# Introduction

## Structure overview

(inspired by Henning & Semenov 2013)

## Planet formation and composition

Why should we care about disk composition?

• Planets acquire their cores and atmospheres from grains and disk gas
• Terrestrial planets in particular will form in the inner disk
• Understanding our own planet and those in other solar systems

## Chemical disk structure

• Upper layers: atoms and ions broken up by radiation
• Warm molecular layer: molecules
• Midplane: molecules frozen onto grains

## Physical disk structure

• Warmer inner disk, cooler outer disk
• Warmer upper layers, cooler midplane
• Inner disk shielded from cosmic rays
• Upper layers more irradiated by star
• Bulk gas motion: accretion, turbulence, mixing

## Structure overview

(inspired by Henning & Semenov 2013)

## Disk midplane

• Formation site of terrestrial planets
• Ions and molecules both in the gas and solid phases provide a rich chemistry to study
• The midplane is a “simple” case

## Accretion flows

• As the gas cloud collapses around a young star:
• Most of the gas gravitates toward the star
• Some gas moves outward to conserve angular momentum
• This naturally forms a disk around the star that spreads with time

## Evolution of young stars

Siess, Dufour, & Forestini (2000) simulated HR diagram

## Drivers of disk chemistry

• Local conditions: temperature, density, etc.
• X-rays, cosmic rays, and UV radiation
• Grains provide a surface on which reactions can occur

# Motivation

## Comparison of timescales

Semenov & Wiebe (2011)

## Global solution

• Can do a single, “global” simulation of disk chemistry
• Pros:
• Can incorporate as much physics as needed
• Cons:
• Very computationally expensive!
• If we have $M$ species and $N$ computational cells and cells are not independent, would need a $M N \times M N$ (sparse) matrix to solve for all chemistry at a single timestep
• Examples given in Haworth et al. (2016)

## Alternative: reduce chemical network

• Can choose a subset of “important” reactions to reduce number of reactions and species being considered
• Pros:
• Can incorporate as much physics as needed
• Cons:
• Lose chemical insight
• Chemical network reduction is a science/art itself!

## Alternative: reduce physics

• Can reduce the physical processes considered
• Common approach is to assume that disk material is not moving radially or vertically
• Pros:
• No loss of chemical complexity
• Cons:
• Missing (possibly) important effects of dynamic processes

## Alternative: local simulation

• Solve for accretion tracks self-consistently and follow chemistry along tracks
• Pros:
• Under some assumptions, no compromise between chemistry and physics
• Approximately equal computational load to a static model
• Can be built upon in future work
• Example: Heinzeller, Nomura, & Walsh (2011)

# Disk model

## Key assumptions

• Azimuthal symmetry
• Small grains are well-coupled to the gas
• X-rays and UV radiation are extincted
• Cosmic rays cannot be neglected
• No 3-body reactions

## $\alpha$-disk model

$$\nu = \alpha c_s h$$

• First appeared in Shakura & Sunyaev (1973)
• Parametrizes unknown viscosity source with small parameter $\alpha < 1$
• $\nu$ = kinematic viscosity
• $c_s$ = local sound speed
• $h$ = disk scale height
• Take $h = c_s / \Omega_\mathrm{Kep}$ for this model

## Temperature structure

$$T = T_0 \left(e^{-\psi t / t_0} + \omega\right) \left(\frac{R}{R_0}\right)^{-\beta}$$

• Chosen to be as flexible as possible without losing the analytic nature of the solution
• Allows space and time variations of $T$
• $\psi$, $\omega$, and $\beta$ determined by fitting radiative transfer output
• Assume that dust and gas temperatures are equal

## Surface density

$$\frac{\partial \Sigma}{\partial t} - \frac{3}{R} \frac{\partial}{\partial R} \left[R^{1/2} \frac{\partial}{\partial R} \left(\nu \Sigma R^{1/2}\right)\right] = 0$$
• From Lynden-Bell & Pringle (1974) and Pringle (1981)
• By combining the Navier-Stokes and mass continuity equation, we find a diffusion-type equation in surface density $\Sigma$

## Putting it all together

• Find non-dimensional form of equation
• Use separation of variables technique
• Find a solution in terms of Bessel functions
• Exploit orthogonality

## Chemistry post-processing

Consider chemistry in a “box”

$$n_i + n_j \rightarrow \cdots$$ $$n_{j1} + n_{j2} \rightarrow n_i + \cdots$$

Chemical network: reactions with rates $P_j$ and $R_j$

$$\frac{\mathrm{d} n_i}{\mathrm{d} t}\bigg|_V = \sum\limits_j P_j n_{j1} n_{j2} - n_i \sum\limits_j R_j n_j$$

# Chemical evolution

## What is the “fiducial” model?

• Track evolves from 5 au to 1 au in about 0.8 Myr
• Stellar evolution is included
• Cosmic ray rate is $\zeta_\mathrm{CR} = 10^{-18}$
• Initial conditions consistent with inheritance chemistry (atoms and molecules)

## What is a “static” model?

• Initial point model = gas parcel stays fixed in position at initial point of track
• Final point model = gas parcel stays fixed in position at final point of track
• Star can still evolve with time

## Selected case studies

How can we understand these behaviors?

## Analysis methods

1. Running perturbed versions of the fiducial model
2. Rate printing
3. Automated pathway finding (Lehmann 2004)

## Case study: C2H2

C2H3+ + NH3 ⟶ NH4+ + C2H2

C + C2H2 ⟶ C3 + H2

CH + C2H2 ⟶ C3H2 + H

When we change the way cosmic rays are attenuated, the rates of all three reactions increase until the end of the track.

## Effect of cosmic rays

A model without cosmic rays

## Case study: NO and C3H3

N + NO ⟶ N2 + O

N + C3H3 ⟶ HC3N + H2

He+ + N2 ⟶ N+ + N + He

He + γCR ⟶ He+ + e-

Atomic N is involved in both destruction reactions; comsic rays are fueling the destruction of both species.

## Initial point vs. fiducial

These two models are very similar.

## Final point vs. fiducial

Enhancement of species in gas and solid phases.

# Conclusions

## Dynamics really do matter

The variation of cosmic rays in particular plays a primary role in driving chemistry

## Remaining work

• Why does molecular oxygen show its behavior?
• Comparing model results to observations

## Future work

• Differential motion of dust and gas?
• Grain growth?

# Extra slides

## Full analytic solution

$$\sigma \propto \xi^{\beta-\frac{3}{2}} \xi_1^{\frac{1}{2}} \left[4 \left(\frac{\xi_1^{\frac{2\beta + 1}{4}}}{2\beta + 1}\right)^2 +\tilde{\tau}\right] \exp\!\left[\frac{-4 \xi^{\frac{2\beta+1}{2}}}{4\xi_1^{\frac{2\beta+1}{2}} + \left(2\beta + 1\right)^2 \tilde{\tau}}\right]$$

$$\tilde{\tau} = \omega \tau - \frac{e^{-\psi \tau}}{\psi} + \frac{1}{\psi}$$

## Solving the chemistry equations

• Assemble the sparse Jacobian matrix $J$
• $N \times N$ matrix, where $N$ is the number of species plus number of physical variables
• $M$ contributions from reactions
• Solve the system $J x = b$ using LU decomposition
• Combines SUNDIALS code with HSL MA48