MIDI. Correlation of MIDI Phase Fluctuations with Fluctuations of Water and Carbon Dioxide, Paranal June 2007

Transcription

1 The Mid-Infrared Interferometric Instrument for the Very Large Telescope Interferometer Sterrewacht Leiden MIDI Correlation of MIDI Phase Fluctuations with Fluctuations of Water and Carbon Dioxide, Paranal June 2007 Doc. No. UL-TRE-MID Issue Date March 4, 2008 Prepared Richard J. Mathar March 4, 2008 Approved Walter J. Jaffe March 4, 2008 Released Signature Signature Signature

2 ii Mathar Jaffe, NEVEC, Leiden Observatory Issue UL-TRE-MID Change Record Issue Date Section/Parag. affected Reason/Initiation/Documents/Remarks Jul-2007 all created (rjm).0 3-Mar-2008 all first official release (rjm) Contents OVERVIEW. Scope Data Sources References Acronyms FITTED DISPERSED MIDI SCANS 5 2. Parameters: times, baselines, meteorological Reduction of Dispersion with Linear Fits to Wavenumber COMPARISON INTERFEROMETRY AND GAS ANALYZER Refractive Index Model Structure Functions and Fried Parameter Power Spectra of Phases Phases Extrapolated to Zero Wavenumber SUMMARY 73 List of Figures associated MIDI data files wind velocity 300 mbar Wind Speed during MIDI nights MIDI data file MIDI data file MIDI data file MIDI data file MIDI data file MIDI data file MIDI data file MIDI data file MIDI data file MIDI data file MIDI data file MIDI data file

3 MIDI Interferom. and Water Vapor Fluct. Issue UL-TRE-MID iii 6 MIDI data file MIDI data file MIDI data file MIDI data file MIDI data file MIDI data file MIDI data file MIDI data file MIDI data file MIDI data file MIDI data file MIDI data file MIDI data file MIDI data file MIDI data file MIDI data file MIDI data file MIDI data file Phase Structure Function, weak points flagged Phase Structure Function D ϕ0 with oscillations, weak points flagged Structure Function D ϕ, weak points flagged, slope N-band water vapor dispersion N-band dry air dispersion N-band carbon dioxide dispersion fitted H2O structure functions fitted CO2 structure functions Digitization error of molar densities Expected phases from VLTI mirror vibrations Comparison Clifford model and MIDI 0 µm PDF st night Comparison Clifford model and MIDI 0 µm PDF 2nd night Comparison Clifford model and MIDI 0 µm PDF 4th night Correlation phase and delay st night Correlation phase and delay 2nd night Correlation phase and delay 3rd night Correlation phase and delay 4th night Comparison Clifford model and MIDI zero-k PDF st night Comparison Clifford model and MIDI zero-k PDF 2nd night Comparison Clifford model and MIDI zero-k PDF 4th night List of Tables Parameters of MIDI data sets: baselines, dates, delays MIDI data delayed start times, star altitudes and azimuths Parameters of data sets: wind angles, structure constants

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5 MIDI Interferom. and Water Vapor Fluct. Issue UL-TRE-MID OVERVIEW. Scope Fluctuations of water and carbon dioxide densities have been measured by W. J. Jaffe and R. S. Le Poole on the roof top of the VLTI control building, in the VLTI tunnel, and the hull of the VST dome with a LI-COR gas analyzer at the end of June 2007 [6]. During four nights in the same period, the fringe tracker of the MIDI/VLTI interferometer collected data on the motion jitter at the mid-infrared wavelength of 0 µm. We combine a subset of 30 of these exposures on visibility calibrators each of a duration of approximately 2 minutes where an overlap with ambient LI-COR data exists. A quantitative comparison between phase power spectra and structure functions is drawn here; the adjective quantitative indicates that not only the power exponents of these spectra are examined which has been done before in OI [2] and will not be reviewed but also the strength of the turbulence, commonly associated with various seeing parameters..2 Data Sources This manuscript has drawn information from a one-time campaign with a loaned LI-COR water/carbon-dioxide gas analyzer, June 25 30, 2007 [6]. an ASCII interface to the temperatures, pressures, relative humidities, wind directions, wind velocities and seeing parameters stored in the Paranal ambient server data base [9]. data on precipitable water vapor, wind speed and wind direction on the 300 mbar level (height) originating from the ECMWF. characterisation of interferometric geometry (including air mass), snapshots of the ambient meteorology and seeing, and stored in FITS files of MIDI raw data of four nights overlapping with the gas analyzer data during 30 exposures of approximately 2 minutes duration (each). data reduction products of the same MIDI detector data with the MIDI prism, equivalent to a spectral resolution of roughly 32 channels in the N band, Temperatures and relative humidities of four sensors in the VLTI tunnel and ducts copied from a directory on ESO s anonymous ftp server..3 References [] Clifford, S. F. 97, J. Opt. Soc. Am., 6, 285 [2] Colavita, M. M., Shao, M., & Staelin, D. H. 987, Appl. Opt., 26, 406

