# Week 12: proof of variation

In quarantine, it doesn’t seem like a lot changes day-to-day. The second part of this blog addresses how I used the linmix package to prove that our TESS-S21 field M-dwarf targets exhibit H-alpha, even in their old age. To begin, however, I’ll address a bunch of changes/improvements/revelations I made to our periodograms and period fitting.

### Understanding is stochastic…

Is what I’d tell my last-blog self. It took a while, but eventually I re-examined a good portion of my periodogram code and recovered some reasonable, meaningful results. I was able to catch most of my errors by reviewing my assumptions, comparing my work to my partner’s, and unit testing. I simplified the functions I was using to generate my periodograms and model fits into one that was able to generate curves for H-alpha and continuum data simultaneously.

Some of what was tripping me up was moving between frequency and period space and getting caught plotting one when what I wanted was the other, this screwed up my plot limits, my model parameter selection, and a few other small pieces of code.

### The kinds of stars in the sky

With my periodograms coming up understandable, I set to work tweaking fit limits in order to finalize a list of period fits for our further analysis.

Because of our nightly cadence, I limited the periodogram to a periods greater than 1.5 days, or a maximum frequency of 0.67. To account for our limited timescale (8 days of observations) I limited the periodogram to periods less than 12 days, or a minimum frequency of 0.08.

The results of running the LombScargle periodogram with these limitations on our TESS-S21 field photometry data are summarized in the table below.

Number | RA | Dec | Type | Ha recovery | Ha-off recovery | Spectral Type | Gaia B-R | Mass (Msun) | ( Don’t use)TESS satellite period (days) | Ha period | Ha-off period | Ha Amplitude | Cont amplitude | Ha amp err | Cont amp err |

M16 | 135.5153622 | 27.92448588 | M-dwarf | 1 | 1 | Unknown | 2.5245 | 0.367 | 0.664257 | 3.596747 | 3.596747 | 0.073079 | 0.094651 | 0.037238 | 0.039717 |

M19 | 135.5450228 | 27.95219364 | VLMS | 1 | 1 | Unknown | 1.9043 | 0.116 | n/a | 3.596747 | 2.550737 | 0.062981 | 0.062578 | 0.026666 | 0.026752 |

M20 | 135.9474267 | 27.95276423 | M-dwarf | 1 | 1 | Unknown | 1.9236 | 0.469 | n/a | 4.007608 | 4.007608 | 0.1731 | 0.159214 | 0.013941 | 0.012779 |

M22 | 135.5884706 | 27.97634018 | M-dwarf | 1 | 1 | Unknown | 2.4557 | 0.306 | 0.72752 | 2.094002 | 2.550737 | 0.371267 | 0.305898 | 0.196167 | 0.233737 |

M26 | 135.6763584 | 28.05036671 | M-dwarf | 1 | 1 | Unknown | 2.313 | 0.536 | n/a | 3.596747 | 4.524441 | 0.037282 | 0.063586 | 0.027947 | 0.028224 |

M27 | 135.846373 | 28.05020411 | M-dwarf | 1 | 1 | Unknown | 2.5615 | 0.476 | 0.611117 | 2.226919 | 2.550737 | 0.150546 | 0.212895 | 0.085413 | 0.088482 |

M28 | 135.5324188 | 28.05821814 | M-dwarf | 1 | 0 | Unknown | 2.384 | 0.4 | 4.524441 | 0.396341 | 0.29016 | ||||

M31 | 135.5860683 | 28.07026933 | M-dwarf | 1 | 0 | Unknown | 3.1366 | 0.162 | 1.14 | 0.399639 | 0.163549 | ||||

M33 | 135.5716284 | 28.09342772 | M-dwarf | 1 | 1 | Unknown | 2.6593 | 0.451 | 0.864788 | 2.226919 | 3.596747 | 0.136379 | 0.100335 | 0.078421 | 0.080169 |

M36 | 135.7813874 | 28.15725462 | M-dwarf | 1 | 0 | Unknown | 1.9223 | 0.469 | 2.094002 | 0.144095 | 0.051123 | ||||

M43 | 135.5698047 | 28.22642501 | M-dwarf | 1 | 0 | Unknown | 2.6969 | 0.319 | 2.226919 | 0.187387 | 0.097253 | ||||

M46 | 135.9206601 | 28.2615831 | M-dwarf | 1 | 0 | Unknown | 2.1817 | 0.43 | 2.377854 | 0.681266 | 0.249835 | ||||

M47 | 135.9032021 | 28.26535994 | M-dwarf | 1 | 0 | Unknown | 2.168 | 0.218 | 2.094002 | 0.401897 | 0.234388 | ||||

M48 | 135.8425684 | 28.28138518 | M-dwarf | 1 | 1 | Unknown | 2.3383 | 0.389 | 0.804101 | 4.524441 | 2.550737 | 0.179248 | 0.181683 | 0.075282 | 0.07984 |

M49 | 135.867911 | 28.31582296 | M-dwarf | 1 | 0 | Unknown | 2.3965 | 0.416 | 2.750729 | 0.378838 | 0.097551 | ||||

M50 | 135.9701439 | 28.3156986 | M-dwarf | 1 | 1 | Unknown | 2.3045 | 0.333 | 1.23875 | 2.377854 | 3.596747 | 0.156172 | 0.19492 | 0.072232 | 0.050623 |

Results varied from inactive rotating stars, like below,

To active rotating stars, like this one,

and these non-detections, which we argue also indicate activity.

Lena made this plot for period vs mass for both of our fields, which shows the spread in period at lower masses, but also interestingly the concentration towards lower values of older targets.

We’ll use these figures use in our analysis for our poster and presentation.

### Proof of variation, a lin-remix

In order to prove the low-mass targets in our TESS field were varying at H-alpha, thereby exhibiting chromospheric activity, I constructed a graph of the amplitude in H-alpha versus the amplitude in the continuum. If the targets were indeed brighter and varying more at H-alpha, the slope of this line would be greater than 1. A slope of one would indicate that both filters varied the same amount, which would indicate inactivity at H-alpha.

Making the plot and doing a simple linear regression seems to prove this, but barely; a slope of 1.12 is certainly greater than 1, but not by an incredibly convincing amount. Alas, if only we could leverage those non-detection results that lay lazily to the far left side of our plot. As it turns out, we can!

This type of data is called censored data, which means that the value of some (or all) of the data is partially constrained. In this case, we know the variation in our continuum emission is less than our error (which we get from our photometry) but otherwise we do not know the value for those data points. Censored data can still be useful and informative; we can imagine visually that if we could include our censored data into our linear regression that we’d likely end up with a slope that’s much greater than unity.

I’m no statistician, but I do like the idea of bayesian statistics, and iterative processes. The linmix package uses an MCMC method to fit linear regressions to data with errors and censored data. The resulting MCMC fit gives a much more convincing result, especially when plotted with error bounds that do not include unity.

linmix was very interesting to play around with and learn to use, and I’m glad I poked into this bit of math unprovoked. Hopefully its something I can use in the future.

Coming up soon is our poster presentation: the conclusion to this epic Astronomy 341 adventure. Until then, clear skies!