This project develops a method to interpolate evolutionary tracks for stars of arbitrary masses using existing published model grids. The method is tested and shown to be accurate to better than 1%. Given observations of a star's radius and luminosity, the project also determines if the star's properties place it in an ambiguous region, requiring a mass range, or a well-defined region, with a single mass value. A point-in-polygon algorithm is used to classify regions. This allows study of how stellar rotation affects evolution by comparing observational properties to evolutionary tracks.

1. Interpolating Evolutionary Tracks of Rapidly Rotating Stars
Danielle Kumpulanian
Department of Physics & Astronomy, Stony Brook University
(Department of Physics, Applied Physics & Astronomy, Rensselaer Polytechnic Institute)
(Dated: August 5, 2005)
This project has two purposes: to provide an accurate method of interpolating data from
published evolutionary models grids, and to solve the problem of inferring the mass or
range of possible mass values of a star given its radius and luminosity. The first is
necessary because the grids only cover a limited set of mass values, and studying an
object of an arbitrary mass requires data for that mass to be interpolated and used.
In this case, the stellar models grids were those published by A. Claret in 2004. These
tracks were plotted on a log(radius) vs. log(luminosity) diagram. The interpolation
method was tested by using the existing tracks and linearly interpolating one intermediate
track. This test showed that this interpolation method could be used, accurate to better
than 1%, for any log(mass) in the range of the log(mass) values given in the grid, and
therefore, new models grids could be accurately generated using existing ones.
The evolutionary tracks plotted on the log(radius) vs. log(luminosity) plot are complicated and
include loops. Because of this, the tracks are divided into three sections, with the middle section
being the loop area. In this area, multiple values for log(mass) can exist, and a range of values can
be determined. In the other areas, one value can be found. So the process requires finding first
which area the [log(radius),log(luminosity)] point in question occupies and, if it is in the loop
region, finding the range of log(mass) involved.
This project fits into a larger context. Two properties of stars, radius and luminosity, are
known to remain unchanged when the stars rotate. These can be found using
observational techniques, and using these two quantities, other properties of the star can
be deduced. This allows for study of the star's evolutionary state and how rotation affects
stellar evolution.
Introduction
project, is the grid published by A.
Stellar evolution is the life history of a Claret1. To be useful, the grids need to
star. Since the lifetimes of stars can be be able to be manipulated to produce
millions or billions of years, a single star tracks for any arbitrary mass. A solution
cannot be observed for its entire lifetime. to this problem is to interpolate data for
All different stages of evolution can be arbitrary masses using the given data
observed. Stars are always being born from the grids.
and always dying, so there are examples Another matter is to estimate the mass
of all stages of stellar life. Using these of a star by comparing observations of L,
observations, detailed calculations can Teff and R with predictions. This can
be made about how the properties of result in a single estimated mass or a
stars of a given mass will change with range of values for mass, depending on
time. These calculations can be used to which segment of the evolutionary track
draw a predicted evolutionary track of a the given star is on.
star with a given mass.
“Grids” of these evolutionary tracks Interpolating Evolutionary Tracks
are published, with a considerable
difference between each mass, so not A C program was written to plot the
every possible mass of a star is data from Claret’s grid. Using
accounted for. Specifically, used in this PGPLOT, a log( R / RSun ) vs.