6 2 Mathar Jaffe, NEVEC, Leiden Observatory Issue UL-TRE-MID [3] d Arcio, L. 995, Differential anisoplanatic OPDs and OPD spectra for the VLTI at Cerro Paranal from PARSCA 992 Balloon Data. VLT-TRE-ESO [4] de Mooij, E. 2006, Atmospheric turbulence measured with MIDI, Tech. rep. [5] Fried, D. L. 965, J. Opt. Soc. Am., 55, 427. E: [6] [6] 966, J. Opt. Soc. Am., 56, 40E [7] Hufnagel, R. E., & Stanley, N. R. 964, J. Opt. Soc. Am., 54, 52 [8] Koehler, B. 2000, Results of OPD measurement on UT#3 in May 00, Test Report. VLT-TRE- ESO [9] Koresko, C., Colavita, M., Serabyn, E., Booth, A., & Garcia, J. 2006, in Advances in Stellar Interferometry, edited by J. D. Monnier, M. Schöller, & W. Danchi (Int. Soc. Optical Engineering), vol. 6268, [0] Labit, P. 998, Meteorological Prediction Software, User and Maintenance Manual. VLT-TRE- ESO [] Lévêque, S. 2002, Temperature sensor network for the VLTI. VLT-TRE-ESO [2] Marchetti, S., & Simili, R. 2006, Infr. Phys. Techn., 47, 263. Presumbably, the factor 0 4 ought read 0 +4 and the temperature 296 C read 23 C in Table. [3] Mathar, R. J. 2006, Astrometric Survey for Extra-Solar Planets with PRIMA, Astrometric dispersion correction. UL-TRE-AOS [4] 2007, Baltic Astronomy, 6, 287 [5] 2007, J. Opt. A: Pure and Appl. Optics, 9, 470 [6] Mathar, R. J., Jaffe, W. J., & Le Poole, R. S. 2007, High Frequency Fluctuations of Water and Carbon Dioxide, Paranal July 25 27, UL-TRE-ESO [7] Roddier, F. 98 (Amsterdam: North Holland), vol. XIX of Prog. Opt., [8] Roddier, F., & Lena, P. 984, J. Optics (Paris), 5, 7 [9] Sarazin, M. 2003, Astroclimatology of Paranal, Tech. rep., European Southern Observatory. URL [20] Schwider, J. 990 (Amsterdam: Elsevier), vol. 28 of Prog. Opt., Acronyms AO DICB DL ECMWF Adaptive Optics Data Interface Definition Board (of ESO) Delay Line European Center for Medium-Range Weather Forecasts

7 MIDI Interferom. and Water Vapor Fluct. Issue UL-TRE-MID ESO FITS FT GPS GUI IDL IR IRIS ISS MDL MIDI NOVA OI OPD OS PDF PWV UT UTC VLT VLTI VST European Southern Observatory Flexible Image Transport System Fourier Transform Global Positioning System Graphical User Interface Interactive Data Language Infrared Infrared Image Sensor Interferometric Supervisor-Software main delay line Mid-Infrared Interferometric Instrument Nederlandse Onderzoekschool voor Astronomie Optical Interferometry optical path difference Observation Software Power Density Function precipitable water vapor Unit Telescope (of the VLTI) Universal Time Coordinated Very Large Telescope Very Large Telescope Interferometer VLT Survey Telescope

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9 MIDI Interferom. and Water Vapor Fluct. Issue UL-TRE-MID FITTED DISPERSED MIDI SCANS 2. Parameters: times, baselines, meteorological An overview of the time allocation of the 30 MIDI exposures in the four nights is in Figure. Some parameters extracted from the primary headers of the first in each group of FITS files of an exposure are in table. The first column is a small integer to avoid tagging with the full standard file name that is given in the second column. Baselines are listed in the third column. Two temperatures along the DL tunnel are in the 4th and 5th column, taken from the keywords ISS TEMP TUN and ISS TEMP TUN4 []. P is the column with the projected baseline, ISS PBL2 START. D and Ḋ are the external path delay (in vacuum) and its time derivative, using LST, OCS ISS RA and OCS ISS DEC as inputs. The sum b 2 = P 2 + D 2 is the squared baseline length and constant for a pair of stations, b = 46.4 m (46.6 m according to the distance between the station coordinates in the OPD model of Mfile22) for UT23 and b = 62.4 m for UT34. The column τ 0 is the mean of AMBI TAU0 START and AMBI TAU0 END. The column T/ h is the value of AMBI LRATE. The column AIRM is the air mass of ISS AIRM START. The detector cycle time (average distance between start of two exposures) was 20.9 ms (49 ) in all cases. These are non-chopped observations which basically ensures that the stream of data fills the time axis almost continuously. The number of piezo steps per scan was 40 in all cases (keyword PIEZ POSNUM), which means block adjustments of the internal or main delay lines initiated by the MIDI fringe tracker could occur in intervals of seconds. (This is roughly equivalent to the mean passage time for a UT diameter at characteristic wind speeds of 0 m/s.) Multiplication with Ḋ of Table which is of the order of mm/s gives an estimate of how much the main delay line moved during that time. Some additional characterization of the seeing conditions is in Figure 2; its data will not be used in the followup analysis.

10 6 Mathar Jaffe, NEVEC, Leiden Observatory Issue UL-TRE-MID delay [mu] delay [mu] delay [mu] delay [mu] delay [mu] /25 00: /26 00: /27 00:00 06/28 00:00 06/29 00:00 Mfile23 Mfile9 Mfile0 Mfile 06/29 04:00 06/29 08:00 Mfile24 Mfile25 Mfile26 Mfile27 Mfile28 Mfile29 06/28 04:00 06/28 08:00 Mfile6 Mfile7 Mfile8 Mfile9 Mfile20 06/27 04:00 06/27 08:00 Mfile Mfile2 Mfile3 Mfile4 Mfile5 06/26 04:00 06/26 08:00 Mfile2 Mfile3 Mfile4 Mfile5 Mfile6 Mfile7 Mfile8 06/25 04:00 06/25 08:00 Mfile2 06/25 2:00 06/26 2:00 06/27 2:00 06/28 2:00 06/29 2:00 06/25 6:00 06/25 20:00 06/26 6:00 06/26 20:00 LI-COR to tunnel 06/27 6:00 06/27 20:00 LI-COR to VST 06/28 6:00 06/28 20:00 06/29 6:00 06/29 20:00 06/30 00:00 06/29 00:00 Mfile22 06/28 00:00 06/27 00:00 06/26 00:00 Mfile0 Figure : Timing overview of the four nights with MIDI data files.

11 MIDI Interferom. and Water Vapor Fluct. Issue UL-TRE-MID ARCFILE TELESCOP TUN TUN4 PBL2 TAU0 LRATE AIRM WINDSP Mfilen start stations T West East P D Ḋ τ0 T/ h ( C) ( C) (m) (m) (mm/s) (ms) ( C/m) (m/s) 0 MIDI T23:53:33 U MIDI T0:42:35 U MIDI T02:58:8 U MIDI T03:4:4 U MIDI T03:49:39 U MIDI T04:56:08 U MIDI T05:54:39 U MIDI T06:35:22 U MIDI T08:5:5 U MIDI T0:24:27 U MIDI T0:50:36 U MIDI T03:04:4 U MIDI T04:26:37 U MIDI T04:3:47 U MIDI T07:06:52 U MIDI T07:27:32 U MIDI T04:50:2 U MIDI T05:42:4 U MIDI T06:27:47 U MIDI T07:5:28 U MIDI T08:23:48 U MIDI T09:42:30 U MIDI T23:5:8 U MIDI T0:29:5 U MIDI T02:46:08 U MIDI T03:58:27 U MIDI T05::09 U MIDI T05:56: U MIDI T07:0:57 U MIDI T08:26:56 U Table : Parameters of the 30 MIDI data sets derived from primary FITS headers.

