Interpolating evolutionary tracks of rapidly rotating stars - presentation

Interpolating Evolutionary Tracks
of Rapidly Rotating Stars

Danielle Kumpulanian
October 20, 2005

Deane Peterson, Stony Brook University

Outline
• Introduction
– Background
– Goals
• Accurately interpolate data
• Determine mass of a star
– Evolutionary track grids
• Description of project
– Interpolation method
– Test & results
– Polygon problem
• Conclusion
– Uses for interpolation
method
• References

Introduction
• Stellar evolution: lifetime of stars
• Stars live for millions or billions of years
• Cannot observe the entire life cycle of a single
star
• Need to piece together observations of stars of
the same mass, at different ages (different
points along evolutionary track)

Introduction
• Can draw predicted evolutionary track for
a star of that mass, based on observations
• Evolutionary tracks are put into “grids”.
• Grids are published for everyone to use.

Evolutionary track grid (2D)
log(M/MSun)=0.0500 … t (Age in years) log(L/LSun) … log(Teff) … …

… …
log(M/MSun)=0.0500 … t (Age in years) log(L/LSun) … log(Teff)







Evolutionary track grid (3D)
log(M/MSun):

0.1250
0.1000
0.0750
0.0500

Evolutionary Tracks
• Luminosity-Radius-Temperature relation:

L = 4πR2σT4
• Use this relation to solve for log(R/RSun):

log(R/RSun) = 0.5{log(L/LSun) – 4[logT – logTSun]}

• Using information from grid, draw
evolutionary tracks on a log(R/RSun) vs.
log(L/LSun) plot.

Evolutionary tracks
Models range from log(M/MSun) = 0.0500 to log(M/MSun) = 0.7000

http://rocinante.colorado.edu/~pja/astr3730/lecture12.pdf

Goals
• Problem: large gap between masses in the
grids…what about models for arbitrary masses?
• Solution: interpolate data for an arbitrary mass
using the existing data in the grids.

Goals
• Problem: given observables such as L, Teff, and
R, what is the mass of the star?
• Solution: use interpolation methods and
determine mass or range of possible masses
depending on which segment of the evolutionary
track the point is on.

Interpolation Method

wt + = (Teff − Teff 2 ) /(Teff 1 − Teff 2 )

wt − = (Teff 1 − Teff ) /(Teff 1 − Teff 2 )

x = x1 wt + + x 2 wt −
x1 < x < x 2

Divide each evolutionary track
into three segments:
1 Teff initial
2 Teff min
3 Teff max
4 Teff final

• Calculate time ratios for each segment of track:

C1 = (t U
min −t U
0 ) /(t L
min −t L
0 )
C 2 = (t U
max −t U
min ) /(t L
max −t L
min )

C 3 = (t U
f −t U
max ) /(t L
f −t L
max )

• Find corresponding points on the upper track for
each point on the lower track
• For each time t Li , a point t Ui is found.
• t Ui most likely does not correspond to a t U in
the grid.

•Interpolate values for log(L/LSun) and
log(Teff/Teff Sun) at points on upper track.
•Calculate log(R/RSun) for these points.

Calculate log(R/RSun) and log(L/LSun) for each
point along the intermediate track:
log( M / M Sun ) L − log(M / M Sun )
wt + =
log(M / M Sun ) L − log( M / M Sun )U
log( M / M Sun ) − log(M / M Sun )U
wt − =
log( M / M Sun ) L − log( M / M Sun )U

x1 = previous log(R/RSun) or log(L/LSun)
interpolated for upper track x = x1 wt + + x 2 wt −
x2 = next log(R/RSun) or log(L/LSun) x1 < x < x 2
interpolated for upper track

Test Results
• Interpolated track matched actual track accurate within
~2%.
• Distance between upper and lower tracks: log(M/MSun) =
0.2000.
• Decrease distance → decrease %error.
• In practice, the interpolated model will be between two
existing models, or a smaller distance between the upper
and lower tracks.

Evolutionary tracks with polygon

Point-In-Polygon
• Threshold: horizontal line with y-coordinate of the test point
• Node: point where the threshold crosses an edge
•of the polygon

• Odd number of nodes: Inside
• Even number of nodes: Outside
• Zero nodes: Outside

• It does not matter which way
(left or right) from the test
point nodes are counted,
the result is the same.

Point-In-Polygon

Works for polygons that cross themselves:

1 node → odd → inside polygon

Works for polygons with holes or
polygons that overlap themselves:

2 nodes → even → outside polygon

Point-In-Polygon
What if the threshold passes through
a vertex of the polygon?

• Only one node can be counted, even though there
are two sides
• Make a rule: If the threshold passes through a vertex,
the point is considered “above” the threshold.
• Each side of the polygon has two endpoints.
• If the threshold passes through the endpoints of two
adjacent sides, only one side will have an endpoint
below the threshold.
•Side a has an endpoint below & an endpoint “on-or-above” the
threshold → count 1 node.
•Side b has both endpoints “on-or-above” → count 0 nodes.

Point-In-Polygon
What if one side of the polygon lies
completely along the threshold?

Treat the same as last example:

• Side c has one endpoint below the threshold and one endpoint
“on-or-above” the threshold → count 1 node
• Side d has two endpoints “on-or-above” → count 0 nodes.
• Side e has two endpoints “on-or-above” → count 0 nodes.

• Total nodes: 1 → inside polygon

Conclusion
• Method of interpolating models was developed and
tested.
– Accurate to better than ~1%.
• Only a portion of an existing models grid was used
– Can easily be translated to use more of this grid or a different
models grid
• Tracks can be divided into three sections, and three
polygons can be drawn with vertices at the endpoints of
these sections.
– Polygons can be used to determine if a single mass or a range
of possible masses can be found.
– Use Point-In-Polygon algorithm to determine which polygon the
point is in.

Conclusion
• After obtaining log(L/LSun) and log(R/RSun) through
observation, remaining properties can be
deduced.
• Evolutionary track can be drawn.
• log(L/LSun) and log(R/RSun) are independent of
rotation.
• Can study star’s evolutionary state and how its
rotation affects its evolution.
• Further work needs to be done on this topic.

References
• A. Claret, Astron. Astrophys. 424, 919 (2004).
• D. R. Finley, Point-In-Polygon Algorithm (1998), URL:
http://www.alienryderflex.com/polygon/.

Aside: Personal Challenges
• Learn how to use Unix, Emacs, etc.
• Learn/Re-learn C programming language
• Figure out solutions to problems
• Figure out how to make programs to carry
out the calculations

Aside: Personal Challenges
• Learn how to use PGPLOT package
– Read manual written in FORTRAN
– Figure out how to change to C
• Make sure code was clearly written and
commented so that others could
understand it

Interpolating evolutionary tracks of rapidly rotating stars - presentation

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