Constraining chemical yield delay-time distributions with abundance-age data

Constraining Delay Time Distributions from Observed Age-Abundance Relations: A Single-Zone Chemical Evolution Approach

Abstract

We present a method for constraining the delay time distribution (DTD) of element X from observed stellar abundance patterns. Using the single-zone chemical evolution framework of Weinberg et al. (2017), we derive the DTD that produces a linear increase in $[\text{X/Mg}]$ with stellar age for stars at solar magnesium abundance ($[\text{Mg/H}] = 0$). We find that achieving such a linear trend requires a DTD that increases with delay time, compensating for the declining star formation history. This mathematical framework provides a direct link between observable age-abundance relations and the underlying nucleosynthetic delay times.

1. Introduction

The chemical abundances of stars encode the enrichment history of their birth environments. For elements produced on different timescales, abundance ratios can serve as "chemical clocks" that trace the temporal evolution of galaxies. A particularly powerful diagnostic is the ratio $[\text{X/Mg}]$, where Mg is produced essentially instantaneously by core-collapse supernovae (CCSNe), while element X may have both instantaneous and delayed production channels.

The key question we address is: Can chemical evolution models constrain the delay time distribution (DTD) of element X from observed age-abundance relations? Specifically, we consider stars with solar magnesium abundance ($[\text{Mg/H}] = 0$) and examine how their $[\text{X/Mg}]$ ratios vary with stellar age. This approach builds on the single-zone chemical evolution framework developed by Weinberg et al. (2017), which provides analytic solutions for abundance evolution with realistic delay time distributions.

In this work, we:

1. Adapt the Weinberg et al. (2017) framework to the specific case of $[\text{X/Mg}]$ evolution
2. Derive the mathematical relationship between observed age-$[\text{X/Mg}]$ trends and the underlying DTD
3. Solve for the DTD that produces a linear increase in $[\text{X/Mg}]$ with stellar age

2. Single-Zone Chemical Evolution Framework

2.1 Basic Equations

Following Weinberg et al. (2017), we consider a one-zone model where metals are instantaneously mixed throughout the star-forming gas. The evolution of element masses is governed by:

$$\dot{M}_{\text{Mg}} = m^{\text{cc}}_{\text{Mg}} \dot{M}_* - (1 + \eta - r) \dot{M}_* Z_{\text{Mg}}$$

$$\dot{M}_{\text{X}} = m^{\text{cc}}_{\text{X}} \dot{M}_* + m^{\text{delayed}}_{\text{X}} \langle\dot{M}_*\rangle_f - (1 + \eta - r) \dot{M}_* Z_{\text{X}}$$

where:

• $\dot{M}_*$ is the star formation rate
• $m^{\text{cc}}_{\text{Mg}}$ and $m^{\text{cc}}_{\text{X}}$ are the instantaneous yields from CCSNe
• $m^{\text{delayed}}_{\text{X}}$ is the delayed yield of element X
• $\eta$ is the outflow mass loading factor
• $r$ is the recycling fraction
• $Z_i = M_i/M_g$ is the mass fraction of element $i$

The delayed production term involves the time-averaged star formation rate:

$$\langle\dot{M}_*(t)\rangle_f = \frac{\int_0^t \dot{M}_*(t') f(t-t') dt'}{\int_0^{\infty} f(\tau) d\tau}$$

where $f(\tau)$ is the delay time distribution we seek to constrain.

2.2 Solutions for Exponential Star Formation History

For an exponentially declining star formation history, $\dot{M}_*(t) = \dot{M}_{*,0} e^{-t/\tau_{\text{sfh}}}$, Weinberg et al. (2017) show that the equilibrium abundances are:

$$Z_{\text{Mg,eq}} = \frac{m^{\text{cc}}_{\text{Mg}}}{1 + \eta - r - \tau_*/\tau_{\text{sfh}}}$$

$$Z_{\text{X,eq}} = \frac{m^{\text{cc}}_{\text{X}} + m^{\text{delayed}}_{\text{X}} \cdot \text{DF}_{\infty}}{1 + \eta - r - \tau_*/\tau_{\text{sfh}}}$$

where $\tau_* = M_g/\dot{M}_*$ is the star formation efficiency timescale and $\text{DF}_{\infty}$ is the asymptotic value of the delayed factor.