12 8 Mathar Jaffe, NEVEC, Leiden Observatory Issue UL-TRE-MID seeing [as] τ 0 [ms] PWV [mm] 300 mbar wind [m/s] Jun 00: Jun 00:00 26 Jun 00:00 Mfile0 Mfile Mfile2 Mfile3Mfile4 Mfile5 Mfile6Mfile7 26 Jun 00:00 Mfile8 27 Jun 00:00 Mfile9 Mfile0 Mfile Mfile3 Mfile2 Mfile4 Mfile5 27 Jun 00:00 28 Jun 00:00 Mfile6 Mfile7 Mfile8 Mfile9 Mfile20 Mfile2 28 Jun 00:00 29 Jun 00:00 Mfile22 Mfile23 Mfile24 Mfile25 Mfile26 Mfile27 Mfile28 Mfile29 29 Jun 00:00 Figure 2: Upper two plots: The wind velocity during the five days of observation at the 300 mbar pressure level, and the precipitable water vapor, from the ECMWF [0, 9]. The PWV plot is a section of Figure 57 in [3]. Lower two plots: seeing and coherence time from the ambient data base server at the same times. The 25th of June is included in the plot to demonstrate implicitly through the PWV that it was rainy.

13 MIDI Interferom. and Water Vapor Fluct. Issue UL-TRE-MID Fig. 3 is a close-up (redundant) view on the wind speeds of Figs. 2 7 of [6] during the times of the 30 MIDI observations, a synopsis of the WINDSP column of Table with the markers of Fig.. The intent is to demonstrate that the wind speed is stable over the approximate 2 minutes of an exposure only to within 0.5 m/s. The WINDSP entry in the FITS headers are (due to some known constraints of data handling procedures in the instrument software) a snapshot taken at some time during the part of the exposure covered by the first FITS file of the series; the nominal accuracy of 0.0 m/s is not achieved.

14 0 Mathar Jaffe, NEVEC, Leiden Observatory Issue UL-TRE-MID wind (m/s) wind (m/s) wind (m/s) wind (m/s) Jun 00:00 Mfile22 28 Jun 00:00 27 Jun 00:00 26 Jun 00:00 Mfile0 26 Jun 0:00 27 Jun 0:00 28 Jun 0:00 29 Jun 0:00 29 Jun 02:00 Mfile23 28 Jun 02:00 27 Jun 02:00 Mfile9 Mfile0 26 Jun 02:00 Mfile 29 Jun 03:00 Mfile24 28 Jun 03:00 27 Jun 03:00 Mfile 26 Jun 03:00 Mfile2 29 Jun 04:00 Mfile25 28 Jun 04:00 27 Jun 04:00 26 Jun 04:00 Mfile3 Mfile4 Mfile2 Mfile3 29 Jun 05:00 Mfile26 28 Jun 05:00 Mfile6 27 Jun 05:00 26 Jun 05:00 Mfile5 29 Jun 06:00 Mfile27 28 Jun 06:00 Mfile7 27 Jun 06:00 26 Jun 06:00 Mfile6 Mfile8 Mfile7 29 Jun 07:00 Mfile28 28 Jun 07:00 Mfile9 27 Jun 07:00 Mfile4 Mfile5 26 Jun 07:00 29 Jun 08:00 Mfile29 28 Jun 08:00 Mfile20 27 Jun 08:00 26 Jun 08:00 Mfile8 26 Jun 09:00 27 Jun 09:00 28 Jun 09:00 29 Jun 09:00 29 Jun 0:00 28 Jun 0:00 Mfile2 27 Jun 0:00 26 Jun 0:00 Figure 3: Ground wind speeds from the ambient server database during the MIDI observations.

15 MIDI Interferom. and Water Vapor Fluct. Issue UL-TRE-MID Reduction of Dispersion with Linear Fits to Wavenumber The interferometric phase ϕ ϕ = n(k)kd i kd e () measured at wave vectors k = 2πσ at internal (stepped) delays D i, external geometric path delay D e for nominal refractive index n(k) is fitted over the MIDI band width to a model of constant interband group delay [9] ϕ = ϕ 0 + kϕ. (2) This is a self-sustained data reduction without astrometric support, which means the input is essentially the cosine of ϕ, so ϕ 0 is only determined modulo 2π. The algorithm is essentially the technique of taking the FT of the interferograms and putting a window around one of the two side bands that arise from splitting the cosine with the Euler formula [20, 3.4]. Some cleansing of the 2-parametric reduced data is done before display and further use of φ 0 and φ : At the start of some MIDI files, the initial time segment shows the behaviour during fringe acquisition, with much larger changes than for the rest of the approximately 2 /2 minutes covered in total. Chopping off some initial frames of the MIDI exposures typically a sequence of fringe acquisition leads to modified start times of the exposures of Table 2. For exposures with index 2, Fig. 6, such a simple way of discarding data is not effective and has not been attempted. Some spikes in ϕ are removed: frames where ϕ differs from the average of the previous 5 frames by more than 20 µm are discarded. This removes data which claim an OPD motion of more than 20 µm within 0.3 seconds. One second is chopped off the end to remove the tail wiggle. In each of the figures 4 33 The top 4 plots show data as a function of UTC. Each small vertical bar represents the spread of values within an individual second, so we have 60 of these per major tic mark on the time axis. the two plots with ordinate ϕ 0 and ϕ show the MIDI parameters reduced to ϕ 0 (in units of cycles, ie, 2π) and reduced to ϕ (in units of µm) after this removal of some optional initial number of frames. the plots carrying ordinate units of some powers of mol/m 3 summarize the LI-COR measurements at its local position (outdoors or tunnel) during that same time. The CO 2 densities are as monitored, the water densities transformed with the linear equation () of [6] for Lfile6 and later. The power spectral density derived for the water vapor is the third plot from below. The enumeration Lfile Lfile0 is the same as in [6]. The MIDI phase fluctuation at λ = 0 µm, the value ϕ(0µ) ϕ 0 + 2πϕ /(0µm) (3) that combines the two variables ϕ 0 and ϕ of each MIDI frame, is transformed to the PDF of ϕ and ϕ 0 with ordinate units rad 2 /. A local maximum in P DF ϕ0 near.2 with occasional ghosts at multiples seems to be some artifact of the data reduction algorithm, where.2 is the inverse of the period of the sawtooth pattern of the phase shifting internal delay line.

