1. Physics 114: Lecture 13 Error Analysis, Part III Dale E. Gary NJIT Physics Department

2. Mar 12, 2010 Designing an Experiment  Let’s take a look at several examples of experiments that one might wish to do, and see how we can use our knowledge of uncertainties and statistics to design the experiment.  The process depends on how much you know about the expected behavior of the system you wish to measure, and the equipment and method you will use to measure it. First example: Detecting a flare star’s radio emission.  The Very Large Array radio telescope has a sensitivity given by where T sys is the system temperature (a measure of the noise in the system), about 30 K, A is the antenna area (  D 2 /4, where D is the diameter, 25 m),  is an efficiency factor (about 0.6),  is the bandwidth (about 50 MHz), and  t is the integration time. k is Boltzmann’s constant.  Question: What is the weakest stellar flare detectable with the VLA?

3. Mar 12, 2010 Designing an Experiment, cont’d First example: Detecting a flare star’s radio emission, cont’d.  Inserting the values for the variables, we have  Here, the units are watts per square meter per Hz ( W m  2 Hz  1 ). Astronomers use a unit called the Jansky, which is 10  26 W m  2 Hz  1. Thus, the sensitivity is more conveniently written as  What this says is that in one second of integration time (  t = 1 s ), the VLA will have a ( 1  ) noise level of 1.04 mJy.  There now arise two questions. The first is, what constitutes a “detection?” Let’s say you want a 10:1 signal to noise measurement. Then you would say that the VLA can measure a stellar flare of 10.4 mJy with a 10:1 SNR in 1 s. The second question is, how long can we integrate?

4. Mar 12, 2010 Designing an Experiment, cont’d First example: Detecting a flare star’s radio emission, cont’d.  If a stellar flare lasts for, say, 1 minute, then we can integrate at most 60 s, so the 10:1 relation becomes  The VLA is now undergoing a major upgrade that, among other things, will give a factor of 10 larger bandwidth (  = 500 MHz ). Going back to the original expression, we can see that depends on bandwidth as 1-over-square-root bandwith. So with the new system, we could measure a stellar flare a factor of root-3 smaller, or 0.77 mJy.

5. Mar 12, 2010 Designing an Experiment, cont’d Second example: Measuring the Position of a Star.  How would you measure the position of a star in an image? Positions of stars are in terms of angles relative to some reference frame. The reference frame used by astronomers is the celestial sphere.  You could measure the angles your telescope makes relative to, say, the local horizon and then use these absolute positions to locate the star in your image, but to do that you would have to control a huge number of variables.  Instead, it is more customary to use tabulated positions for background stars, and measure the position of the target star relative to other, nearby stars.  This is the differential measurement technique I have mentioned before.  Let’s look at the practical method of measuring one star relative to others in the same image.

6. Mar 12, 2010 Designing an Experiment, cont’d Second example: Measuring the Position of a Star, cont’d.  The first step is to do a “plate solution.” This is a term from the days when photographic (glass) plates were used. The positions of all stars in the image are determined in terms of pixels of the image, and compared with the known positions from star catalogs.  One can restrict the solution, for example if you only allow a shift in x and y, then the new positions would be x’ = x + a, and y’ = y + b, where a and b and the (signed) shifts in position.  You can also allow (and in general need to allow) a rotation. A general rotation is given by x’ = x cos  + y sin  and y’ =  x sin  + y cos . where  is the angle of rotation to apply. Combining these, of course, gives x’ = x cos  + y sin  + a and y’ =  x sin  + y cos  + b.  In addition, you can apply a scaling (like a uniform stretching), which is done by multiplying by some scale factors in each dimension: x’ = A x (x cos  + y sin  )+ a and y’ = A y (  x sin  + y cos  + b