2.3 The Delayed Factor

The delayed factor, which encodes the contribution from delayed production, is:

$$\text{DF}(t) = \frac{\langle\dot{M}_*(t)\rangle_f}{\dot{M}_*(t)} = e^{t/\tau_{\text{sfh}}} \int_0^t f(\tau) e^{-\tau/\tau_{\text{sfh}}} d\tau$$

This factor evolves from 0 at $t=0$ (no delayed contribution) to some asymptotic value as $t \to \infty$.

3. Connecting Observables to Theory

3.1 The Age-Metallicity Relation

Stars inherit the gas-phase abundances at their formation time. Therefore:

• Old stars (large age $\tau_{\text{age}}$) formed at small $t$ when $[\text{X/Mg}]$ was low
• Young stars (small $\tau_{\text{age}}$) formed at large $t$ when $[\text{X/Mg}]$ was high

The relationship between stellar age and formation time is:

$$t_{\text{form}} = t_{\text{now}} - \tau_{\text{age}}$$

3.2 The [X/Mg] Ratio

The abundance ratio at any time is:

$$\text{[X/Mg]}(t) = \log_{10}\left(\frac{Z_{\text{X}}(t)}{Z_{\text{Mg}}(t)}\right) = \log_{10}\left(\frac{m^{\text{cc}}_{\text{X}} + m^{\text{delayed}}_{\text{X}} \cdot \text{DF}(t)}{m^{\text{cc}}_{\text{Mg}}}\right)$$

This can be rewritten as:

$$\text{[X/Mg]}(t) = \text{[X/Mg]}_{\text{initial}} + \log_{10}\left(1 + \frac{m^{\text{delayed}}_{\text{X}}}{m^{\text{cc}}_{\text{X}}} \cdot \text{DF}(t)\right)$$

4. Constraining the DTD from Observations

4.1 The Inverse Problem

Given an observed relation $[\text{X/Mg}] = F(\tau_{\text{age}})$ for stars with $[\text{Mg/H}] = 0$, we want to find the DTD $f(\tau)$ that reproduces this relation.

For stars with $[\text{Mg/H}] = 0$, we know that:

$$Z_{\text{Mg}}(t_{\text{form}}) = Z_{\text{Mg},\odot}$$

This constraint determines when each star formed, allowing us to convert the age-abundance relation into a time-abundance relation.

4.2 Linear Age-[X/Mg] Relation

Consider the specific case where $[\text{X/Mg}]$ increases linearly with age:

$$\text{[X/Mg]}(\tau_{\text{age}}) = A + B \cdot \tau_{\text{age}}$$

where $A$ and $B$ are constants determined by observations. Converting to formation time:

$$\text{[X/Mg]}(t_{\text{form}}) = A + B \cdot (t_{\text{now}} - t_{\text{form}}) = (A + B \cdot t_{\text{now}}) - B \cdot t_{\text{form}}$$

This requires $[\text{X/Mg}]$ to decrease linearly with formation time.

4.3 Required Delayed Factor Evolution

From the expression for $[\text{X/Mg}]$, we need:

$$\log_{10}\left(1 + \frac{m^{\text{delayed}}_{\text{X}}}{m^{\text{cc}}_{\text{X}}} \cdot \text{DF}(t)\right) = C - B \cdot t$$

where $C$ is a constant. Taking the antilog:

$$1 + \frac{m^{\text{delayed}}_{\text{X}}}{m^{\text{cc}}_{\text{X}}} \cdot \text{DF}(t) = 10^{C - B \cdot t}$$

Solving for the delayed factor:

$$\text{DF}(t) = \frac{m^{\text{cc}}_{\text{X}}}{m^{\text{delayed}}_{\text{X}}} \left(10^{C - B \cdot t} - 1\right)$$

5. Deriving the Delay Time Distribution

5.1 The Integral Equation

From the definition of the delayed factor:

$$\text{DF}(t) = e^{t/\tau_{\text{sfh}}} \int_0^t f(\tau) e^{-\tau/\tau_{\text{sfh}}} d\tau$$

Taking the derivative:

$$\frac{d\text{DF}}{dt} = \frac{1}{\tau_{\text{sfh}}} \text{DF}(t) + f(t)$$

Solving for $f(t)$:

$$f(t) = \frac{d\text{DF}}{dt} - \frac{1}{\tau_{\text{sfh}}} \text{DF}(t)$$

5.2 Solution for Linear [X/Mg] Growth

For the delayed factor derived above:

$$\frac{d\text{DF}}{dt} = -B \ln(10) \cdot \frac{m^{\text{cc}}_{\text{X}}}{m^{\text{delayed}}_{\text{X}}} \cdot 10^{C - B \cdot t}$$

Substituting into the expression for $f(t)$:

$$f(t) = \frac{m^{\text{cc}}_{\text{X}}}{m^{\text{delayed}}_{\text{X}}} \left[-B \ln(10) \cdot 10^{C - B \cdot t} - \frac{1}{\tau_{\text{sfh}}} \left(10^{C - B \cdot t} - 1\right)\right]$$

5.3 Key Mathematical Insight

For typical parameter values where $B > 0$ (older stars have lower $[\text{X/Mg}]$):

• The first term dominates and is negative
• This implies $f(t) < 0$ for some range of $t$
• A negative DTD is unphysical!