16 2 Mathar Jaffe, NEVEC, Leiden Observatory Issue UL-TRE-MID In PDF plots, the frequency abscissa axis is accompanied by a second logarithmic axis, which shows wave numbers σ ν/v (4) in units of inverse meters associated with the frequencies ν by the individual wind velocity v during that observation (Fig. 3 and Table ). They are inverse length scales of the density fluctuations assuming constant speed during the exposure. As mentioned earlier, the Nyquist frequency of the LI-COR data is 0, the Nyquist frequency of the MIDI data 25. In all spectra and structure functions, a linear drift term has been removed before the Fourier transform; the sub-plots with the data as a function of UTC time show the data prior to this transformation. The spectra appear noisier than those of Fig in [6] because an additional averaging on the time axes has been introduced in Fig of [6] for display (but not for calculation of the derived quantities like autocorrelation and structure function) but not here. The time axis of LI-COR is a PC clock manually adjusted to UTC, whereas the MIDI exposures are clocked with a GPS based system with sub-second accuracy. A relative shift of the two time axes of the order of 2 seconds is expected. The PDF of the molecular LI-COR water density is calculated over the time interval shown in the upper part of the figures. A linear drift has been removed, and the water density correction () of [6] is made for all measurements in the tunnel and in the VST. The UTC keyword of the MIDI primary headers does not show the start of the exposures but a time when ISS received some specific message from the MIDI OS. This is typically 8 seconds ahead of the values of MJD-OBS and OBS-DATE which should be used for accurate reference of the start of exposures.

17 MIDI Interferom. and Water Vapor Fluct. Issue UL-TRE-MID start ALT AZ Mfilen (deg) (deg) T23:53: T0:42: T02:58: T03:4: T03:49: T04:56: T05:54: T06:35: T08:5: T0:24: T0:50: T03:04: T04:27: T04:32: T07:07: T07:27: T04:50: T05:42: T06:27: T07:5: T08:23: T09:42: T23:5: T0:30: T02:46: T03:58: T05:: T05:56: T07:02: T08:27: Table 2: Time stamps of the first exposures in the MIDI data files of Table after optional removal of a variable initial portion (of typically up to 550 frames) which show large variation of the delay parameter. ALT and AZ are the pointing coordinates in degrees in the local coordinate system following DICB conventions, from the FITS headers.

18 4 Mathar Jaffe, NEVEC, Leiden Observatory Issue UL-TRE-MID CO 2 (mmol/m 3 ) LI-COR water (mol/m 3 ) ϕ 0 (cycl) Lfile3 Lfile3 Mfile0 PDF ((mol/m 3 ) 2 /) MIDI ϕ (µm) e-6 e-8 e-0 UTC Mfile0 23:54 23:55 23: ν/v (/m) 0. Lfile e+2 e+0 e-2 Mfile Mfile ν () 0 Figure 4: Correlation of MIDI data file 0 with LI-COR densities.

19 MIDI Interferom. and Water Vapor Fluct. Issue UL-TRE-MID CO 2 (mmol/m 3 ) Lfile3 LI-COR water (mol/m 3 ) ϕ 0 (cycl) Lfile3 Mfile MIDI ϕ (µm) PDF ((mol/m 3 ) 2 /) 2.0 UTC Mfile :43 0:44 0: ν/v (/m) 0. e-6 Lfile3 e-8 e e+2 e+0 e-2 Mfile Mfile ν () 0 Figure 5: Correlation of MIDI data file with LI-COR densities.

20 6 Mathar Jaffe, NEVEC, Leiden Observatory Issue UL-TRE-MID CO 2 (mmol/m 3 ) Lfile3 LI-COR water (mol/m 3 ) ϕ 0 (cycl) Lfile3 Mfile2 PDF ((mol/m 3 ) 2 /) MIDI ϕ (µm) -2.0 UTC Mfile :59 03:00 03: ν/v (/m) 0. e-6 Lfile3 e-8 e e+2 e+0 e-2 Mfile Mfile ν () 0 Figure 6: Correlation of MIDI data file 2 with LI-COR densities.

21 MIDI Interferom. and Water Vapor Fluct. Issue UL-TRE-MID CO 2 (mmol/m 3 ) Lfile3 LI-COR water (mol/m 3 ) ϕ 0 (cycl) Lfile3 Mfile3 PDF ((mol/m 3 ) 2 /) MIDI ϕ (µm).0 UTC Mfile :42 03: ν/v (/m) 0. e-6 Lfile3 e-8 e e+2 e+0 e-2 Mfile Mfile ν () 0 Figure 7: Correlation of MIDI data file 3 with LI-COR densities.

22 8 Mathar Jaffe, NEVEC, Leiden Observatory Issue UL-TRE-MID CO 2 (mmol/m 3 ) Lfile3 LI-COR water (mol/m 3 ) ϕ 0 (cycl) Lfile3 Mfile4 MIDI ϕ (µm) PDF ((mol/m 3 ) 2 /) e-6 e-8 e-0 UTC Mfile4 03:50 03:5 03:52 ν/v (/m) Lfile e+2 e+0 e-2 Mfile Mfile ν () 0 Figure 8: Correlation of MIDI data file 4 with LI-COR densities.

23 MIDI Interferom. and Water Vapor Fluct. Issue UL-TRE-MID CO 2 (mmol/m 3 ) Lfile3 LI-COR water (mol/m 3 ) ϕ 0 (cycl) Lfile3 Mfile5 PDF ((mol/m 3 ) 2 /) MIDI ϕ (µm) 2.0 UTC Mfile :57 04: ν/v (/m) 0. e-6 Lfile3 e-8 e e+2 e+0 e-2 Mfile Mfile ν () 0 Figure 9: Correlation of MIDI data file 5 with LI-COR densities.

