Maximal Masses of White Dwarfs for Polytropes in $R^2$ Gravity and Theoretical Constraints

We examine the Chandrasekhar limit for white dwarfs in $f(R)$ gravity, with a simple polytropic equation of state describing stellar matter. We use the most popular $f(R)$ gravity model, namely the $f(R)=R+\alpha R^2$ gravity, and calculate the parameters of the stellar configurations with polytropic equation of state of the form $p=K\rho^{1+1/n}$ for various values of the parameter $n$. In order to simplify our analysis we use the equivalent Einstein frame form of $R^2$-gravity which is basically a scalar-tensor theory with well-known potential for the scalar field. In this description one can use simple approximations for the scalar field $\phi$ leaving only the potential term for it. Our analysis indicates that for the non-relativistic case with $n=3/2$, discrepancies between the $R^2$-gravity and General Relativity can appear only when the parameter $\alpha$ of the $R^2$ term, takes values close to maximal limit derived from the binary pulsar data namely $\alpha_{max}=5\times 10^{15}$ cm$^2$. Thus, the study of low-mass white dwarfs can hardly give restrictions on the parameter $\alpha$. For relativistic polytropes with $n=3$ we found that Chandrasekhar limit can in principle change for smaller $\alpha$ values. The main conclusion from our calculations is the existence of white dwarfs with large masses $\sim 1.33 M_{\odot}$, which can impose more strict limits on the parameter $\alpha$ for the $R^2$ gravity model. Specifically, our estimations on the parameter $\alpha$ of the $R^2$ model is $\alpha \sim 10^{13}$ cm$^2$.