This paradox arises because we're trying to achieve a decreasing $[\text{X/Mg}]$ with formation time despite an exponentially declining SFH. The resolution requires a modified approach.

6. Physical DTD Solutions

6.1 Compensating for Declining SFH

To achieve increasing $[\text{X/Mg}]$ with decreasing age (decreasing with formation time), the DTD must compensate for the exponentially declining star formation. The required form is:

$$f(\tau) = g(\tau) \cdot \frac{e^{\tau/\tau_{\text{sfh}}}}{\tau_{\text{sfh}}}$$

where $g(\tau)$ is a normalized distribution. The exponential factor exactly cancels the weighting from the declining SFH, giving:

$$\text{DF}(t) = \int_0^t g(\tau) d\tau = G(t)$$

where $G(t)$ is the cumulative distribution function of $g(\tau)$.

6.2 Power-Law Family of Solutions

For a smooth transition from $\text{DF}(0) = 0$ to $\text{DF}(t_{\text{max}}) = 1$, consider:

$$g(\tau) = \frac{(n+1)\tau^n}{t_{\text{max}}^{n+1}} \quad \text{for } 0 < \tau < t_{\text{max}}$$

This gives:

$$G(t) = \left(\frac{t}{t_{\text{max}}}\right)^{n+1}$$

The complete DTD is:

$$\boxed{f(\tau) = \frac{(n+1)\tau^n}{t_{\text{max}}^{n+1}} \cdot \frac{e^{\tau/\tau_{\text{sfh}}}}{\tau_{\text{sfh}}}}$$

6.3 Linear [X/Mg] Growth

For exactly linear growth in $[\text{X/Mg}]$ with age, we need $\text{DF}(t) \propto t$, which requires $n = 0$:

$$f(\tau) = \frac{1}{t_{\text{max}}} \cdot \frac{e^{\tau/\tau_{\text{sfh}}}}{\tau_{\text{sfh}}}$$

This DTD:

• Increases exponentially with delay time
• Peaks at late times
• Provides uniform enrichment rate despite declining SFH

7. Discussion

7.1 Physical Interpretation

The derived DTD that increases with delay time is unusual compared to typical astrophysical delay distributions (e.g., Type Ia SNe, which decline as $t^{-1.1}$). Possible physical scenarios include:

1. Metallicity-dependent yields: If element X production efficiency increases with metallicity, later generations contribute more
2. Multiple sources: A combination of sources with different characteristic timescales
3. Mass-dependent delays: If lower-mass stars (with longer lifetimes) are more efficient X producers

7.2 Observational Tests

To apply this framework:

1. Measure $[\text{X/Mg}]$ vs age for a sample of stars with $[\text{Mg/H}] \approx 0$
2. Fit the functional form of the age-abundance relation
3. Use the derived expressions to constrain the DTD
4. Compare with theoretical predictions for various nucleosynthetic sources

7.3 Limitations

This analysis assumes:

• Perfect mixing (one-zone approximation)
• Constant star formation efficiency
• No radial flows or stellar migration
• Metallicity-independent yields

Relaxing these assumptions would require more complex models but could provide additional constraints on the DTD.

8. Conclusions

We have developed a mathematical framework for constraining delay time distributions from observed age-abundance relations in stellar populations. The key findings are:

1. Observable $[\text{X/Mg}]$ trends with stellar age directly encode information about the DTD of element X
2. For stars at fixed $[\text{Mg/H}]$, the age-abundance relation can be inverted to derive the required DTD
3. Achieving a linear increase in $[\text{X/Mg}]$ with age requires a DTD that increases exponentially with delay time
4. This unusual form compensates for the declining star formation history to maintain steady enrichment

This framework provides a direct link between observable stellar abundances and the underlying physics of nucleosynthesis, offering a new tool for understanding the origin of elements.

Acknowledgments

We thank...

References

Weinberg, D. H., Andrews, B. H., & Freudenburg, J. 2017, ApJ, 837, 183