[Paper Review] Rapid differential rotation of protoneutron stars and constraints on radio pulsars periods

J. O. Goussard, P. Haensel+1 more|ArXiv.org|Nov 28, 1997

Pulsars and Gravitational Waves Research6 references16 citations

TL;DR

This paper models differentially rotating protoneutron stars using general relativistic equations and spectral methods, showing that thermal structure significantly affects maximum angular momentum. For a 1.5 M⊙ protoneutron star with a shocked high-entropy envelope, the minimum possible period of the resulting cold neutron star is ≈1.7 ms, implying millisecond pulsars likely require accretion spin-up rather than direct formation.

ABSTRACT

Models of differentially rotating protoneutron stars are calculated, using realistic equations of state of dense hot matter. Various conditions within the stellar interior, corresponding to different stages of protoneutron star evolution, are considered. Families of differentially rotating models of a given baryon mass are calculated, using a two-parameter formula describing the angular velocity profile within a rotating protoneutron star. Maximum angular momentum, which can be accommodated by a protoneutron star is calculated, for various thermal conditions in stellar interior, for a baryon mass of $1.5 M_\odot$. In the case of a thermally homogeneous (isentropic or isothermal) neutrino-opaque interior this maximum angular momentum turns out to be somewhat higher than that of a cold neutron star of the same baryon mass, rotating uniformly at the mass shedding angular velocity. However, if the protoneutron star has a thermal structure characteristic of initial state, with a low entropy (unshocked) core, and a high entropy (shocked) outer half of baryon mass, the maximum angular momentum is significantly lower. This leads to a minimum period of uniform rotation of cold neutron stars of baryon mass $\sim 1.5~M_\odot$, formed directly (i.e. without a subsequent significant accretion of mass) from protoneutron stars with shocked envelope, of about 1.7 ms and strengthens the hypothesis that millisecond pulsars are accretion accelerated neutron stars.

Motivation & Objective

• To investigate how differential rotation and thermal structure in protoneutron stars affect their maximum angular momentum capacity.
• To determine the minimum possible spin period of cold neutron stars formed from protoneutron stars with shocked envelopes.
• To assess whether direct formation of millisecond pulsars is possible without accretion, based on protoneutron star evolution.
• To compare angular momentum limits in differentially rotating protoneutron stars with those of uniformly rotating cold neutron stars.
• To evaluate the role of thermal structure—particularly low-entropy core and high-entropy envelope—on rotational constraints.

Proposed method

• Numerical models of differentially rotating protoneutron stars are computed using general relativistic equations for stationary, axially symmetric equilibrium.
• A two-parameter formula describes the angular velocity profile within the star, allowing systematic exploration of differential rotation.
• Spectral methods are employed in a numerical code to solve the relativistic equations of motion under the barotropic approximation.
• Models are calculated for a fixed baryon mass of 1.5 M⊙ under two thermal scenarios: shocked envelope (high-entropy outer half) and isentropic (mixed) interior.
• The mass shedding limit and a newly defined 'minimal mass limit' are used to constrain stable configurations.
• Angular momentum is calculated for each configuration, comparing results to the maximum angular momentum of uniformly rotating cold neutron stars.

Experimental results

Research questions

• RQ1 What is the maximum angular momentum a differentially rotating protoneutron star can sustain, given realistic equations of state and thermal conditions?
• RQ2 How does the presence of a shocked high-entropy envelope affect the minimum possible spin period of the resulting cold neutron star?
• RQ3 Can protoneutron stars with a shocked envelope produce cold neutron stars with periods shorter than 1.7 ms without accretion?
• RQ4 How does differential rotation in protoneutron stars compare to uniform rotation in terms of angular momentum capacity?
• RQ5 What constraints does the thermal structure of the protoneutron star interior place on the final spin period of the neutron star?

Key findings

• For a 1.5 M⊙ protoneutron star with a shocked high-entropy envelope, the minimum possible period of the resulting cold neutron star is approximately 1.7 ms.
• This minimum period is significantly longer than the periods observed in the fastest known millisecond pulsars, such as PSR 1937+21 and PSR 0034-0534.
• Protoneutron stars with a shocked envelope have a much lower maximum angular momentum capacity than those with isentropic interiors, due to thermal structure effects.
• In contrast, protoneutron stars with isentropic or isothermal interiors can sustain slightly higher angular momentum than cold uniformly rotating neutron stars of the same mass.
• The results imply that direct formation of millisecond pulsars from protoneutron stars is unlikely without significant accretion, supporting the accretion spin-up hypothesis.
• The study introduces a 'minimal mass limit' for differentially rotating protoneutron stars, arising from structural constraints in the shocked envelope scenario.