Hydrostatic equilibrium and stellar structure in $f(R)$ gravity

We investigate the hydrostatic equilibrium of stellar structure by taking into account the modified Lané-Emden equation coming out from $f(R)$ gravity. Such an equation is obtained in a metric approach by considering the Newtonian limit of $f(R)$ gravity, which gives rise to a modified Poisson equation, and then introducing a relation between pressure and density with polytropic index $n$. The modified equation results an integro-differential equation, which, in the limit $f(R)\to R$, becomes the standard Lané-Emden equation. We find the radial profiles of the gravitational potential by solving for some values of $n$. The comparison of solutions with those coming from general relativity shows that they are compatible and physically relevant.

Figure 1

Plot of solutions (blue lines) of standard Lané-Emden Eq. (8): $w_{\mathrm{GR}}^{(0)}(z)$ (dotted line) and $w_{\mathrm{GR}}^{(1)}(z)$ (dashed line). The green line corresponds to $w_{\mathrm{GR}}^{(5)}(z)$. The red lines are the solutions of modified Lané-Emden Eq. (21): $w_{f(R)}^{(0)}(z)$ (dotted line) and $w_{f(R)}^{(1)}(z)$ (dashed line). The blue dashed-dotted line is the potential derived from GR ($w_{\mathrm{GR}}(z)$) and the red dashed-dotted line is the potential derived from $f(R)$ gravity ($w_{f(R)}(z)$) for a uniform spherically symmetric mass distribution. The assumed values are $m\xi =1$ and $m{\xi }_0=.4$. 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.