This will be short. I’ve provisionally decided to proceed with the Progeny based SSP model libraries I’ve discussed over the last several posts. I’ve picked two versions for model runs: a “small” one with 5 metallicity bins and 42 age bins from log(T) = 6 to 10.1 in 0.1 dex intervals, and a “medium” sized one with just 3 metallicities (log(Z/Z☉) = {-0.25, 0, +0.5}) and 74 age bins with log(T) = 6.0, 6.5 and 6.55, … , 10.1 in 0.05 dex intervals. These all use the MIST isochrones, Kroupa IMF, and the recommended stellar ingredients from the first Progeny paper. As discussed in a previous post the wavelength interval is limited to 3300 – 9000Ã… because of the prevalence of terrestrial night sky lines and calibration issues in the near IR portion of MaNGA spectra.
I’ve decided to take one more, maybe final, look at a sample of SDSS selected galaxies in MaNGA. I remembered recently that I’ve made several attempts to select post-starburst samples with various queries of SDSS databases. One I did some time ago had nearly 5800 hits in DR8, with 104 cross matches in MaNGA. Part of the query is pasted below:
select into mydb.mylargerka
s.ra,
s.dec,
s.plate,
s.mjd,
s.fiberid,
s.z,
s.zErr,
from specObj s
left outer join galSpecline as g on s.specObjid = g.specObjid
left outer join galSpecIndx as gi on s.specObjid = gi.specObjid
left outer join galSpecExtra as ge on s.specObjid = ge.specObjid
where
(g.oii_3729_eqw > -5 and g.oii_3729_eqw_err > 0) and
(gi.lick_hd_a_sub > 4 and gi.lick_hd_a_sub_err > 0) and
s.z >= .02 and
(s.snMedian > 10) and
(s.zWarning = 0 or s.zWarning = 16)
order by
s.plate, s.mjd, s.fiberid
So basically this is just a standard sort of post-starburst selection with relaxed limits on both Balmer absorption and emission line strength. The line index data were from the MPA/JHU pipeline, which was last run on DR8.
I had run models for about 1/4 of the 104 galaxy sample when a heat wave arrived, and I decided for the sake of our electric bills not to continue intensive computing 24/7. Temperatures are currently below normal, so I may be able to resume soon. About all I can say so far is the sample contains a mix of known PSBs and false positives — which are mostly ordinary star forming galaxies.
A few brief comments about the simulations, of which I’ve done a few more but will probably bring to an end. First, here is a plot I mentioned but didn’t display last time of the bias in stellar mass estimate against the lookback time to the burst. Points are color coded by burst strength.
Bias is stellar mass estimate vs. lookback time to burst (simulations)
There appears to be a weak trend with burst age up to about 1 Gyr, but at all burst ages and strengths there are biases on both sides of zero. It isn’t clear to me what, if anything else, is driving biases in either direction. The one thing I can say for sure is that the models are overconfident in their ability to estimate the stellar mass since the typical 1σ error bar is under 0.02 dex while the scatter is around ±0.1 dex. I actually think 25% uncertainty in stellar mass estimates is optimistic.
I remembered a short while ago that “outshining” is the term of art in the industry for the situation in which light from recent star formation overwhelms that of the older population. This seems to be a fairly major concern in the literature. A full text search of ADS found 722 instances of its use in the astronomical literature with an explosion of usage after 2006. A quick scan of titles suggests perhaps half of the papers are about SFH modeling. Of course the word is often used in other contexts.
As a slightly more stringent test for outshining I ran one more simulation with a more recent and stronger starburst (tb=0.25 Gyr, fb = 1) than the earlier simulations. Even though the light of the burst population dominates the old base population the latter does have some effect of the combined spectrum (in red below, and offset vertically for clarity): it is redder and the line strengths are altered somewhat relative to the burst population.
Synthetic spectra – strong recent starburst. Fluxes are logarithmically scaled. Total flux (red) is vertically offset.
The model actually captures both the ancient and recent star formation history rather well. The mass growth marginal confidence band at old ages covers the input right up to the beginning of the burst build up, and the post-burst SFR is modeled accurately. The total mass in the model slightly exceeds the input: log(M*) = 4.88±0.02 model, 4.84 input. The model specific star formation rate is nearly identical: -9.33±0.08 model, -9.36 input.
Simulation – strong recent starburst
Shortly after starting these simulations I noticed a paper by Suess et al. (2022) describing simulations with a similar objective of testing the ability of a code named PROSPECTOR to recover star formation histories of post-starburst galaxies in the ideal case of inputs matched to the model, i.e. the inputs are used to generate the mock data and then to fit it. I’m not going to say a lot about either the code or paper. IIRC the first published description of the code (Leja et al. 2017) claimed it to be the first to model non-parametric star formation histories in a fully Bayesian framework. As far as I know this is true in the published literature but they only could use a few very broad time bins; the version used in Suess uses 9. I was already using the full time resolution of my adopted SSP model libraries by then.
The 2022 paper only shows a single, no doubt cherry picked example of a fit to mock data. Like mine their model star formation histories fail to cover the inputs for some age ranges. On the other hand their fits to a mock spectrum appear to be rather poor with large systematic errors. In every model run of mine residuals look very much like 0 mean Gaussian white noise with the expected deviance. They appear to show a similar range of deviations from input stellar masses with no significant error in the mean. Another striking similarity is they find a definite floor to late time star formation rates. As I’ve noted many times my models will always include some contribution from very young populations and there seems to be a floor around 10-11.5 /yr in specific star formation rate.
A much more recent paper by the same group (Wang et al. 2025) looked at simulated data from quite different systems, namely ones with bursty star formation on short time scales. Their work was motivated by yet more simulations of galaxy formation in the early universe. I’m again not going to comment much on this paper except to notice that they concluded that “given the correct SFH model, it is indeed possible to infer the SFH by performing SED fitting.” In other words they had to fine tune their prior to get to the right posterior. I’m sure it’s not as tautological as that sentence appears. Anyway, this motivated me to take a brief look at a few multiple burst simulations. The one shown below has two very sharply peaked ones with roughly the same peak SFR but more total mass in the older one. The model has spread out star formation over the entire interval between the input bursts with a slower rise and decay. Once again the maximum likelihood fit obtained with non-negative least squares captures the timing and relative magnitude of the bursts rather well.