24 20 Mathar Jaffe, NEVEC, Leiden Observatory Issue UL-TRE-MID CO 2 (mmol/m 3 ) Lfile3 LI-COR water (mol/m 3 ) ϕ 0 (cycl) Lfile3 Mfile6 MIDI ϕ (µm) PDF ((mol/m 3 ) 2 /) -.0 UTC Mfile :55 05:56 05: ν/v (/m) 0. e-6 Lfile3 e-8 e e+2 e+0 e-2 Mfile Mfile ν () 0 Figure 0: Correlation of MIDI data file 6 with LI-COR densities.

25 MIDI Interferom. and Water Vapor Fluct. Issue UL-TRE-MID CO 2 (mmol/m 3 ) LI-COR water (mol/m 3 ) ϕ 0 (cycl) Lfile3 Lfile3 Mfile7 MIDI ϕ (µm) PDF ((mol/m 3 ) 2 /) -2.0 UTC Mfile :36 06:37 06: ν/v (/m) 0. e-6 Lfile3 e-8 e e+2 e+0 e-2 Mfile Mfile ν () 0 Figure : Correlation of MIDI data file 7 with LI-COR densities.

26 22 Mathar Jaffe, NEVEC, Leiden Observatory Issue UL-TRE-MID CO 2 (mmol/m 3 ) LI-COR water (mol/m 3 ) ϕ 0 (cycl) Lfile3 Lfile3 Mfile8 MIDI ϕ (µm) PDF ((mol/m 3 ) 2 /) e-6 e-8 e-0 UTC Mfile8 08:6 08:7 08:8 ν/v (/m) Lfile e+2 e+0 e-2 Mfile Mfile ν () 0 Figure 2: Correlation of MIDI data file 8 with LI-COR densities.

27 MIDI Interferom. and Water Vapor Fluct. Issue UL-TRE-MID CO 2 (mmol/m 3 ) Lfile5 PDF ((mol/m 3 ) 2 /) MIDI ϕ (µm) ϕ 0 (cycl) LI-COR water (mol/m 3 ) e-6 e-8 e-0 UTC Lfile5 Mfile9 Mfile9 0:25 0:26 0:27 ν/v (/m) Lfile e+2 e+0 e-2 Mfile Mfile ν () 0 Figure 3: Correlation of MIDI data file 9 with LI-COR densities.

28 24 Mathar Jaffe, NEVEC, Leiden Observatory Issue UL-TRE-MID CO 2 (mmol/m 3 ) Lfile5 PDF ((mol/m 3 ) 2 /) MIDI ϕ (µm) ϕ 0 (cycl) LI-COR water (mol/m 3 ) e-6 e-8 e-0 UTC Lfile5 Mfile0 Mfile0 0:5 0:52 0:53 ν/v (/m) Lfile e+2 e+0 e-2 Mfile Mfile ν () 0 Figure 4: Correlation of MIDI data file 0 with LI-COR densities.

29 MIDI Interferom. and Water Vapor Fluct. Issue UL-TRE-MID CO 2 (mmol/m 3 ) Lfile5 LI-COR water (mol/m 3 ) ϕ 0 (cycl) MIDI ϕ (µm) PDF ((mol/m 3 ) 2 /) e-6 e-8 e-0 UTC :05 03:06 03:07 ν/v (/m) Lfile5 Mfile Mfile Lfile e+2 e+0 e-2 Mfile Mfile ν () 0 Figure 5: Correlation of MIDI data file with LI-COR densities.

30 26 Mathar Jaffe, NEVEC, Leiden Observatory Issue UL-TRE-MID CO 2 (mmol/m 3 ) Lfile5 PDF ((mol/m 3 ) 2 /) MIDI ϕ (µm) ϕ 0 (cycl) LI-COR water (mol/m 3 ) e-6 e-8 e-0 UTC :28 04:29 ν/v (/m) Lfile5 Mfile2 Mfile2 Lfile e+2 e+0 e-2 Mfile Mfile ν () 0 Figure 6: Correlation of MIDI data file 2 with LI-COR densities.

31 MIDI Interferom. and Water Vapor Fluct. Issue UL-TRE-MID CO 2 (mmol/m 3 ) Lfile5 PDF ((mol/m 3 ) 2 /) MIDI ϕ (µm) ϕ 0 (cycl) LI-COR water (mol/m 3 ) e-6 e-8 e-0 UTC :33 04:34 ν/v (/m) Lfile5 Mfile3 Mfile3 Lfile e+2 e+0 e-2 Mfile Mfile ν () 0 Figure 7: Correlation of MIDI data file 3 with LI-COR densities.

32 28 Mathar Jaffe, NEVEC, Leiden Observatory Issue UL-TRE-MID CO 2 (mmol/m 3 ) Lfile5 LI-COR water (mol/m 3 ) ϕ 0 (cycl) Lfile5 Mfile4 PDF ((mol/m 3 ) 2 /) MIDI ϕ (µm) 0.0 UTC Mfile :08 07: ν/v (/m) 0. e-6 Lfile5 e-8 e e+2 e+0 e-2 Mfile Mfile ν () 0 Figure 8: Correlation of MIDI data file 4 with LI-COR densities.

33 MIDI Interferom. and Water Vapor Fluct. Issue UL-TRE-MID CO 2 (mmol/m 3 ) Lfile5 LI-COR water (mol/m 3 ) ϕ 0 (cycl) MIDI ϕ (µm) PDF ((mol/m 3 ) 2 /) 0. Lfile Mfile UTC Mfile :28 07:29 07: ν/v (/m) 0. e-6 Lfile5 e-8 e e+2 e+0 e-2 Mfile Mfile ν () 0 Figure 9: Correlation of MIDI data file 5 with LI-COR densities.