2. log( L / LSun ) diagram was made, which
showed an evolutionary track for each of
the log( M / M Sun ) models. Adding to
this program, diagonal lines indicating
constant Teff were drawn on the diagram
for reference. Noting that many of the
points gathered around and below the FIG. 3: Three sections of track: 1 to 2, 2 to 3,
and 3 to 4, where 2 identifies a minimum in the
Teff = 5500 Kelvin line, and that this
surface temperature and 3 identifies a local Teff
made it difficult to distinguish the
maximum.
evolutionary tracks, a decision was made
to end the tracks at Teff = 5500 K (Fig. Using these ratios given by equations (1)
1). through (3), for each point on the lower
The models on this grid are only for track, a corresponding point on the upper
specific masses. To produce track can be found. This corresponding
evolutionary tracks for other masses, an point does not necessarily exist in the
accurate way of interpolating was grid for that model. It is calculated based
developed. Using linear interpolation on the relevant time ratio and times on
and two of the existing models, an upper the lower track:
track and a lower track, a new,
intermediate track can be made (Fig. 2). (4)
One of the properties tabulated in the t U i = C1 (t L i − t L 0 ) + t U 0
grids is the age in years, which increases (5)
from left to right on the diagram. Using t U i = C 2 (t L i − t L min ) + t U min
this property, the ratio of the time scales (6)
between the two tracks can be t U i = C 3 (t L i − t L max ) + t U max .
established. However, since the
evolutionary tracks are not simple
In order to use equations (1) through
curves, they need to be divided into three
(6), the time values at the start and end
sections, and the ratio can be calculated
points of the segments must be found for
for each section (Fig. 3).
each of the upper and lower tracks.
These sections are defined as follows:
After using a C program to find the Teff
from the initial point to the first Teff
for each of these points, the value for
minimum, from the Teff minimum to the
time on these rows in the grid was
following Teff maximum, and from the recorded and used to find the time
Teff maximum to the end of the track. constants and the new times for the
upper track.
(1) Next, the calculated t U i were used to
C1 = (t U min − t U 0 ) /(t L min − t L 0 ) interpolate Teff and log( L / LSun ) for
(2) these newly created points along the
C 2 = (t U max − t U min ) /(t L max − t L min ) upper track.
(3) Given values of the quantities of
C 3 = (t U f − t U max ) /(t L f − t L max ) interest at the corresponding time points
on the upper and lower tracks, the track

3. for the intermediate mass value can be track and the two tracks used to compose
produced by linearly interpolating in it.
log( M / M Sun ) .
(7) Results: Interpolation of Evolutionary
log( M / M Sun ) − log( M / M Sun )
L Tracks
wt + =
log( M / M Sun ) L − log( M / M Sun )U Testing the interpolation method
(8)
showed it to be accurate to better than
log( M / M Sun ) − log( M / M Sun )U 2% for a difference of 0.2000
wt − =
log( M / M Sun ) L − log( M / M Sun )U log( M / M Sun ) between the two models
(9)
used to interpolate the third model. In
x = x1 wt + + x 2 wt − using this method for interpolating an
arbitrary track, it should be accurate to
In equation (9), x represents the quantity better than 1% as the difference between
being interpolated. the upper and lower tracks decreases.
To test this method of interpolation, an
existing evolutionary track was
interpolated using the tracks above and Finding Mass, given Radius and
below it, and all three were plotted on a Luminosity
log( R / RSun ) vs. log( L / LSun ) diagram,
along with the actual model for the Consider a log( R / RSun ) vs.
intermediate value. Specifically, the log( L / LSun ) diagram where the space
log( M / M Sun ) =0.6000 track was between each track is much smaller than
reconstructed using the on the diagram in Fig. 2. Not only does
log( M / M Sun ) =0.5000 and each track loop over itself, it loops over
other tracks as well. In this ambiguous
log( M / M Sun ) =0.7000 tracks (Fig. 2).
region of the diagram, for a given
It appears that the intermediate track is [ log( R / RSun ) , log( L / LSun ) ], an infinite
reproduced to within ~2%. One expects
number of tracks pass through this point.
interpolation between adjacent tracks to
This means that there is a range of
be much more accurate. Satisfied this
possible masses for a star with these
method will work, a C subroutine was
properties. One significant problem to
written to make this possible. Given a
solve is how to tell if a point is in a
log( M / M Sun ) within the range of those particular region given values for its
in the grid, the subroutine used the luminosity and the radius.
interpolation method described above to If the tracks are divided into three
create a new evolutionary track, with all sections, a polygon can be drawn by
of the properties, not limited to Teff , connecting the Teff min points, connecting
log( R / RSun ) , and log( L / LSun ) . the Teff max points, and connecting the
Another subroutine gives the option of end points of these two groups (Fig. 4).
writing this data to a file, and yet another If a [ log( R / RSun ) , log( L / LSun ) ] is
subroutine uses PGPLOT to plot the new
within this polygon, a range of values for
its log( M / M Sun ) can be found. If it is