Simulation – 2 short starbursts
Recall that my stellar contribution estimates are parametrized as an N-simplex with an implicit Dirichlet prior with concentration parameter α = 1, which is uniform on the simplex. In principle adopting an explicit prior with a concentration parameter < 1 should encourage a more bursty star formation history without favoring any particular ages, and it did (this run used α = 0.25):
Simulation – 2 short starbursts, modified prior
Here are histograms of a few summary quantities I track: the present day stellar mass, specific star formation rate (100 Myr average), and the summed log-likelihood of the fits to the spectra. Both runs underestimated the sSSFR because the recent burst was more spread out in the models.
Two burst simulation: comparing two priors on stellar contributions for sampled stellar mass, SSFR, and log-likelihood of fits
Finally, I did a few runs with two other libraries: EMILES, which has been my base SSP model library for some time, and BPASS with single star evolution and an upper mass limit of 100 M☉. The parameters I used for the star formation history resulted in a gentle late time revival rather than a burst. Both model runs had late time bursts, although the mass added was negligible. The EMILES run has the characteristic jumps at ages where the age intervals change. Although it’s small the BPASS run has the jump at 1.6 Gyr that I noted previously.
Simulation with C3K input and EMILES as test library
As with real data the EMILES models have some systematic errors around the trough around 7000-7300Ã…, and also in the blue near the Balmer break.
The BPASS models fit the data surprisingly well, despite using completely different sources for stellar atmospheres.
Comparison PP fit with C3K as test library:
The table below summarizes a couple of the quantities I track. The Progeny C3K models that were used to create the inputs recovers them flawlessly. The other two recover the mass (BPASS is low by more than its nominal uncertainty), but are biased on the high side in late time star formation rate estimates.
Stellar mass
sSFR
input
4.78
-9.91
Progeny C3K
4.78 (0.017)
-9.90 (0.054)
EMILES
4.82 (0.012)
-9.62 (0.028)
BPASS
4.67 (0.03)
-9.73 (0.049)
I’m going to get back to real data now using the Progeny generated libraries. The simulations were a useful exercise if for no other reason than to show that timing faded starbursts can’t be done very accurately, at least with full spectrum fitting at visual wavelengths. I did get some ideas for small code improvements, and the idea of stacking star formation rate and mass growth histories seems like a useful visualization tool.
I’ve decided to take a break from real galaxy data and run some simulations of synthetic galaxy spectra. The main motivation was to help make an informed decision about which subset of the ProGeny based SSP libraries to use: the ones based on the MIST isochrones have ages uniformly logarithmically spaced at 0.05 dex intervals and 15 metallicity bins. Most of the latter are considerably sub-solar and aren’t likely to be major contributors to the light of nearby massive galaxies. For possibly metal poor dwarf galaxies it’s always possible to pick a metal poor subset — I have a subset of EMILES that I’ve used for that purpose for some time. More important to me is the choice of age bins. As I mentioned in the last post I did model runs with both 42 (0.1 dex age intervals) and 83 ages, with SSP models younger than log(T) = 6 and older than the currently accepted age of the universe excluded. Since execution time of a Stan model is roughly proportional to the number of parameters and these models are rather time consuming I’d like to know if the coarser age spacing has sufficient resolution. I will show below why it’s correct to exclude the very youngest populations.
Other reasons to do these kinds of exercises are to test the accuracy of retrieval of key quantities in idealized conditions where model assumptions are met by the inputs, to compare different model libraries, or to compare different modeling codes (something I don’t plan to do at present). Performing simulations is de rigueur in the SFH modeling industry and there are many examples in the literature, both as appendices and standalone papers — to name one the second of the two ProGeny papers I linked last time falls in this category. I’ve even done some limited simulations as far back as 2018, but the only case I wrote about was a very simple one.
This time I’ve decided to be more systematic and at least slightly more realistic. Since I remain interested in post-starburst galaxies I am creating simulations with two components: a base population that begins forming stars at the beginning of the universe, which I take to be a lookback time of \(10^{10.125}\) yr. = \(13.36\) Gyr. The second component is a “burst” population that commences star formation at a lookback time of \(\mathsf{t}_b\) with a relative strength of \(\mathsf{f}_b\). The base population has solar metallicity (Z = 0.02) while the burst is slightly subsolar (log(Z/Z☉) = -0.25). This obviously isn’t meant to represent a realistic model for chemical evolution — the purpose again is to test the ability to recover the input stellar metallicities in idealized circumstances. In keeping with my usual attitude of treating this as an afterthought I haven’t gotten around to exploring this issue yet.
Both the base and burst populations have time varying star formation rates proportional to Gamma probability distributions. The Gamma distribution is a two parameter family of univariate functions that can have a fairly wide variety of shapes, all of which have a single mode at \(t \ge 0\) followed by a monotonic decline asymptotically approaching 0. See the table below for the most salient properties. I don’t know of any real astrophysical justification1Abramson et al. (2016) argued for a log-normal form for the evolution of the cosmic SFR density and suggested that this describes the evolution of individual galaxies as well. More recently Katsianis, Yang, and Zheng (2021) argued for a gamma distribution fit to the evolution of the CSFRD and again claimed with some physical motivation that this form holds for individual galaxies. I was unaware of the latter work when I started this exercise. for this choice but it’s notable that it includes two of the most popular parametric forms for modeling star formation histories: exponentially declining (\(\alpha = 1\)), and what’s generally called the “delayed-\(\tau\)” model (\(\alpha =2\)) in the literature. Note that using a probability distribution has nothing to do with probability per se, but a probability density has two useful properties: it’s non-negative on its support and has a finite integral (equal to one when properly normalized).
Given shape and scale parameters the mass in each time bin is proportional to the difference in the cumulative gamma distribution at the beginning and end of each interval. The star formation history for each component is just the mass formed in all age bins, with the burst component multiplied by the burst strength. These are scaled to sum to 1, and the spectra are straightforwardly calculated as matrix products of the input SSP model fluxes with the mass histories. The “flux” values are multiplied by 105 for the sole purpose of making them approximately unit scaled. This has the effect of making the total mass born in the simulated “galaxy” equal to 105 M☉. The fluxes are interpolated to the same logarithmic wavelength grid used by SDSS, convolved with a gaussian to simulate stellar velocity dispersion, then truncated to the wavelength range (3621 Ã…, 9000 Ã…). The velocity dispersion is a settable parameter, but so far I have only used 140 km/sec. The base and burst spectra, which were calculated separately for visualization purposes are added, then multiplied by Calzetti attenuation with a user selected optical depth, then finally perturbed with Gaussian random noise. The flux variances are Poisson-like, that is they are proportional to the “counts” in each wavelength bin and scaled to a target average signal to noise. Again this is a settable parameter, but so far I have used a target SNR of 40, which is fairly typical for a fiber spectrum near the center of the IFU of a bright MaNGA galaxy. So far at least I do not add emission lines, but they are allowed in the models.