34 30 Mathar Jaffe, NEVEC, Leiden Observatory Issue UL-TRE-MID CO 2 (mmol/m 3 ) Lfile6 LI-COR water (mol/m 3 ) ϕ 0 (cycl) Lfile6 Mfile6 PDF ((mol/m 3 ) 2 /) MIDI ϕ (µm) -.0 UTC Mfile :5 04: ν/v (/m) 0. e-6 Lfile6 e-8 e e+2 e+0 e-2 Mfile Mfile ν () 0 Figure 20: Correlation of MIDI data file 6 with LI-COR densities. The water densities in Figs are those reduced with Equation () of [6]. The wavenumber scale ν/v on the LI-COR densities does not make much sense in these figures because they have been measured in the tunnel and are scaled here with the weather pole wind velocity.

35 MIDI Interferom. and Water Vapor Fluct. Issue UL-TRE-MID CO 2 (mmol/m 3 ) Lfile6 LI-COR water (mol/m 3 ) ϕ 0 (cycl) Lfile6 Mfile7 PDF ((mol/m 3 ) 2 /) MIDI ϕ (µm) -4.0 UTC Mfile :43 05:44 05: ν/v (/m) 0. e-6 Lfile6 e-8 e e+2 e+0 e-2 Mfile Mfile ν () 0 Figure 2: Correlation of MIDI data file 7 with LI-COR densities. Note that the wavenumber scale on the LI-COR densities does not make much sense here because they have been measured in the tunnel and are scaled here with the weather pole wind velocity.

36 32 Mathar Jaffe, NEVEC, Leiden Observatory Issue UL-TRE-MID CO 2 (mmol/m 3 ) Lfile6 LI-COR water (mol/m 3 ) ϕ 0 (cycl) Lfile6 Mfile8 PDF ((mol/m 3 ) 2 /) MIDI ϕ (µm) -3.0 UTC Mfile :28 06:29 06: ν/v (/m) 0. e-6 Lfile6 e-8 e e+2 e+0 e-2 Mfile Mfile ν () 0 Figure 22: Correlation of MIDI data file 8 with LI-COR densities. Note that the wavenumber scale on the LI-COR densities does not make much sense here because they have been measured in the tunnel and are scaled here with the weather pole wind velocity.

37 MIDI Interferom. and Water Vapor Fluct. Issue UL-TRE-MID CO 2 (mmol/m 3 ) Lfile6 LI-COR water (mol/m 3 ) ϕ 0 (cycl) Lfile6 Mfile9 MIDI ϕ (µm) PDF ((mol/m 3 ) 2 /) -.0 UTC Mfile :6 07:7 07:8 0.0 ν/v (/m) 0. e-6 Lfile6 e-8 e e+2 e+0 e-2 Mfile Mfile ν () 0 Figure 23: Correlation of MIDI data file 9 with LI-COR densities. Note that the wavenumber scale on the LI-COR densities does not make much sense here because they have been measured in the tunnel and are scaled here with the weather pole wind velocity.

38 34 Mathar Jaffe, NEVEC, Leiden Observatory Issue UL-TRE-MID CO 2 (mmol/m 3 ) Lfile6 LI-COR water (mol/m 3 ) ϕ 0 (cycl) Lfile6 Mfile20 PDF ((mol/m 3 ) 2 /) MIDI ϕ (µm) -.0 UTC Mfile :24 08:25 08: ν/v (/m) 0. e-6 Lfile6 e-8 e e+2 e+0 e-2 Mfile Mfile ν () 0 Figure 24: Correlation of MIDI data file 20 with LI-COR densities. Note that the wavenumber scale on the LI-COR densities does not make much sense here because they have been measured in the tunnel and are scaled here with the weather pole wind velocity.

39 MIDI Interferom. and Water Vapor Fluct. Issue UL-TRE-MID CO 2 (mmol/m 3 ) Lfile6 PDF ((mol/m 3 ) 2 /) MIDI ϕ (µm) ϕ 0 (cycl) LI-COR water (mol/m 3 ) UTC Lfile6 Mfile Mfile :43 09:44 09:45 ν/v (/m) e-6 e-8 e-0 Lfile e+2 e+0 e-2 Mfile Mfile ν () 0 Figure 25: Correlation of MIDI data file 2 with LI-COR densities. Note that the wavenumber scale on the LI-COR densities does not make much sense here because they have been measured in the tunnel and are scaled here with the weather pole wind velocity. The origin of the sudden piston near 09:43:40 and of the even more obvious hub in Fig. 27 is unknown.

40 36 Mathar Jaffe, NEVEC, Leiden Observatory Issue UL-TRE-MID CO 2 (mmol/m 3 ) Lfile8 LI-COR water (mol/m 3 ) ϕ 0 (cycl) MIDI ϕ (µm) PDF ((mol/m 3 ) 2 /) e-6 e-8 e-0 UTC Lfile8 Mfile22 Mfile22 23:52 23:53 23:54 ν/v (/m) Lfile e+2 e+0 e-2 Mfile Mfile ν () 0 Figure 26: Correlation of MIDI data file 22 with LI-COR densities.

41 MIDI Interferom. and Water Vapor Fluct. Issue UL-TRE-MID CO 2 (mmol/m 3 ) Lfile8 LI-COR water (mol/m 3 ) ϕ 0 (cycl) Lfile8 Mfile23 MIDI ϕ (µm) PDF ((mol/m 3 ) 2 /) -3.0 UTC Mfile :3 0:32 ν/v (/m) e-6 e-8 e-0 Lfile e+2 e+0 e-2 Mfile Mfile ν () 0 Figure 27: Correlation of MIDI data file 23 with LI-COR densities.

42 38 Mathar Jaffe, NEVEC, Leiden Observatory Issue UL-TRE-MID CO 2 (mmol/m 3 ) Lfile8 LI-COR water (mol/m 3 ) ϕ 0 (cycl) Lfile8 Mfile24 PDF ((mol/m 3 ) 2 /) MIDI ϕ (µm) -.0 UTC Mfile :47 02: ν/v (/m) 0. e-6 Lfile8 e-8 e e+2 e+0 e-2 Mfile Mfile ν () 0 Figure 28: Correlation of MIDI data file 24 with LI-COR densities.

