Background & Central Ideas: Avoid isochrones (why?)

There are no 1-D curves in real HR diagrams asymmetrical sequences, see distortions in (e.g.) turnoff region

Isochrones focus on ‘where’ - what about ‘how many’?

Background and Central Ideas (continued)

Isochrones can fold back on themselves (even self-intersect) (undermines the concept of distance from isochrone)

Isochrones are not well suited to deal with multiple stars (or with distributions in composition, rotation, etc.)

Wouldn’t it be nice to get standard errors and correlation coefficients, and of course impersonal answers?

To sum up - in a 2-D problem (like HRD), it’s hard to operate objectively when theory and observation differ topologically (like lines vs. dots)

So what’s an alternative? Straightforward would be areal density (observed “stars” per sq. mag. - pixels)

Direct inclusion of multiple stars (need a statistical algorithm)

Impersonal solutions for parameters (say by Least Squares)

Spinoff

An alternative to Monte Carlo (called FSA)

A useful scheme for precise numerical derivatives (answers independent of the increment)

Problems Related to Stellar Structure and Evolution (partial and cryptic list)

Mixing length convection, convective overshooting and semi-convection, the multitude of composition parameters (it’s not just X, Y, Z), uncertainties in opacities and nuclear energy generation rates, gravitational settling and radiative acceleration of species, mixing by thermal pulses, mass loss via winds, binary evolution

Parameterize and solve for the parameters? (but then we have a long parameter list) The situation worsens greatly at advanced evolutionary stages

So solutions for real clusters have to be approached with great caution, lest the results be take too seriously!

Let’s make sure we can do synthetic clusters first

Now: the analytic Probbbbbblems! (ouch)

A mountain of evolutionary computing - Lotsa stars over a mass range, including companions Wait, that’s not enough stars - need smooth AD’s (so do maybe 30 times more)

Binaries, triples, . . . . (also distributed in mass) (get factor 6 [1 single, 5 wide binaries]) (or 36 [add 5 close binaries, 25 triples])

Not only mass, but composition, interstellar extinction, and other parameters will lie on distributions

May want to differentiate AD’s wrt parameters (so number of IMF stars is multiplied by [Np+1])

. . . and we will have to iterate

How to generate theoretical stars over their parameter distributions and do the fitting in AD . . . Efficiently How about: FSA = Functional Statistics Algorithm

Observed stars are subject to observational error and synthesized stars are not - what to do about that? . . . need error modeling

Analytic problems (continued)

Evolutionary motion over pixel boundaries can give discontinuities that make life difficult (so let’s have precise differentiation via pixel sharing)

Analytic problems(continued)

Do close binary evolution? Can do now, but probably not so urgent (and more parameters!)

. . . and then programming problems you don’t want to know about (like how to avoid preposterous memory needs)

Speed and efficiency:

Too many low mass stars; too few high mass stars (overkill, underkill) Remedy: have two IMF’s (a real one and a phony one)

Evolutionary computation is too slow - need 106 times faster

Remedies: Evolution by approximation functions (Eggleton, Tout, Pols, Hurley) Downers - Add-ons are difficult There’s only one such program (can’t explore physics) Stored evolution tracks: Downer: binary star evolution introduces many dimensions

Evolution by approximation functions also helps with precision of nulmerical derivatives

Data windows are convenient

Stepwise window expansion - experiment the easy way

How’s to do the Least Squares?

Let’s stick to standard adjustment theory. Why? It’s 200 years old and the bugs are out of it Get parameter correlations Get standard error estimates

Comparison is between observed and theoretical Areal Density

But what specific algorithm? DC can do non-linear problems without sacrificing correlations or standard errors (only cost is iteration)

Input: real or simulated cluster magnitudes and color indices plus initial parameter estimates.

A direct program synthesizes an HRD, with multiple stars, various distributions (Z, AV, etc.), and simulated errors.

Actual Procedure (somewhat abbreviated)

1. Compute observational A’s for all pixels in active window. 2. Generate Main Sequence stars according to an assumed form of IMF, but much less steep (with input parameters). Do R times as many stars as indicated by input N (to achieve smoothness).

3. Generate companions to make binaries and triples (with associated weights - see FSA later).

4. Compute contributions to all pixels by all singles and multiples (one system at a time). Correct each contribution to correspond to the “real” (i.e. input) IMF.

5. Re-scale pixel A’s (divide by R).

6. Increment adjusted parameters and compute numerical derivatives, dA/dp.

7. Solve DC equation of condition for parameter corrections

8. Correct parameters and go back to #2 for next iteration.

FSA - how’s it work? 1. Adopt a distribution (Z, Av, companion mass, etc.) 2. Space Z’s (or whatever) either a. uniformly, with weights proportional to probablility of occurrence, or b. non-uniformly with spacing contrived to make weights equal (by integrating under the distribution)

3. Run the input through an operator (e.g. stellar evolution program) to get weighted theoretical output distribution

4. Match theory to observation according to a fitting criterion (e.g. Least Squares) and solution algorithm (e.g. DC, Simplex, etc.)

Note that random numbers are not involved - efficiency improvement over Monte Carlo

The only FSA distributions used so far are companion mass and photometric errors (magnitude and color index)

Parameters1. Z (composition)2. Gaussian S.D. of Z3. AV, interstellar extinction in V (or other ‘vertical’ coordinate

magnitude)4. Gaussian S.D. of AV

5. Ratio of AV to color excess in B-V6. Intercept in Pclose= aclose + bcloseqclose

7. Slope, bclose, in above distribution8. Intercept in Pwide= awide + bwideqwide

9. Slope, bwide, in above distribution10. Reimers mass loss parameter (winds)11. Binary enhanced mass loss parameter12. Lowest mass for which the IMF is defined13. Mass at the first of two breaks in the IMF14. Mass at the second break in the IMF15. Exponent (of mass) in the KTG IMF for low mass stars16. Exponent (of mass) in the KTG IMF for intermediate mass stars17. Exponent (of mass) in the KTG IMF for high mass stars18. Cluster V band distance modulus19. Cluster age20. Lowest mass on the observed Main Sequence21. Number of stars above the low mass IMF cutoff (in sampled part of

cluster)

Z V-MV

AVCluster Age

Companion Probability Number of Stars

Sidelights: Luminosity functions - a special case (only a few trials so far)

The field star problem (mainly for open clusters)

Practical issues: Systems with primaries outside the window can be insidethe window, even initially, counting companions (so safety margin needed in IMF lower limit)

But safety margins shouldn’t be overly big or computation time goes way up!

Multiple systems can evolve in and out of windows in apparently strange ways (including jumps)

Only limited experiments so far on pixel size (Typical has been 0.05 in magnitude and 0.008 in CI.)

Conversion from [Teff, Mbol] to bandpass quantities is by Legendre polynomials fitted to Kurucz atmospheres (Van Hamme & Wilson, 2003

Where do we go from here? Gain . . .

consistent convergence to correct results for synthetic clusters (open & globular)

reliable standard errors

ability to tap multiple evolution models

experience in various overall situations

experience in fitting real clusters

more specific parameterization of heavy element abundances (and Y=helium).

And also - give luminosity functions a serious tryout

To crib a line from B. Franklin . . .