The subsequent modeling workflow is essentially the same as I use with real data, with some very minor tweaks reflecting the fact that the flux values are arbitrarily scaled. The Stan code is exactly the same as my current working version. Model runs use the same number of warmup and sampling iterations as my real data runs (250 and 750 respectively with 4 independent chains). Post processing is also essentially the same. I extract the posterior predictive fits to the spectra, the model star formation histories, the optical depth of attenuation and the reddening curve slope parameter. I compute the present day stellar mass, the 100 Myr averaged star formation rates, and specific SFR. The stellar mass and SFR are arbitrarily scaled but would scale by the same factor if these were real data from a distant galaxy, so the SSFR would be the same.I also track a few observables: the 4000 Å break index Dn(4000) and HδA.
This time I am exploring a wider range of star formation histories than my previous effort, which only examined a case or two with instantaneous bursts. With seven parameters (plus two more that I’ve held constant so far) it isn’t really practical to do a comprehensive grid of models, so for the most part I have randomly selected parameter values in the ranges shown below. At the time of writing I’ve run a few more than a hundred models. Most of these use the “small” version of the ProGeny C3K base SSP models as the test library, with a few using the input library as the test. I’ve also done a few with the “large” version (see the previous post for details of the ingredients). I plan to do some simulations with the other model libraries I’ve used, especially EMILES, but that’s somewhat of a different project. For now I’m examining the accuracy of outputs for the idealized case where the inputs satisfy the assumptions of the model.
\( \alpha_0 \in [1, 2] \)
\(\theta_0 \in [2, 6] \)
\(\mathsf{t}_b \in [0.5, 2]\)
\(\mathsf{f}_b \in [0.1, 0.75]\)
\( \alpha_b \in [1, 2] \)
\( \theta_b \in [0.01, 0.25] \)
\( \tau_V \in [0.0, 0.5] \)
Here is a cherry-picked example of a fairly strong and recent burst, using the small C3K based library as the test. First is the “observed” spectrum and the replicated values from the posterior (top pane), and the posterior residuals (bottom pane). The residuals look very much like — and pass standard statistical tests for — Gaussian white noise. This is rather different than I usually see with real data, of which a number of examples can be found by scrolling down through previous posts, where there are often large regions of spectra that are systematically misfit. This reflects the fact that all model assumptions are met by the data by construction, so fitting the data is expected.
Simulated spectrum and “posterior predictive” fit and residuals.
The model star formation history from the same simulated data contains some surprises, and these turn out to be fairly generic features of most model runs. This is a bit of a new graphical display for me. I’ve put the star formation rate history above the mass growth history with the same age scale for each so I can get different views of the same sequence of events. The input model parameters are on a line in between. The blue lines are the input star (mass) formation history. The red line in the second plot is from the preliminary maximum likelihood fit. The black lines and grey ribbons are the marginal means and 95% confidence intervals. Reading from right to left there are several points to note:
The details of the pre-burst SFH are largely lost, although the total mass formed in the model is roughly the same.
The ramp up to the peak of the burst begins sooner and is more gradual than the input. Note that both the input SFR and mass growth are well outside the model confidence intervals in the ramp up period.
Most significantly in my mind at least, the peak of the burst is much later (more recent) in the model than the input. This turned out to be a fairly generic feature of these models.
In this run the post-peak decay is rather steeper on average than the input. In model runs with a rapid input decay the model will usually have a more gradual one.
The input and model SFR eventually converge and the late time model SFH agrees very well with the input, except
There’s an odd upturn in SFR at the very youngest ages (< 3 Myr). This happens to varying degrees in every simulation model run and in all the real data I’ve looked at. It turns out there’s a simple explanation for this. There is virtually no spectroscopic evolution in the relevant wavelength range in the first few Myr of a coeval population’s life. Since the spectra are indistinguishable there is no way to constrain the individual contributions without a more informative prior, and the models are correctly estimating the same marginal distributions for the youngest few populations. Since the time intervals get shorter with decreasing age this produces an apparent upturn in SFR.
Black line and grey ribbon: model results for SFH and MGH. Blue line: input. Red line: MGH from maximum likelihood fit.ProGeny generated SSP model spectra — C3K base models – ages 1-10 Myr, solar metallicityProGeny generated SSP model spectra — C3K base models – ages 1-2 Myr, solar metallicity. Spectra are vertically offset.
There are two surprises in the simulations I’ve run so far. First is that, like the example above, the time of peak star formation in the model burst tends to be biased downwards (more recent) from the input. The bias gets larger in magnitude with increasing lookback time to burst onset and can be as large as several hundred Myr as seen below. The few outliers here were weak bursts with very large variations in the time of maximum SFR.
Bias in time of peak SFR vs. input burst onset time
The other surprise is that the 95% confidence bands of the model star formation histories, whether viewed as SFR or mass growth histories, fail to cover the inputs for at least some epochs around the time of the burst. In other words the models are saying that the actual input star formation histories are highly unlikely. Why is this?
It’s not, I think, because of any convergence failure. Stan is quite aggressive at flagging potential issues and there were virtually none in any of the model runs. I also don’t think it’s because the model runs were too short. I’ve occasionally gathered much larger samples with longer warmup times as well and except for exploring the tails of the distributions a little more thoroughly there is little change in marginal 95% confidence intervals.
Most of the model runs were done with the reduced size SSP library with just half of the available age bins (0.1 dex intervals). This doesn’t seem to have reduced the real effective age resolution, but in the small number of runs I did with the full set of ages the marginal contributions have larger confidence intervals, so this could increase the coverage.
