞ we recover the GR case	Solutions of the standard andmodiﬁed Lanè-Emden equations	

Plot of solutions (blue lines) of standard	Lane´-Emden: w(0)GR(z) (dotted line)
and w(1)GR(z) (dashed line). The green
line corresponds to w(5)GR(z)	

The red lines are the solutions ofmodiﬁed Lane´-Emden: w(0)f(R)(z)(dotted line) and w(1)f(R)(z) (dashed line).	

The blue dashed-dotted line is the potential derived from GR wGR(z) and the red dasheddotted line is the potential derived from f(R) gravity for a uniform spherically symmetricmass distribution	

From a rapid inspection of these plots, the diﬀerences between GR and f(R) gravitational
potentials are clear and the tendency is that at larger radius z they become more evident.	

Discussion and Conclusions	The hydrostatic equilibrium of a stellar structure in the framework of f (R)gravity has been considered.	

Adopting a polytropic equation of state relating the mass density to thepressure, we derive the modiﬁed Lane´-Emden equation and its solutions for
n = 0,1 which can be compared to the analogous solutions coming from theNewtonian limit of GR	

When we consider the limit f(R)àR, we obtain the standard hydrostaticequilibrium theory coming from GR	A peculiarity of f(R) gravity is the nonviability of the Gauss theorem, andthen the modiﬁed Lane´-Emden equation is an integro-diﬀerential equation
where the mass distribution plays a crucial role	

The correlation between two points in the star is given by a Yukawa-liketerm of the corresponding Green function	

Discussion and Conclusions	

These solutions have been matched with those coming from GR and thecorresponding density radial proﬁles have been derived	

In the case n = 0, we ﬁnd an exact solution, while, for n= 1, we used aperturbative analysis with respect to the solution coming from GR	

It is possible to demonstrate that density radial proﬁles coming from f (R)gravity analytic models and close to those coming from GR are compatible	

This result rules out some wrong claims in the literature stating that f(R)gravity is not compatible with self-gravitating systems	Next Steps	

The next step is to derive self-consistent numerical solutions of the modiﬁed
Lane´-Emden equation and build up realistic star models where further
values of the polytropic index n and other physical parameters, e.g.	

temperature, opacity, transport of energy, are considered.	

These models are a challenging task, since, up to now, there is no self-consistent,
ﬁnal explanation for compact objects (e.g. neutron stars) with masses larger
than Volkoﬀ mass, while observational evidence widely indicates these objects.
Another important issue is that such an approach could allow to address
dynamics of systems like Wolf-Rayet stars, magnetars and oscillating stars.	

Work in progress…see Mariafelicia Talk!