[Paper Review] The rates of Type Ia Supernovae. I. Analytical Formulations

L. Greggio|ArXiv.org|Apr 18, 2005

Gamma-ray bursts and supernovaePhysics and Astronomy63 references159 citations

TL;DR

This paper presents an analytical formalism to compute Type Ia supernova (SNIa) rates from stellar populations, linking the realization probability $A_{\rm Ia}$ and delay time distribution $f_{\rm Ia}(\tau)$ to progenitor models. It shows that current SNIa rates in late-type galaxies constrain $A_{\rm Ia} \sim 10^{-3}$ per $M_\odot$ of star formation, and favors the double-degenerate channel over single-degenerate due to its flatter delay-time distribution, with half of SNIa events occurring within 0.3–3 Gyr after a burst of star formation.

ABSTRACT

This paper provides a handy tool to compute the impact of Type Ia Supernova (SNIa) on the evolution of stellar systems. An effective formalism is presented to couple the SNIa rate to the star formation history, which rests upon the definition of two key properties of the progenitor's model: the realization probability of the SNIa event from a single stellar generation and the distribution function of the delay times. It is shown that the current SNIa rate in late type galaxies implies that the realization probability is on the order of 0.001. Analytical formulations for the distribution function of the delay times for Single (SD) and Double Degenerate (DD) progenitors are derived, based on stellar evolution arguments. These formulations, which agree well with the results of Monte Carlo simulations for the evolution of close binaries, have a built in parametrization of the key properties of the alternative candidates. The various models for the progenitors have different impact on the large scales. In particular, the paper examines the systematic trend of the SNIa rate per unit mass with the color of the parent galaxy, and shows that the recent observations favor the DD model. The SD scenario can reproduce the data only if the distribution of the primordial mass ratios is flat, and the accretion efficiency onto the WD is close to 100%. The timescale for the Fe release from SNIa to the interstellar medium ranges between 0.3 and 3 Gyr for a wide variety of hypothesis on the SNIa progenitors. (ABRIDGED)

Motivation & Objective

• To develop a tractable analytical framework for computing SNIa rates from stellar populations, independent of full population synthesis codes.
• To link the realization probability $A_{\rm Ia}$ and delay time distribution $f_{\rm Ia}(\tau)$ to physical properties of binary progenitor systems.
• To constrain progenitor models (single vs. double degenerate) using observational constraints on SNIa rates in early- and late-type galaxies.
• To quantify the timescale for iron release from SNIa events following a starburst, relevant for chemical evolution and ICM enrichment.
• To enable direct comparison of progenitor models via their impact on the shape of $f_{\rm Ia}(\tau)$ and the resulting SNIa rate evolution.

Proposed method

• Derives analytical expressions for $f_{\rm Ia}(\tau)$ based on binary evolution, assuming schematized mass transfer and merger outcomes.
• Parametrizes progenitor models via key physical inputs: minimum/maximum masses, mass ratio distribution, orbital separation distribution, and accretion efficiency.
• Uses the realization probability $A_{\rm Ia}$ to normalize the total SNIa rate per unit mass of stars formed.
• Compares analytical $f_{\rm Ia}(\tau)$ functions to results from population synthesis codes under identical parameters, showing good agreement.
• Applies the formalism to constrain progenitor models using observed SNIa rate trends across galaxy types.
• Evaluates the secular evolution of SNIa rates relative to mass return in elliptical galaxies, using power-law approximations for $f_{\rm Ia}(\tau)$.

Experimental results

Research questions

• RQ1 What is the analytical relationship between the delay time distribution $f_{\rm Ia}(\tau)$ and the physical parameters of binary progenitor systems?
• RQ2 How do different progenitor models (single vs. double degenerate) predict distinct $f_{\rm Ia}(\tau)$ shapes and SNIa rate evolutions?
• RQ3 What constraints does the observed SNIa rate in late-type galaxies place on the realization probability $A_{\rm Ia}$?
• RQ4 How does the observed trend of SNIa rates across galaxy types favor the double-degenerate over the single-degenerate channel?
• RQ5 What is the timescale for half of the SNIa events to occur after a single burst of star formation across different progenitor models?

Key findings

• The current SNIa rate in late-type galaxies constrains the realization probability to $A_{\rm Ia} \sim 10^{-3}$, i.e., one SNIa per 1000 $M_\odot$ of stars formed.
• The comparison of SNIa rates in early- and late-type galaxies implies that the delay time distribution $f_{\rm Ia}(\tau)$ must be more populated at short delays, favoring the double-degenerate channel.
• The double-degenerate model provides a better fit to observations than the single-degenerate model, which predicts too steep a decline in $f_{\rm Ia}(\tau)$.
• The single-degenerate scenario can only reconcile with observations if the mass ratio distribution is flat and accretion efficiency is near 100%.
• The timescale for half of the SNIa events to occur after a starburst ranges from 0.3 to 3 Gyr across all considered progenitor models.
• The shape of $f_{\rm Ia}(\tau)$ is not a simple power law, with a prominent peak at intermediate delays that affects gas flow dynamics in elliptical galaxies.