I suspect though that the prior on the star formation history is having a larger effect than I would have suspected. The main clues here are the maximum likelihood fits. which often produce mass growth histories (the red lines in the plot above and the ones below the fold below) that are rather close to the input MGH, also falling outside the 95% model confidence bands for at least some epochs. In my current workflow the slightly perturbed maximum likelihood solution is used to initialize the Stan model. In most of the model runs the samples quickly move away from the initial values. While the fits to the data are good in all model runs the summed log-likelihood of the sample spectra is always less than the maximum likelihood solution, in the sample shown below by about 7%.
Log-likelihood of posterior samples and maximum likelihood solution for a simulated model run
Below the fold I’ve posted some more cherry picked examples of model star formation histories. A few general comments about them:
The late time star formation rate is generally well captured by the models.
The ancient (pre burst) star formation history is sometimes captured well with weak bursts. Particulary noteworthy is the second example below, which has a recent and weak burst with a gradual ramp up and decay. The entire star formation history is modeled accurately. In the literature there are sometimes references to “outshining” or a “dazzle” effect in (post)-starburst galaxies. This seems to be a real thing.
Early time bursts aren’t necessarily recognizable as such. In some cases the model just shows an extended period of star formation followed by more or less gradual quenching. I believe this is consistent with some claims in the literature, but I don’t care to look for the relevant papers right now.
I’ll conclude with a look at results for all simulation runs (so far) for some of the summary quantities I track. First, the present day stellar mass is on average unbiased but in individual runs there are errors as large as ±0.1 dex. The typical posterior uncertainties are only ~0.01 dex, so the models are consistently too optimistic. I’ve noted this in real data, which shows similar nominal uncertainties with moderately high S/N data.
Not shown here but I did notice a weak dependence on the input burst onset time or time of maximum SFR in the sense that recent bursts tend to underestimate the stellar mass. This is another possible manifestation of the “outshining” effect.
Error in model stellar mass vs. input stellar mass
The most dramatic bias is a tendency to overestimate the specific star formation rate when the input SSFR is low. This is also something I’ve seen in real data: as I’ve noted a time or two even the reddest and deadest of ellipticals seem to have a floor of about -11.5 for the estimated SSFR.
One thing I didn’t think about until finishing this analysis is that the contribution estimates tend to be skewed to the right when they’re near 0, so the median might be a better estimate of central tendency than the mean. That might make a (probably) small difference. I will look into this.
Error in model specific star formation rate vs. input SSFR
Finally, there’s a nearly linear relation between the error in the posterior estimate of the optical depth of attenuation and the input. Note also the stratification of the reddening slope parameter δ: at relatively high input optical depths the largest departures have the reddest attenuation curves. I think this behavior is directly related to the prior. Since I don’t think the attenuation is very well constrained by the data I use fairly informative priors on both the optical depth and slope parameter; both priors are truncated normals with a lower bound of 0 for Ï„V and a range of (-0.5,1.2) for δ. This is an easy thing to test simply by changing the priors for these parameters.
Mean and standard deviation of model optical depth – input optical depth vs. input optical depth
I am not quite finished with these simulations. I want to expand the range of input parameters a little bit, and I need to look a little more carefully at subsets of the ProGeny SSP models. The main result so far is that my hopes of timing key events in post-starburst galaxies’ histories were probably too optimistic.
After my not so insightful realization recounted last time that my attempt to modify the prior on star formation histories wasn’t actually doing anything I thought a little further about how to specify one. Gaussians are always popular choices for priors, so why not give them a try? For a first cut I added the following lines to the “transformed data” section of the Stan model:
Even though the prior allows values of the stellar contributions that are infeasible given the simplex declaration this causes no technical problems: the models sample without complaint and parameters satisfy all constraints. Execution times are comparable to the original model formulation and convergence diagnostics are OK. But, the model runs had some unexpected features. I did a set of model runs for a single MaNGA galaxy from the post-starburst ancillary sample — mangaid 1-201936 (plateifu 10220-3703). Model star formation histories compared to the original model are shown below for the 58 binned spectra:
MaNGA plateifu 10220-3703 (mangaid 1-201936). Model star formation histories with two different SFH priors.
What’s striking here is that several of the spectra in the low S/N outskirts of the galaxy have nearly constant star formation rates with very little sample variation. In other words the models are basically returning the priors. The cause of this behavior isn’t quite clear. Relatively low signal to noise seems to be necessary, but not sufficient since similarly noisy spectra have essentially the same SFH’s as the original model formulation. It also isn’t due to convergence failure because much longer runs with more adaptation iterations show the same behavior. It is possible perhaps that the posteriors are significantly multimodal and Stan is preferentially falling into one of them. Notable also is that the fits to the data measured by log-likelihood are virtually identical even for the runs with the anomalous SFH’s. At the very least this tells us that uncertainties in quantities derived from the models are considerably larger than within model run variations — of course I have always believed this and said so a number of times.
After trying several variations on this theme that either had none at all or undesirable effects on sampling, and after some additional consideration I think that, given the model parametrization, the uniform on the simplex prior for stellar contributions is actually the one I want. That leaves the question of what, if anything, to do about the abrupt jumps in model star formation rates.
One possibility is simply to redefine the endpoints of the age bins to be, say, halfway between nominal SSP ages instead of at the model ages as is my current practice. In the case of the EMILES library this would mean for example that the 3.75, 4, 4.5 Gyr bins would have widths of 0.25, 0.375, 0.5 Gyr instead of the present 0.25, 0.25, 0.5. This involves no change to the actual model runs at all, so most quantities derived from the models are unchanged.
Another solution is to adopt a library with a more uniform age progression. One with approximately equal increments in log age seems preferred. As yet there have been no published updates to the MaStar based SSP libraries mentioned last time, so I’m waiting for them, while still considering generating my own.
I’m going to briefly return to BPASS based models. After that I’m not sure.
One thing I’ve wondered about for a while is the extent that priors on SSP model contributions affect modeled star formation histories. Previous experience suggests not much at all unless the prior is highly constraining. To be a little more specific in my current workflow I normalize the SSP model spectra to have average flux values of 1 in (approximately) the V band, and also adjust the galaxy fluxes in the same way. In the Stan model the stellar contribution parameters are declared as a simplex, that is a vector with non-zero elements summing to 1. That makes the parameter values the fractional contributions to the (unreddened) galaxy flux at V. My current working code doesn’t provide an explicit prior for the stellar parameters, but they have an implicit proper prior of a uniform distribution on the appropriate dimensional simplex. More technically the prior is a Dirichlet distribution with all concentration parameters equal to 1. Note the marginal distributions aren’t uniform, but they are all the same. One implication of that is that a typical draw from the prior1Stan doesn’t sample from the prior even for initialization, but of course the prior influences the posterior through Bayes’ rule. will have jumps in the star formation rate at exactly the times where the width of the age bins jump, a problem that I’ve noted several times before.
