Coupling physics and chemistry in protoplanetary disks

Ellen M. Price

Research exam, 10 May 2018


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

  • Temperature gradient
    • Warmer inner disk, cooler outer disk
    • Warmer upper layers, cooler midplane
  • Local density gradient
  • Radiation
    • 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

Evolution of a solar mass star

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


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

Disk model overview

$\alpha$-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

Temperature evolution

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 $\Sigma$

Surface density evolution

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

Velocity field

Track variables

Disk model overview

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

Static models

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.


Important differences at larger radii.


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?
    • Possibly linked to cyanide radical
  • 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

Results: 10 au track

Results: 30 au track

Results: 50 au track