4. outside this region, on either side, in Now, count all points on the threshold to
principle a unique value for its be considered “on-or-above” the
log( M / M Sun ) can be obtained. threshold. Then one side generates a
node because one of its endpoints is
There are many point-in-polygon
below the threshold and the other is on-
algorithms. The one used here2 involves
or-above it. The other side does not
drawing a straight horizontal line, from
generate a node because both of its
the point, through the polygon, and out
endpoints are on-or-above the threshold.
the other side (Fig. 5).
FIG. 7: Side a has an endpoint below and an
endpoint on-or-above the y-threshold, and both
endpoints of side b are on-or-above. Only one
node will be counted, due to side a.
FIG. 5: There are an odd number of blue nodes,
so the red test point is inside the polygon.
Notice the end result is independent of which
direction (left or right of the test point) the node
count takes place.
The y-coordinate of the test point FIG. 8: Side d lies entirely along the y-threshold,
becomes the y-threshold, and the points and neither of its endpoints will be counted.
Side e does not generate a node because both
where the threshold crosses the edge of
endpoints are on-or-above the threshold, but side
the polygon are called nodes. If there c, having one endpoint below the threshold, does
are an odd number of nodes, then the generate a node.
point is inside the polygon. If there is an
even number or zero nodes, the point is A similar problem occurs when one side
outside the polygon. This works for of the polygon lies entirely on the
polygons with holes in them or overlap threshold (Fig. 8). Neither of the
themselves, and also for polygons that endpoints of this side will be counted. If
have lines that cross (Fig. 6). the test point falls on an edge of the
polygon, the results are unpredictable
and depend on the orientation of the
polygon and the coordinate system.
FIG. 6: The algorithm still works for overlapping Results: Points Contained in a
polygons and polygons that cross themselves. Polygon
In the case of the threshold passing The point-in-polygon algorithm is fast
through a vertex of the polygon (Fig. 7), and easily programmed. It can be used
that node can only be counted once for for a variety of tasks, including the case
the algorithm to work properly. One of of obtaining values of mass for points
the sides has a point below and a point given their luminosity and radius.
on the threshold, and the other side has a
point above and a point on the threshold.

5. Conclusions track for that star can be drawn. Since
log( R / RSun ) and log( L / LSun ) are
A method of interpolating data from independent of rotation, this allows for
existing evolutionary models grids was study of the star’s evolutionary state and
developed and was tested, accurate to how its rotation affects its evolution.
better than 1%.
Noting the complexity of the
evolutionary tracks, they can be divided Acknowledgements
up into three sections: two simple
sections and one uncertain section. The author would like to thank Prof.
Polygons containing these sections can Deane Peterson for advising her on this
be drawn using multiple tracks, with project. Also, thanks to Joseph Yasi for
vertices at the endpoints of the track his help with debugging. This project
sections. If a point was made possible by a grant from the
[ log( R / RSun ) , log( L / LSun ) ] falls within National Science Foundation (Phy–
one of the two simple sections, its 0243935).
log( M / M Sun ) can be calculated. If it
falls within the middle, multivalued area,
References
a range of values for its log( M / M Sun )
can be found. In order to decide which 1. A. Claret, Astron. Astrophys. 424, 919 (2004).
area the point is in, a point-in-polygon 2. D. R. Finley, Point-In-Polygon Algorithm
(1998), URL:
algorithm must be used.
http://www.alienryderflex.com/polygon/.
After obtaining the log( R / RSun ) and
the log( L / LSun ) of a star through
observation, the remaining properties
can be deduced, and an evolutionary

6. FIG. 1: Evolutionary tracks with Teff 5500K, log(M/MSun) range 0.0500 to 0.7000.
FIG. 2: Three real tracks (from grid) are drawn in black, with the interpolated track in red and the re-
created upper track in green. This test proved the interpolation method to be accurate to better than ~2%.

7. FIG. 4: Polygon enclosing ambiguous middle region of each track, with vertices at Teff min and Teff max .