A possible solution to this problem is simply to alter the prior to encourage smoothly varying star formation rate rather than smoothly varying light contributions. It turns out I’ve been feeding my Stan code the data I need to do that: the initial mass in a given model SSP is inversely proportional to the normalization factor applied to the spectrum, and the star formation rate is just the mass divided by the time interval assigned to the SSP. Both of those quantities are passed as data to the Stan model even though they weren’t used in any way previously. I added just 3 lines to the code to change the prior on the stellar contributions. In the “transfored data” section
If I did this right the prior is completely agnostic to any star formation history, whereas the previous implicit prior was completely agnostic to any run of light contributions.
The modified code compiled without complaint and there’s no discernible difference in either execution time or convergence diagnostics.
I’ve only run this on one set of galaxy data, from MaNGA plateifu 8565-3703 (mangaid 1-92735). This is one of Schawinski’s “blue early type galaxies,” chosen mostly because the data were binned to just 34 spectra so a complete set of model runs only took a few hours. And, as seen below the other things that don’t change are the modeled star formation histories.
Plateifu 8565-3703 (mangaid 1-92735). Star formation history models with two distinct priors on stellar contributions.
I need to do some more validation exercises, but it appears my long ago conclusion that the choice of prior on the star formation history has little effect was correct. The data dominates the model outcome through the likelihood.
Update
Since I hit publish a few days ago I realized two things. First, the weights I applied to the stellar contributions to “encourage” a smoothly varying star formation rate were inverted. What I should have used was:
The second thing I realized is this makes no difference at all given the form of the prior. The transformation simply maps a point on the simplex to another point on the simplex that has exactly the same probability density since by construction the prior is uniform on the simplex.
So, it should have been expected, and it’s a good thing that it in fact happened that the model runs produced the same results. What happens if I use the following prior, with the weights supplying the parameters for the Dirichlet?
target += dirichlet_lupdf(b_st_s | norm_sfr);
This failed to sample almost always. I’m not entirely sure why, but I suspect this turns a problem with relatively simple geometry at least with regards to the prior to one with a complex and troublesome geometry.
This little experiment actually told me nothing about the effect of priors on model star formation histories. Two of the priors are actually the same, and the third fails for reasons that aren’t completely clear. I may experiment with different forms of prior. I’m still, of course, looking for a new SSP model library.
When I was doing my initial fits to the M31 MaNGA spectra I noted two that I initially thought were contaminated by foreground stars, and therefore I masked them to prevent further analysis. One of the two, in MaNGA plateifu 9677-12701 (mangaid 52-8) is a certain foreground star and won’t be discussed further. The other, in plateifu 9678-12703 (mangaid 52-23), turns out to be a luminous red supergiant that’s a genuine resident of M31. This is confirmed by two nearly contemporaneous catalogs of M31 red supergiants: the one by Ren et al. (2021) that I noted previously and Massey et al. (2021), which I stumbled upon more recently.
In fact there are two comparably bright red supergiants in this IFU. One that’s about 9″ north of the masked one probably should have been masked by whatever criteria I used, but it’s likely I failed to notice the fit to the data since I don’t have the patience to look at every spectrum and data fit that pops up. So, here is the spectrum, displayed in (negative) magnitudes with arbitrary zero point. The blue spectrum is the closest match in my SSP library, a 10 Myr old population with the highest metallicity I used (2.5 Z☉. This is one of the theoretical spectra from PyPopstar). This sorta looks right except it’s much too blue. The solution to that is, of course, to add some reddening through dust attenuation.
plateifu 9678-12703 (M31 10 kpc ring)
Spectrum contaminated with red supergiant and closest match SSP model spectrum
The maximum likelihood (non-negative weighted least squares) fit did just that, with only a single stellar contributor and a very high dust attenuation of Ï„V = 3.0. This still doesn’t quite work: the residuals are rather strongly sloped in the blue and the details of the absorption features in the red aren’t quite right.
plateifu 9678-12703 – NNLS fit to spectrum contaminated with red supergiant
I still use a Calzetti attenuation relation in my NNLS fits. The Bayesian fits using Stan have the more flexible attenuation prescription that I described back in this post, and that helped considerably with the continuum as seen in the plot below. The absorption features in the red still aren’t fit well. The model has an even more extreme attenuation estimate with a much “grayer” than Calzetti slope, with Ï„V = 4.38 ± 0.05 and δ = -0.33 ± 0.011see the link above for the meaning of these parameters.
9678-12703 – posterior predictive fit to spectrum
The model star formation history (displayed as a mass growth history below) isn’t completely implausible. The presence of a very luminous evolved star indicates the region is at least some Myr old, and a rapid onset and decline of star formation is typical for star forming regions in mature spirals. The recent episode of star formation added about 7% to the present day stellar mass, while at least 60% was in place by 8 Gyr ago (per the model).
plateifu 9678-12703 (M31 10 kpc ring)
Model mass growth history for a region containing a bright red supergiant
Nevertheless I consider the model results to be highly suspect, mainly based on the very large optical depth. Massey estimates the attenuation for the red supergiant to be AV ≈ 1.18, although this is apparently based on a formula rather than a direct empirical estimate. But another piece of evidence that’s close to a smoking gun is an estimate based on the Balmer decrement of emission. Despite the lack of apparent ionizing sources outside the bright H II regions in the west there is widespread diffuse emission in this region with star-forming like line ratios. The estimated optical depth derived from the Balmer decrement for this region is Ï„V, bd = 1.47 ± 0.23 (1σ), reasonably consistent with the Massey estimate and with the values derived for the rest of the IFU.
How widespread a problem is this? Below I plot the Balmer decrement derived optical depths against the stellar based estimates for all spectra in M31 MaNGA with star forming emission line ratios, about 11% of the entire sample. The 5 most extreme outliers are in this IFU in the regions surrounding the two bright red supergiants (the masked spectrum would also be in this region of the plot). The same 5 regions are also extreme outliers in the SFR vs. stellar mass and SFR vs. Hα plots that I showed early on. So, even though there are many cataloged supergiants in the study region these two appear to be uniquely bright and to have had the largest impact on model results.