43 MIDI Interferom. and Water Vapor Fluct. Issue UL-TRE-MID CO 2 (mmol/m 3 ) Lfile8 LI-COR water (mol/m 3 ) ϕ 0 (cycl) Lfile8 Mfile25 MIDI ϕ (µm) PDF ((mol/m 3 ) 2 /) 0.0 UTC Mfile :59 04:00 04:0 0.0 ν/v (/m) 0. e-6 Lfile8 e-8 e e+2 e+0 e-2 Mfile Mfile ν () 0 Figure 29: Correlation of MIDI data file 25 with LI-COR densities.

44 40 Mathar Jaffe, NEVEC, Leiden Observatory Issue UL-TRE-MID CO 2 (mmol/m 3 ) Lfile8 LI-COR water (mol/m 3 ) ϕ 0 (cycl) Lfile8 Mfile26 PDF ((mol/m 3 ) 2 /) MIDI ϕ (µm).0 UTC Mfile :2 05:3 0.0 ν/v (/m) 0. e-6 Lfile8 e-8 e e+2 e+0 e-2 Mfile Mfile ν () 0 Figure 30: Correlation of MIDI data file 26 with LI-COR densities.

45 MIDI Interferom. and Water Vapor Fluct. Issue UL-TRE-MID CO 2 (mmol/m 3 ) Lfile8 LI-COR water (mol/m 3 ) ϕ 0 (cycl) Lfile8 Mfile27 PDF ((mol/m 3 ) 2 /) MIDI ϕ (µm) 0.0 UTC Mfile :57 05: ν/v (/m) 0. e-6 Lfile8 e-8 e e+2 e+0 e-2 Mfile Mfile ν () 0 Figure 3: Correlation of MIDI data file 27 with LI-COR densities.

46 42 Mathar Jaffe, NEVEC, Leiden Observatory Issue UL-TRE-MID CO 2 (mmol/m 3 ) LI-COR water (mol/m 3 ) ϕ 0 (cycl) Lfile8 Lfile8 Mfile28 PDF ((mol/m 3 ) 2 /) MIDI ϕ (µm) -2.0 UTC Mfile :03 07: ν/v (/m) 0. e-6 Lfile8 e-8 e e+2 e+0 e-2 Mfile Mfile ν () 0 Figure 32: Correlation of MIDI data file 28 with LI-COR densities.

47 MIDI Interferom. and Water Vapor Fluct. Issue UL-TRE-MID CO 2 (mmol/m 3 ) Lfile8 LI-COR water (mol/m 3 ) ϕ 0 (cycl) Lfile8 Mfile29 PDF ((mol/m 3 ) 2 /) MIDI ϕ (µm) -2.0 UTC Mfile : ν/v (/m) 0. e-6 Lfile8 e-8 e e+2 e+0 e-2 Mfile Mfile ν () 0 Figure 33: Correlation of MIDI data file 29 with LI-COR densities.

48 44 Mathar Jaffe, NEVEC, Leiden Observatory Issue UL-TRE-MID Figures show the structure function of ϕ 0 or ϕ for Mfile25 Mfile28 after flagging of weak points. Figure 34: D ϕ0 in units of rad 2.

49 MIDI Interferom. and Water Vapor Fluct. Issue UL-TRE-MID Figure 35: D ϕ0 in units of rad 2. The detector cycle time has been the same for all 30 runs, and is not any different for run 5 shown here (see page 5). It is therefore not possible to evaluate from the data in this report whether the oscillations are an artifact of instrumental hardware, or some part of the data evaluation, for example Gibb s oscillations of some Fourier Transform. Figure 36: D ϕ in units of µm 2.

50 46 Mathar Jaffe, NEVEC, Leiden Observatory Issue UL-TRE-MID

51 MIDI Interferom. and Water Vapor Fluct. Issue UL-TRE-MID COMPARISON INTERFEROMETRY AND GAS ANALYZER 3. Refractive Index Model An instrument like MIDI with some spectral dispersion may differentiate the chemical components in the optical paths if their refractive indices offer sufficient differences in the instrument s spectral band. We employ a linearized phase model of the dispersion [4] k ˆχ dry (λ) = a dry + b dry k; k ˆχ wet (λ) = a wet + b wet k. (5) We use the variables χ for the susceptibility, and ˆχ for its value ( intrinsic polarizability) divided by the molecular number density. Differentiation of (5) with respect to k shows that the parameters b dry and b wet represent the augmented group refractive indices n g [4, (7)] divided by the number densities. 2 Numerical examples of the theory are shown in Fig. 37, Fig. 38 and 39. The coefficients in the fits quoted there are supposedly all proportional to the densities that is, for our purposes here we may ignore the higher-order effects [5] because we do not need to achieve astrometric accuracy and are divided through the partial pressures to generate the intrinsic, density-independent numbers. 3 In the N-band, the intrinsic linear fitting coefficients are m3 b dry mol m 3 mol ; a dry cm 3.77mol m m 3 mol cm ; b wet m mol m 3 mol (6) a wet cm 0.66mol m m 3 mol cm (7) This means the two b coefficients (inclinations of the fit) are almost the same for the two components, whereas the a coefficients (constant terms, axis intersections) differ by a factor of on a permolecule basis. 2 This means (5) is the solution to the differential equation n g(k) n(k) + kdn/dk = b, assigning a constant group refractive index n g = + b to the dispersion. 3 Doing this quickly is supported by the water vapor saturation curve, Fig. 37 in [3] and the GUI in strw.leidenuniv.nl/~mathar/progs/prwaterweb.html.

52 48 Mathar Jaffe, NEVEC, Leiden Observatory Issue UL-TRE-MID e e Pa pure H2O, K, 0.66 mole/m 3 8.0e e e-07 n- 5.0e e e e-07.0e e wavenumber σ(cm - ) Pa pure H2O, K, 0.66 mole/m (n-)k (rad/cm - ) wavenumber σ(cm - ) Figure 37: Upper plot: pure water vapor susceptibility χ wet at an average ambient partial pressure in the N band. Lower plot: product of this susceptibility χ wet and k = 2π/λ = 2πσ. A linear fit to the lower plot is χk = cm k, which means the group refractive index is and enhanced by a factor of two relative to the of the phase refractive index of the upper plot.

