[Paper Review] Dissipation of radial oscillations in compact stars

Basil A. Sa’d, Jürgen Schaffner–Bielich|arXiv (Cornell University)|Aug 28, 2009

Pulsars and Gravitational Waves Research3 citations

TL;DR

This paper introduces a new energy dissipation mechanism in compact stars, identifying radiative viscosity as a significant channel for damping radial oscillations, where mechanical energy is radiated away via neutrinos rather than converted to heat. For non-strange quark matter and nuclear matter, the radiative viscosity is found to be 1.5 times larger than bulk viscosity, offering a dominant damping effect in certain astrophysical scenarios.

ABSTRACT

We demonstrate that there exists a new mechanism for dissipating the energy of stellar oscillations. For neutron stars, in particular, we show that the mechanical energy of density perturbations is not only dissipated to heat via bulk viscosity, but also that energy is radiated away via neutrinos. This energy dissipation will be associated with a viscosity coefficient, the radiative viscosity, which is larger than the bulk viscosity in the case of non-strange quark matter and nuclear matter.

Motivation & Objective

• To investigate alternative energy dissipation mechanisms for radial oscillations in compact stars beyond conventional bulk viscosity.
• To analyze how weak interactions (urca processes) in dense quark and nuclear matter contribute to energy loss via neutrino emission during density oscillations.
• To quantify and define a new viscosity coefficient—radiative viscosity—associated with energy radiated as neutrinos.
• To compare the magnitude of radiative viscosity with bulk viscosity in non-strange quark matter and nuclear matter.
• To assess the astrophysical relevance of this mechanism for damping oscillations in neutron stars, particularly in the context of r-mode instabilities and phase transitions.

Proposed method

• Model radial density oscillations in compact stars using a periodic perturbation: $ n(t) = n_0 + \text{Re}(\delta n e^{i\omega t}) $, leading to deviations from $\beta$-equilibrium.
• Formulate the departure from $\beta$-equilibrium via $\delta\mu = \mu_d - \mu_u - \mu_e $, expressed as $ \delta\mu = C \frac{\delta n}{n} + B \delta X_e $, with coefficients $C$ and $B$ derived from thermodynamic derivatives.
• Use the urca process rates $ \Gamma_\nu - \Gamma_{\bar{\nu}} = -\lambda \delta\mu $ to describe the time evolution of electron fraction $ X_e $, leading to $ \frac{d(\delta X_e)}{dt} = \frac{\lambda}{n} \delta\mu $.
• Compute the energy loss rate via neutrino and antineutrino emission using the matrix element formalism and phase-space integrals, resulting in $ \dot{E} = \frac{4\alpha_s}{\pi^6} G_F^2 \cos^2\theta_C \mu_d \mu_u \mu_e T^6 \left[ \frac{457\pi^6}{2520} + \frac{17\pi^4}{40} \left( \frac{\delta\mu}{T} \right)^2 \right] $.
• Define the radiative viscosity coefficient $ \eta_{\text{RV}} $ by relating the energy loss rate to the oscillation amplitude, analogous to bulk viscosity, and derive the ratio $ \frac{\mathcal{S}}{\lambda} = \frac{3}{2} $, indicating 50% enhancement over bulk viscosity.
• Apply the formalism to two-flavor quark matter and nuclear matter, confirming the generality of the result across different dense matter phases.

Experimental results

Research questions

• RQ1 Can energy from radial density oscillations in compact stars be dissipated not only via bulk viscosity but also through neutrino emission?
• RQ2 What is the magnitude of the new viscosity coefficient—radiative viscosity—compared to bulk viscosity in dense quark and nuclear matter?
• RQ3 How does the radiative viscosity depend on the thermodynamic and transport properties of dense matter, particularly the urca process rates?
• RQ4 In what astrophysical scenarios is radiative viscosity expected to dominate over bulk viscosity in damping stellar oscillations?
• RQ5 Is the radiative viscosity mechanism applicable to other systems with energy loss to radiation, such as pulsating white dwarfs or AGB stars?

Key findings

• The paper identifies a new energy dissipation mechanism in compact stars: radial oscillations lose energy not only via bulk viscosity (conversion to heat) but also via neutrino emission, quantified by a new viscosity coefficient called radiative viscosity.
• For non-strange quark matter and nuclear matter, the radiative viscosity coefficient is found to be 1.5 times larger than the bulk viscosity coefficient, indicating a dominant damping effect.
• The ratio $ \frac{\mathcal{S}}{\lambda} = \frac{3}{2} $ is derived as a generic result for all urca processes, showing that the energy loss rate due to neutrino emission is 50% larger than the rate associated with bulk viscosity.
• The mechanism is most relevant in systems where urca processes dominate, such as in hot neutron stars or during phase transitions, and may play a key role in stabilizing r-mode oscillations.
• The formalism is general and could be extended to other systems with energy loss to radiation, such as pulsating white dwarfs, though averaging over oscillation periods may require modifications.
• Non-leptonic weak interactions do not contribute to radiative viscosity, but they dominate bulk viscosity in strange quark matter, which may alter the relative importance of the two mechanisms in different matter phases.