M31 MaNGA – Optical depth off attenuation estimated from Balmer decrement vs. model values of Ï„V
Of course there are hotter bright stars in the study area and these could affect results in different and possibly unexpected ways. For example the outermost IFU contains one bright star that GAIA estimates has a surface temperature of 5500 C, which would make it a G supergiant if it’s in M31. I noted in the last post that the model star formation history for that region looks like a post starburst with an age around 800 Myr. This is, I think, several galactic rotation periods, and stars born that long ago should have dispersed by now unless they’re gravitationally bound. There’s no sign of a star cluster there nor is there a cataloged one nearby, so it seems likely to me that the “starburst” is an artifact. As I noted in the last post though the fit to the data is quite good.
These examples illustrate an issue that’s fairly well known, which is that using simple stellar populations as building blocks of low mass stellar systems are potentially affected by so-called “stochastic” effects, which simply means that the distribution of stellar masses can vary randomly from what’s assumed in the SSP models. Specifically, in M31 there are individual stars luminous enough to affect spectra. One possible solution might be to add some stellar spectra to the library. I might give that a try some day.
I’m going away and won’t be writing for a while. I’m hoping to acquire or build a SSP model library based on SDSS MaStar spectra yet this year. This is a much larger collection of stellar spectra than has been previously available and it has the advantage of having the same flux calibration and (approximately) spectral resolution as the SDSS galaxy spectra. I also plan to return to my study of post-starburst galaxies.
After a fairly long break I want to get back to M31 and MaNGA for one, or perhaps several posts and take a more detailed look at my model results. I still haven’t decided where I’m going to take this investigation. I may examine every IFU or just the ones that I found most interesting, and I’m not sure which of the many quantities that I estimate I’ll discuss. Besides my models I’ve retrieved a number of catalogs of interesting objects using Aladin. These include in particular H II regions (Azimlu et al. 2011), OB associations (Magnier et al. 1993), and red supergiants (Ren et al. 2021). All of these are products of recent or ongoing star formation. There are of course a huge number of catalogs of just about every type of astronomical object found in galaxies, and I may examine some more depending on what interests me.
For orientation here’s a screencap of the Legacy Survey sky browser’s false color GALEX image of the northern half of M31 with the IFU positions overlaid and labelled with MaNGA’s plateifu identifiers. As a reminder these are all located within the PHAT survey footprint and specifically within the region for which star formation histories were estimated by Williams et al. (2017).
Screen capture of Legacy Survey Galex image of M31 with MaNGA IFU overlay
Before getting to individual IFU’s here is one more set of IFU-wide results. The following three graphs are model mass growth histories in units of present day solar mass per kiloparsec2. These are uncorrected for projection effects.
There are a couple interesting points here. There’s a clear stratification of mass density with projected radius, with about a factor 30 decline from the innermost to outermost IFU. This is in fairly good agreement with Williams’ estimate in their Figure 14.
The other thing to note is that all regions had most (> 55%) of their stellar mass in place by 8 Gyr ago and 92-99% in place by 1 Gyr ago. The largest fraction of recent star formation is in the IFU 9678-12703, which is very close to the region with the highest SFR in this half of the galaxy. There is also a trend towards later mass build up with increasing radius, which is completely consistent with the “inside-out” growth paradigm. The outermost IFU, 9678-12701 at about 16kpc radius has formed about 5% of its present day stellar mass in the past Gyr.
As I said in the previous post I don’t see clear evidence for a widespread burst of star formation that’s widely believed to have occurred around 2-4 Gyr ago. A confounding factor in my models is that they invariably show jumps in SFR at times when the interval between SSP model ages change and the two oldest of these occur at 1 and 4 Gyr, so this produces a possibly spurious period of apparently accelerated star formation. I hope to find (or perhaps produce) a set of SSP models with a better age distribution this year.
Growth of stellar mass density – inner disk M31 MaNGA IFU’sGrowth of stellar mass density – M31 MaNGA IFU’s in 10 kpc ringGrowth of stellar mass density – outer disk M31 MaNGA IFU’s
I think I’m going to hit publish now and resume with inner disk IFU’s next time.
In this post I’m going to compare IFU wide star formation histories from my models to those of Williams et al. (2017) in the nearest 83″ by 83″ PHAT tile to each MaNGA IFU in the study. I picked the Williams paper for comparison mostly because it’s possible to! They give a complete tabulation of model results for all regions and all 4 sets of isochrones that they used, and these are available through the Vizier service. Specifically I used their Table 2, which provides star formation rate densities summed over all metallicities. Since the SSP model spectra I use are based on BaSTI isochrones I initially compared to their BaSTI based models. One problem with the Williams comparison is the authors had a very wide youngest time bin of 300 Myr, which is where my models should generally have the highest precision (I make no strong claims about accuracy). It would be nice to do a similar comparison to the earlier companion paper on recent star formation by Lewis et al., which gives a much finer grained view of the last ~half Gyr, but unfortunately there is no published tabulation of their results.
At the other end of the timeline the oldest bin is also very wide, from 8 to 14 Gyr lookback time. This isn’t a surprise: the limits for reliable photometry of individual stars were rather shallow, no fainter than m = 28 or Mg ≈ 3.6 according to Lewis. This is brighter than the main sequence turnoff at 8 Gyr, so any information about the truly ancient star formation history is coming from giant branch stars which have very similar evolutionary tracks at old ages1Checked by downloading a few isochrones from the BaSTI website.
At the end of my last post I mentioned the necessity to correct densities for the rather large inclination of M31’s disk. It turns out though that I reproduce Williams’ Table 3 from their Table 2 if the densities are uncorrected. Their tabulated SFR densities are in units of 10-4 M☉arcmin-2/year. One arcminute at their adopted M31 distance is about 0.227 kpc, so to convert to star formation rates per kpc2 the values in table 2 are multiplied by 19.8 × 10-4. From my models I sum the star formation rates over all modeled spectra in each IFU and divide by the total area in fibers, with each fiber covering a projected area of 42.78 pc2. Note that I do not try to analyze a single composite spectrum summed over the entire IFU since dust attenuation is quite patchy.
