Thoughts on Maser Case

Transcription

1 School of Statistics University of Minnesota March 5, 2007

2 Maser case SiO masers orbiting variable star. Could be radiatively pumped or shockwave pumped. If radiatively pumped, all fluxes should go up and down together. If shock pumped, fluxes may peak at different times. Velocity centroid should be correlated with flux if shock pumped.

3 Client questions Base question: radiation or shock pumping? Specific question: should I look at lagged cross correlation plots? I gave you guys some specific questions.

4 What further information would you like to have about the set up? I d like to know about the velocity centroid. I guess that velocity is the orbital velocity of the maser around the star, but it could be the velocity of the maser relative to earth. (Both based on Dopler shift of the radio frequency?) Velocity centroid is some kind of mean velocity.

5 Set up continued Kepler s third law implies that orbital velocity decreases with distance to the star, so VC may be an inverse measure of distance. Presumably greater flux at lesser distance, so greater flux at greater velocity. But I d expect this for both radiative and shock pumped masers, so what am I missing? This is all guess work, so I may be completely wrong about what the velocity centroid is.

6 What further information would you like to have about the data? How often do we observe multiple masers on the same night? How smooth is the temporal pattern? Does the temporal pattern look sinusoidal? Is the periodic pattern stable? How much noise is there? Do we have similar amounts of data for all masers? Do we have flux data for the star?

7 Lagged cross-correlation Two sequences of data x 1, x 2,..., x n and y 1, y 2,..., y n. We can correlate x i and y i ; this is a cross-correlation. We can correlate x i and y i k ; this is a lag-k cross-correlation. For example, if we have hourly temperatures at two stations, we could correlate x i and y i 24. If they are two time zones apart, might want to correlate x i and y i 22.

8 Lagged cross-correlation, continued By changing the lag k, find the shift k m with highest correlation. If k m = 0, the sequences are in phase. Otherwise, out of phase by lag amount. So lagged cross-correlation gets at what she wants.

9 But... We need concurrent data for lagged cross-correlation. Equal spacing is even better. We have neither. So, either we abandon the approach, or we have to impute to equal-spaced concurrent data. Imputing should give a decent answer if temporal patterns are smooth and regular. There may be problems if relative lag changes over time, because the statistic is for all the data.

10 What do we tell her? If data are reasonably smooth, regular, and dense, peak of lagged cross-correlation after imputation should work OK. However, don t make confidence/reliability statements about the correlations. Otherwise, need to work with statistician.

11 OK, now with data Always begin with graphics!

12 Three with most data

13 Magnitude of star Data from the AAVSO International data base,

14 What do we see? Four masers, two with lots of data, one with very little. Some drift in flux. Seem to go up and down together at first, perhaps not later. Period should be 1 (length of period of star), but seems longer at the end. At least at first, masers peak after the star. Star is fairly regular, but a bit less bright near the end.

15 Sinuosids Let t be time in solar phase cycles, and let y t = µ t + A t cos(2π(t φ t )) A t is amplitude and φ t is phase (here, in cycles). Equivalent to y t = µ t + A t cos(2πφ t ) cos(2πt) + sin(2πφ t ) sin(2πt) or y t = µ t + w 0t cos(2πt) + w 1t sin(2πt) where A t = w 2 0t + w 2 1t φ t = atan(w 0t, w 1t )/(2π)

16 Local Sinusoidal Regression For general data, A t and φ t will change over time, but they may change smoothly. 1 Take a window of data, say one nominal cycle. 2 Regress y t on cos(2πt), sin(2πt), and t. 3 Convert coefficients to amplitude and phase. 4 Shift window over (can overlap current window) and repeat. This gives a series of amplitude and phase. Note: I added a trend term in the regression (for drift in flux) and require at least 10 data points in the window to do the fit.

17 Phase tells us... If masers are radiatively pumped, they should all have phase zero (ie, in lock step with the star). If masers all have the same non-zero phase, they could be shock pumped and the same distance from the star If phase increases, period of maser is lengthening (vice versa for decreases). Increasing phase could be due to period of star changing (does not seem to be happening here, but does for some variable stars). Increasing phase might suggest that maser has eccentric elliptical orbit and is moving away from the star (and the shock waves).

18 The envelope please

19 Mixed results 1 All masers have the same phase up to cumulative phase 3. 2 This phase is about 1/3 to 1/5 of a cycle behind the star, decreasing over the first two cycles. 3 After 3, first maser continues slight phase decrease but is fairly stable. 4 After 3, the other two masers increase phase (lengthen period). 5 This seems to argue that all three are shock pumped.

20 Constant term

21 Slope term

22 Constant and slope and star By size, largest peaks in star brightness are at 1, 3, 2, and 4 (much smaller). Constant terms follow this pattern: higher mean when star is brighter. At 1, we must have a general decrease to have a lower peak at 2; at 2 we must have a general increase to have a higher peak at three; etc. Slope terms follow this pattern.

23 Amplitude term Difficult to generalize here:

24 Velocity centroid

25 Who knows? Overall negative correlation between velocity and flux. Maser with smallest flux has highest velocities, and masers with greastest flux have smallest velocities. No correlation between flux and velocity for masers 1 and 3. Positive correlation between flux and velocity for maser 2. I don t know what to say about this.

26 Summary Masers seem to be shock pumped. Behavior changes after cumulative phase 3. Velocity centroid is still a mystery.