Constraints between equations of state and mass-radius relations in general clusters of stellar systems

Abstract

We prove three obstruction results on the existence of equations of state in clusters of stellar systems fulfilling mass-radius relations and some additional bound (on the mass, on the radius or a causal bound). The theorems are proved in great generality. We start with a motivating example of TOV systems and apply our results to stellar systems arising from experimental data.

Existence and classification of pseudo-asymptotic solutions for Tolman–Oppenheimer–Volkoff systems

“…In the next section we will analyze the topology of the surface S c corresponding to rows 1, 2 and 7 of Table (1), showing that they admit one, two and one topology changes, respectively. This means that Theorem 1.1 and Corollary 1 can be improved, giving the theorem below.…”

“…This means asking if for every given k, l there exists at least one nonzero pair (p, ρ) for which equation (1) is satisfied when assuming (2). Before answering this question, notice that by means of isolating ρ in (2) and substituting the expression found in (1) we see that the TOV equation is of Riccati type [6,7]: p (r) = A(r) + B(r)p(r) + C(r)p(r) 2 (3)…”

“…Since the seminal work of Chandrasekhar, the axiomatization problem of astrophysics has been neglected. In [1] the authors reintroduced the problem, showing that arbitrary clusters of stellar systems experience fundamental constraints. In this paper we continue this work, focusing on a specific class of stellar systems: the TOV systems.…”

“…In order to state precisely the question being considered and our main results, we recall some definitions presented in [1]. A stellar system of degree (k, l) is a pair (p, ρ) of real functions, with p piecewise C k -differentiable and ρ piecewise C l -differentiable, both defined in some union of intervals I ⊂ R, possibly unbounded.…”

“…In a general cluster Stellar k,l,m I we define a continuity equation as an initial value problem M = F (M, ρ, r) in Stellar l,m I × I with initial condition M (a) = M a . Since the vector space of piecewise differentiable functions is locally convex but generally neither Banach nor Fréchet [1,3], the problem of general existence of solutions for continuity equations is much more delicate [4,5]. Therefore one generally works with continuity equations which are integrable.…”