The graphs below overlay my modeled star formation rate densities on those of Williams in the tile with the nearest center to that of each IFU. The ribbons indicate the nominal 95% credible limits of SFR. These are certainly wildly optimistic. Table 2 of the paper includes uncertainty estimates, which I chose not to include. SFR densities are linearly scaled with different limits for each plot. The time scale is logarithmic. Something in between linear and logarithmic would seem more appropriate since this perhaps gives too much space to very recent times, but I haven’t found a suitable scaling.
These are ordered into three groups by location: the inner disk which is everything inside the 10 kpc ring, the 10 kpc ring, and the outer disk which is everything outside the 10 kpc ring. There’s some ambiguity about the locations of the plateifu’s 9678-9101 and 9678-12704. The first of these is about 12 kpc from the nucleus in what could be either a wide section of the 10 kpc ring or a separate structure. 9678-12704 appears in projection to be just outside the 10 kpc ring but it may be considerably farther out in a segment of spiral arm at ~15 kpc.
Commentary will continue after the graphs. I will discuss the individual IFU’s in more detail in later post(s).
Inner Disk
M31 MaNGA ancillary program – my star formation history models summed over each IFU compared to nearest tile in Williams et al. (2017) with BaSTI isochrones. Inner disk IFU’s.
10 Kpc ring
M31 MaNGA ancillary program – my star formation history models summed over each IFU compared to nearest tile in Williams et al. (2017) with BaSTI isochrones. 10 kpc ring IFU’s.
Outer disk
M31 MaNGA ancillary program – my star formation history models summed over each IFU compared to nearest tile in Williams et al. (2017) with BaSTI isochrones. Outer disk IFU’s.
The first two things I noticed were that star formation in every region declines monotonically from very early times to at least 4 Gyr ago. It also starts out lower than in the PHAT team’s models. Because of this early time mass deficit all of my models have smaller current day stellar mass densities by varying amounts. I don’t really have a pat explanation for this. Some authors have posited a “dazzle effect”2I’m going to discuss this a little further at the end of the post where recent star formation obscures the contribution of old populations. It’s certainly likely that this occurs, but if these Bayesian models are behaving as I hoped this lack of information should manifest as larger uncertainties rather than a systematic bias. Well, my hope could be wrong. On the other hand I don’t see strong evidence in these models for such an effect. From my eyeball analysis I don’t see an obvious correlation between present day star formation and the size of the early time deficit.
Another possibility is a systematic difference in the amount or shape of attenuation between my models and theirs. There is another well known “degeneracy” between stellar age and attenuation in SFH modeling, but I haven’t yet investigated whether this could be occurring here.
The PHAT models have a very long interval from 8 to ~2-3 Gyr ago with very little star formation. Some authors find evidence for a large increase from about 2-4 Gyr which is usually attributed to a merger or perhaps close encounter with M33. This isn’t seen in the BaSTI based models but there is a large more recent burst from about 1-2 Gyr lookback time. My models see neither a cessation of star formation nor a particularly large burst at intermediate ages. As I’ve noted before my models “want” to have smoothly time varying light3and therefore mass contributions and this might make a modest burst at moderately large ages difficult to discern. Another confounding factor arises from the abrupt changes in age intervals (at 0.1, 0.5, 1, and 4 Gyr) which results in the sawtooth pattern in SFR that’s obvious in every plot above.
At ages younger than 1 Gyr there’s generally good agreement about the course of star formation up until the youngest age bin of width 300Myr in the PHAT models. My models have anywhere from slightly to dramatically higher SFR densities averaged over the most recent age bin. I suspect this is because many of the IFU positions were chosen to be in regions with active star formation. In particular the plateifu 9678-12703 (mangaid 52-23) is very close to the region in the 10 kpc ring with the highest density of ongoing star formation in the northern half of the disk.
I plan to discuss the individual IFUs in more detail in a later post. Below the fold are some more graphics: mass growth histories and SFR densities compared to the PADOVA isochrone based models.
One of the ancillary programs (with principal investigator Julianne Dalcanton) in the final MaNGA release targeted 18 fields in the disk of the Andromeda galaxy M31. The targets were selected from within the footprint of the “Panchromatic Hubble Andromeda Treasury,” aka PHAT1not my coinage., also with PI Dalcanton. The initial PHAT survey description was in Dalcanton et al. (2012) and was followed by a lengthy series of papers. Especially relevant for this discussion are two papers describing estimates of the recent and ancient star formation histories of the disk outside the area dominated by bulge light: Lewis et al. (2015), “The Panchromatic Hubble Andromeda Treasury. XI. The Spatially Resolved Recent Star Formation History of M31” and Williams et al. (2017), “PHAT. XIX. The Ancient Star Formation History of the M31 Disk.” For reference here is a mosaic of HST images in the F475W filter with the IFU locations overlaid:
Mosaic of HST F475W images of PHAT study region with M31 MaNGA IFU positions overlaid
Zooming out to show the whole disk here they are overlaid on a false color FUV+NUV image from GALEX, which gives a pretty good picture of where stars are actually forming:
GALEX false color NUV+FUV image of M31 with MaNGA IFU positions overlaid – screencap from Aladin
This data set provides an excellent opportunity to compare my SFH modeling code to a completely different, more direct, method of inferring star formation histories namely counting resolved stars in color magnitude diagrams. I recently completed model runs for all 18 IFU’s with the same Voronoi binning of stacked RSS spectra, the same modeling code and SSP model spectra as I’ve used for a while now.
There’s no redshift listed in the DRP catalog; NED gives a heliocentric redshift of -0.001, but for purposes of calculating intrinsic quantities I need the “Hubble flow” redshift. I adopted a distance of 761 (± 11) kpc or distance modulus of (m-M)0 = 24.407 from Li et al. (2021), which is the most recent and according to the authors most precise determination to date. With my adopted Hubble constant of H0 = 70 km/sec/Mpc this makes the Hubble flow recession velocity 53.27 km/sec or zdist = 0.0001777. The angular scale is 3.69 pc/arc-second. This distance estimate is a few percent smaller than the PHAT team authors and most other recent literature I reviewed, but fortunately most other sources of uncertainty are much larger.
