MESA equations, time steps
and meshes
Michael Palmer
Overview of steps
• Takes input files and also initializes the modules
• These are then stored in memory
• Prepares to take new time step but first checks to see
if remeshing needs to be performed (more later)
• Adjusts model to reflect mass loss or gain,
abundance changes, calculates convective diffusion
coefficients. Solves for this structure using NewtonRaphson solver
• Next time step estimated (more later)
• Output files generated (more later)
In the Beginning
• Can used predefined models for solar
metallicity in a variety of mass ranges and
evolutionary stages
• ZAMS - 32 masses 0.08 - 100Msun
• Low mass PMS - 0.001 - 0.025Msun
• WD - He cores 0.15 - 0.45Msun, CO cores 0.496 1.025Msun and ONe cores 1.259 - 1.376 Msun
• Create PMS, specify mass, uniform
composition, luminosity and temperature.
Uses modules and NR-solver to find c for
that model.
• Works best for a mass range of 0.02 - 50Msun
Structure
•
1-D spherically symmetric
• Divided into cells, number
depends on complexity
• Numbered from outside in
•
Each cell has variables that
are mass averaged or
defined at the outer
boundary
• Tk, Xi,k, and k are mass
averaged variables
• Boundary variables mass
interior to mk,rk,Lk, and vk
• Inner boundary of
innermost cell usually
defined as centre of star,
Rc,Lc and vc zero unless
defined otherwise (special
circumstances)
•
How does it handle
equations of stellar
evolution?
Figure 9 Paxton et al. 2010
Equations of Stellar Evolution
• Ones we are used to
• An example from the MESA code
Paxton et al. 2010
(MESA Paper)
ESE continued
• For enhanced stability pressure equation is
divided by
P_k = (Pk-1dmk + Pkdmk-1)/(dmk + dmk-1)
• Boundary conditions constructed using Ps and
Ts. The difference from surface to centre of
first cell is found by
Paxton et al. 2010
(MESA Paper)
• This gives us our first boundary conditions for
both P1 and T1
Mesh Adjustment
• Cells may be merged or split at the
beginning of each time step, however,
remeshing designed so most cells are
left alone during remeshing
• Two stages for remeshing, planning and
adjustment stage
Planning
• Decides which cells may be split or merged
• Minimizes splits but maximizes mergers
• Must ensure magnitude between two adjacent cells
are below a certain threshold, ex ∆log(P) < P where
the default for P is 1/30
• Local reductions in magnitudes will result in higher
resolution in the desired region of the star. Example
when ∆log(P) is small compared to ∆log(nuc)
• Also increase near boundaries of convective zone, over a
distance of the pressure scale height
• Where ∆log(Xi) is large compared to ∆log(P)
• Near locations where spatial gradients exist in the most
abundant species
Adjustment
• Executes remesh plan
• Cells needing to be split have an interpolation in
mass performed to obtain luminosity and enclosed
volume at new cell boundary. From this a new cells
density found by
• New composition mass fractions calculated. For cell
mergers simple, for cell divisions neighbouring cells
used to linearly approximate mass fraction as a
function of mass coordinate. Slopes adjusted so
mass fractions sum to one everywhere and functions
are then integrated over new cell mass to find mass
fraction
Adjustment continued
• Temperature calculations depend on degeneracy of
electrons. For degenerate electrons, split cells take
on their former temperature (from before they were
split). While merging cells take a mass average of
their temperatures
• For non degenerate electrons, a parabola is formed
for the specific internal energy of neighbouring and
original cell. The parabola is integrated over the new
cell to find total internal energy. EOS module is called
with new parameters (composition and density) until
a temperature that predicts this internal energy is
found
Time Step
• Time step based on digital control theory
• Uses both previous and current time step to calculate the
next
• c unweighted average of the relative changes in log()
log(T) and log(R) across all cells, also uses previous c
• Target value of t = 10-4
• f(x) = 1 + 2tan-1[0.5(x-1)]
• Next time step calculated by
• Where t-1, t and t+1 are the previous, current and next
time step respectively
• Proposed step can be reduced based on
tests
• Can set both hard and soft limits, if change exceeds hard
limit, solution is rejected and retries. If change exceeds
soft limit next time step is reduced proportionally
Mass Adjustments
• Mass adjustments, for either loss or gain, are
done at the beginning of a new time step,
before structure is solved
• Several ways to account for dM/dt
• Section 6.6 - To many to list
• Write your own module for a mass loss or gain, example
given is for mass loss of carbon stars
• Total mass is M = Mc + Mm
• Mc is core mass and Mm is modeled mass
• Relative cell mass, dqk = dmk/Mm, and relative
mass interior to cell qk = mk/Mk or 1-∑dqi ,i =
1,..,k-1, must be defined as this for accuracy
close to outer envelope
Mass part 2
• Once change, M found, mass structure changed.
This does not effect cell number.
• Changes mass location of cells and revises
composition based on location
• Does not effect initial solutions, T,,..
• Mass structure divided into 3 regions
• Inner, intermediate and outer
• Inner region and outer boundaries defined by
temperature, logT = 6 (inner) logT = 5 (outer)
• Region expanded for stability if intermediate mass is
not much larger than M. First expanded by moving
outer boundary to surface and if mass in intermediate
is still to small can move inner boundary in (to a
point). Intermediate mass is not allowed to change by
more than 10% or by more than a factor of 2
Mass finale
• When regions defined, dqk rescaled by
M / (M + M) in inner region
• Outer region dqk left alone
• Intermediate scaled so ∑dqk = 1
• Composition for last two are than
updated
Convergence
• yi is trial solution, F(yi) is residual, yi is the correction
term and [dF/dy]i is the Jacobian
• Jacobain gets its derivatives from modules like the
eos module
• Converges on a solution by building upon yi
• Solution accepted when solution meets convergence
criteria. However if convergence cannot be achieved
• Retry => Uses a smaller time step to see if this results in
convergence
• Backup => If retry fails. Returns to previous model with smaller
time step than was used to find it.
• Keeps doing this until convergence happens or time step goes
below specified threshold, sequence terminated then
Output
• Two types
• Photo and save
• Photo - Binary file that has complete current
state of star. There will be no changes in
evolution if you restart from this
• Save - Not as complete as binary
• Two types - log and profiles
• Log - contain properties like stellar age
• Profiles - contains properties at specified time steps at
each zone from centre to surface
• Other options to, can produce images using
PGPLOT or output into another format that a
different code uses
Finally The End?
• Please see resolution sensitivity (Not
fair to Athira to take her time!)
• Questions?