53 MIDI Interferom. and Water Vapor Fluct. Issue UL-TRE-MID e Pa dry air, K, 3.77 mole/m 3 2.0e-04.5e-04 n-.0e e e wavenumber σ(cm - ) Pa dry air, K, 3.77 mole/m 3.4 (n-)k (rad/cm - ) wavenumber σ(cm - ) Figure 38: Upper plot: dry air susceptibility χ dry in the N band at a typical Paranal pressure. Lower plot: product of this susceptibility χ dry and k = 2π/λ = 2πσ. A linear fit to the lower plot is χk = cm k, which means the group refractive index is and close to the phase refractive index of the upper plot.

54 50 Mathar Jaffe, NEVEC, Leiden Observatory Issue UL-TRE-MID e Pa pure CO 2, K, 0.03 mole/m 3.2e-07.0e e-08 n- 6.0e e e e wavenumber σ(cm - ) 27.4 Pa pure CO 2, K, 0.03 mole/m (n-)k (rad/cm - ) wavenumber σ(cm - ) Figure 39: Upper plot: pure CO 2 susceptibility χ c at a typical partial pressure in the N band. Lower plot: product of this susceptibility χ c and k = 2π/λ = 2πσ. Each molecule of Carbon Dioxide has a polarizability which is approximately.53 times the polarizability of the average air molecule, in accordance with 0.5 µm measurements [2, Tabl 2].

55 MIDI Interferom. and Water Vapor Fluct. Issue UL-TRE-MID Structure Functions and Fried Parameter Density fluctuations C 2 ρ are converted to line-of-sight-integrated refractive index structure functions and corresponding phase structure functions D ϕ with C 2 n = C 2 ρ ˆχ 2 ; (8) D ϕ = k 2 ˆχ 2 C (0) (α)c 2 ρkp +α / sin a (9) at scale height K, projected baseline length P, Kolmogorov power α = 2/3, and star altitude a above the horizon. The sum of the fits (5) for the product k ˆχ at k = 2π/(0 µm), and the definition of the Fried parameter r 0 [5, (6.3)] yield D ϕ = (k ˆχ) 2 C (0) (α)c 2 ρkp +α / sin a = 6.884(P/r 0 ) +α. (0) The constants are C (0) (α) = 2.94 [7] and k ˆχ ( π/ π/ 0 5 ) m 2 /mol 6.5 m 2 /mol as discussed in the other parts of the manuscript. ( ) 3/5 ( ) 3/ sin a sin a mol 6/5 r 0 = (k ˆχ) 2 C (0) (α)cρk CρK 2. () m2/5 The fit of the water structure functions D H2O to a Kolmogorov 2/3 power law within the spatial distances from to 0 m is illustrated in Figure On doubly-logarithmic axes, these are straightline fits with fixed inclination. For Mfile6 Mfile2 this computation is skipped because the LICOR measurements occurred in the VLTI tunnel during these observations. 4 The only difference to Table 2 in [6] is that 4 hours of observation were summarized there, whereas the observations here cover approximately 2 minutes each.

56 52 Mathar Jaffe, NEVEC, Leiden Observatory Issue UL-TRE-MID D H2O ((mole/m 3 ) 2 ) e-06 e-07 e e-06 e-07 e x (m) 0 0 x (m) D H2O ((mole/m 3 ) 2 ) e-06 e-07 e x (m) 0 Figure 40: Kolmogorov fits to the LICOR water vapor structure functions. One panel is shown per night, curves enumerated with the file numbers as in Table 3. The colored curves are the structure functions; the black straight lines are the fits ( x) 2/3, each with CH2O 2 as the single free parameter.

57 MIDI Interferom. and Water Vapor Fluct. Issue UL-TRE-MID The procedure is repeated for the carbon dioxide data in Figure 4. The structure function is flatter than the exponent 2/3 if D CO2 is small, particularly in the fourth night. The main reason for this is uncovered in Figure 42: the digitization of the LI-COR data is quantized in units of 0 5 mol/m 3, which leads to a constant, distance-independent background noise level of 0 0 (mol/m 3 ) 2 in the structure function. At the time intervals at which the fit is attempted here, this is most of the value of the structure function itself, and not providing information on the actual air density fluctuations. In detail, the connection between the density and refractive index structure functions is with (8) and scaling with C 2 dry /C2 CO2 = /( ) 2, C 2 n = ( m 3 ) 2 C 2 m6 dry 4. 0 mol mol 2 C2 4 m6 dry mol 2 C2 CO2. (2) For Paranal, Cn 2 is in the range /m 2/ /m 2/3 [3], which means the expected range of CCO2 2 is from /m 2/3 to /m 2/3, not 0 /m 2/3. If we reverse-engineer the Cdry 2 from the R0 MEAN data of the primary FITS headers at 500 nm with () for all 30 MIDI files, C 2 dry = r 5/3 0 sin a K k 2 ˆχ 2 C (0) (α), (3) a maximum of /m 2/3 is found at MFILE22. Multiplied by ( ) 2 to extrapolate to the 380 ppm level of CO 2, C 2 CO2 is actually not supposed to be larger than /m 2/3, in obvious discrepancy with the naïve fits to the noise level of the LI-COR CO 2 data in Table 3.

58 54 Mathar Jaffe, NEVEC, Leiden Observatory Issue UL-TRE-MID D CO2 ((mole/m 3 ) 2 ) e e D CO2 ((mole/m 3 ) 2 ) e- e-0 e- x (m) x (m) e- x (m) 0 Figure 4: Kolmogorov fits to the LICOR CO 2 vapor structure functions. One panel is shown per night, curves enumerated with the file numbers as in Table 3. The colored curves are the structure functions; the black straight lines are the fits ( x) 2/3, each with CCO2 2 as the single free parameter.

59 MIDI Interferom. and Water Vapor Fluct. Issue UL-TRE-MID H CO H CO 2 (mmol/m 3 2 O (mol/m 3 2 (mmol/m 3 2 O (mol/m 3 ) ) ) ) Mfile0, Lfile3 Mfile0, Lfile3 Mfile, Lfile3 Mfile, Lfile time since MJD-OBS (sec) Figure 42: These replots of the LI-COR molar densities of Figures 4 and 5 demonstrate that a digitization (quantisation) noise of 0 5 mol/m 3 is inherent (and dominant at short time scales) to the CO 2 measurements, and a noise of mol/m 3 is inherent (and relatively less important) to the water measurements.