An issue I noticed early on was the modelled values of the optical depth of attenuation were right at 0 for almost all spectra with only a few much larger exceptions. A quick check of the metadata showed that the values adopted for the foreground galactic extinction almost certainly were taken from the SFD dust maps which faithfully capture the intrinsic dust content of M31 albeit at rather low resolution. These hugely overestimate the actual foreground galactic extinction and that has multiple undesirable consequences. So, I assigned a single extinction value of E(B-V) = 0.055 to all IFU’s, consistent with the NED value of AV = 0.17 mag. The preliminary runs were redone with the newly adopted extinction value.
After binning to a minimum mean SNR of 5 there were 2,624 spectra in the 18 IFUs, of which I ran models for 2,621. Three spectra had apparent foreground stars, although one of those might actually be a red supergiant in M31. The fibers are basically sampling star cluster size and stellar mass regions so a single extremely luminous star could potentially affect a spectrum.
I’m only going to show a few summary results for the entire sample in this post. My goal is to do a more detailed quantitative comparison to (at least) the SFH models of Wilson, for which there are extensive results tabulated. There are of course many catalogs of interesting objects within M31, and I plan to look at some of them.
First, here is a plot of the (100 Myr averaged) star formation rate density against stellar mass density, color coded by BPT diagnostic. The solid line is my estimate of the “spatially resolved star forming main sequence” based on a small sample of non-barred spiral galaxies. The dashed line is the estimate of Bluck et al. (2020), which I commented previously appears to mark approximately the location of the green valley at least with regard to my models. A striking feature of this plot is the apparent stratification into at least three distinct groups that can be interpreted as starforming, quiescently evolving, and passively evolving. I suspect this observed stratification is just the result of hand picking a small number of “interesting” regions. Most or perhaps all of the points in the passively evolving group are in the IFU closest to the bulge, while most of those along and above the SFMS lie near the most vigorously star forming regions in the PHAT footprint. Especially noteworthy are 5 outliers that are well above any others in the plot in terms of SFR density. These are all in the same IFU (plateifu 9678-12703) which is located within the largest star forming region in that quadrant of the “10 kpc ring.”
100 Myr average star formation rate density vs. stellar mass density for 2621 binned spectra in M31 disk. Solid and dashed lines are my and Bluck’s central estimates of the “spatially resolved star forming main sequence.
Next are plots of star formation rate density against Hα luminosity density. The left panel is for all spectra color coded by BPT diagnostic, with Hα adjusted by the modeled amount of stellar attenuation. The right panel shows regions with star forming BPT diagnostics only, with Hα corrected by the observed Balmer decrement. The solid line in both panels is Calzetti’s calibration of the Hα – SFR relationship. The relationships plotted here are consistent with what I’ve seen in other MaNGA samples and with published values, which is encouraging.
Star formation rate density vs. Hα luminosity density for 2621 binned spectra in M31 disk. (L) Emission corrected for modeled stellar attenuation. (R) For regions with star forming emission line ratios only: emission corrected from estimated Balmer decrement.
The obvious point of comparison to my models are the detailed star formation histories in the two PHAT papers mentioned at the top. Unfortunately there is no detailed tabulation of model results in the paper by Lewis et al. The paper by Williams et al. has extensive tables, but there are still a few obstacles to detailed comparisons which I will discuss next time.
A few more items from my handwritten notes that I want to get in pixels. I have never previously tried to correct surface densities for inclination in disk galaxies, but for comparison purposes and because of the large inclination of M31’s disk I need to do so here. I adopted an inclination angle of 77°, so a 1″ radius fiber covers a 3.69 x 16.4 pc (semi major and minor axes) elliptical region, or 190 pc2. Densities need to be adjusted downward by a factor 4.45 or -0.648 dex2This adjustment was not made in the plots above. Since these are plots of densities against densities all points would just shift downwards along lines of slope one..
In order to achieve 100% coverage of the IFU footprints the exposures were dithered to three different positions with overlapping fiber positions. Comparing the area in fibers to the area in spaxels in the cubes the overfilling factor averages 0.217 dex or 65%. The total area in all cubes is 10,731 arcsec2, or a deprojected area of 0.65 kpc2. The most distant IFU from the nucleus is at a projected radius of about 16 kpc. A simple extrapolation to the ≈800 kpc2 area of the disk within that radius is probably unsafe.
One final map to anticipate the next post(s). Wilson provides tables of model star formation rates for 16 age bins, 826 regions, and 4 different sources of isochrones including the same BASTI isochrones I use. The complete data set is available through Vizier. In the plot below I created a map of the recent star formation rate density interpolated to nominal 10″ resolution from their Table 2 models with BASTI isochrones. This should be compared to their Figure 16 (they use logarithmic scaling).
Current (300 Myr average) star formation rate density in the PHAT footprint per models of Wilson et al. (2017) with positions of MaNGA IFUs overlaid.
I just have a quick comment about my last two subjects. I mentioned both of them have exceptionally strong Balmer absorption as measured by the Lick index HδA. They also have similar 4000Å break strengths:
IC 0976: Dn4000 = 1.308±0.005, HδA = 8.05±0.31
MCG +07-33-040: Dn4000 = 1.153±0.009, HδA = 8.06±0.41
For context here’s a variation of the same plot I’ve shown several times of the MPA-JHU measurements for a large sample of SDSS galaxy spectra with their locations overlaid:
Dn4000 – Hδ of SDSS spectra of post-starburst galaxies IC 976 and MCG +07-33-040 overlaid on measurements for a large sample of SDSS spectra
Both galaxies have HδA indexes near the upper limits of any measurements in SDSS, and both are clearly in the post-starburst area of the HδA-Dn4000 plane. Depending on your interpretation of the 4000Å break strength index IC 976 could be slightly older or have a slightly lower specific star formation rate, but the difference is small. Using the toy evolutionary models that people often use these two galaxies could easily be at slightly different stages of the same evolutionary trajectory.
In fact though the detailed star formation history models show rather different trends over the last ~Gyr, with recall MCG+07-33-040 having a more extended and more recently terminated period of enhanced star formation than IC 976, while the latter had considerably more stellar mass added by the starburst.
This nicely illustrates a point I raised 3 posts ago, which is that this particular pair of indexes can’t break the “burst age – burst mass” degeneracy. Full spectrum fitting with non-parametric star formation histories potentially can. I’m still not prepared to take these models too literally.