7. Mar 12, 2010 Designing an Experiment, cont’d Second example: Measuring the Position of a Star, cont’d.  If the stretching is not uniform, you could apply two other kinds of distortion, one that depends on a power of x and y, and one that depends on the cross-term, xy. So the most general form of transformation we will consider is x’ = B x x 2 + C x y 2 + A x (x cos  + y sin  + D x xy + a y’ = B y y 2 + C y y 2  A y (x sin  + y cos  ) + D y xy + b  This allows for non-uniform stretching, rotations, and shifts. We can rewrite this completely generally as x’ = a 1 x 2 + a 2 y 2 + a 3 x + a 4 y + a 5 xy + a 6 y’ = b 1 x 2 + b 2 y 2 + b 3 x + b 4 y + b 5 xy + b 6  There are techniques for determining the coefficients a i and b i, given the positions (x, y) of, say, 100 stars from a catalog, needed to put them in observed positions (x’, y’) in the image.  To allow only shift and rotation, you would set a 1 = a 2 = a 5 = b 1 = b 2 = b 5 = 0.

8. Mar 12, 2010 Designing an Experiment, cont’d Second example: Measuring the Position of a Star, cont’d.  Here is an exerpt from the log file of a plate solution:  19:21:04 - Object List for Image 1 (HAT-P-3R.fit): 113 Detections (51 Stars, 51 Ref. Stars, 0 Movers)  19:21:04 - Astrometry of Image 1 (HAT-P-3R.fit): 46 of 51 Reference Stars used: dRA = 0.34", dDe = 0.31" X = -6.657060632E-7 +1.075686677E-6*x' -5.482967694E-6*y' +2.692809401E-12*x'^2 - 4.352329042E-12*x'*y' +7.838462073E-12*y'^2 Y = -3.516684632E-7 -5.480012973E-6*x' -1.075532987E-6*y' +3.408279133E-12*x'^2 - 8.329083983E-12*x'*y' +2.354389221E-12*y'^2 Origin: x0 = 640.0, y0 = 512.0 Center Coordinates: RA = 13h 44m 34.81s, De = +48° 03' 29.9" Focal Length = 2864.3mm, Rotation = -78.89° Pixel Size: 1.15" x 1.15", Field of View: 24.6' x 19.7'

9. Mar 12, 2010 Designing an Experiment, cont’d Second example: Measuring the Position of a Star, cont’d.  RA dRA Dec. dDec R dR x y Flux FWHM Peak Fit  h m s " ° ' " " mag mag ADU " SNR RMS  ------------------------------------------------------------------------------------------------------------  13 43 37.887 -0.58 +48 01 45.64 -0.08 16.74 +0.34 823.69 43.50 48 3.4 2.0 0.132  13 43 38.167 -0.00 +47 53 02.82 -0.24 13.13 +0.03 1269.25 132.18 1328 4.0 11.1 0.073  13 43 40.239 +0.12 +48 01 38.71 +0.29 15.00 +0.10 825.70 64.72 238 3.9 4.6 0.083  13 43 52.028 -0.49 +48 05 18.18 -0.17 13.73 +0.13 619.01 129.02 762 3.9 8.5 0.067  13 43 52.330 +0.56 +48 04 57.97 -0.53 14.12 +0.12 635.74 134.93 536 3.9 7.0 0.065  13 43 53.019 +0.29 +48 05 34.80 +0.45 14.22 +0.02 603.20 134.72 487 3.9 6.7 0.070  13 43 56.306 +0.65 +48 03 12.91 -0.13 16.65 -0.05 718.73 186.22 52 3.9 1.8 0.140  13 44 03.720 -0.03 +47 53 48.70 +0.59 15.38 +0.18 1187.46 343.16 168 3.9 3.9 0.076  13 44 04.484 +0.09 +47 59 40.38 -0.04 16.36 +0.06 886.33 291.27 68 4.1 2.3 0.090  13 44 05.544 (+1.23) +48 10 08.52 (-0.71) 12.84 -0.16 349.11 196.13 1741 4.0 12.8 0.065  13 44 12.245 -0.29 +48 09 58.95 +0.90 14.88 +0.28 346.16 254.78 265 3.9 4.9 0.069  13 44 13.600 +0.20 +47 52 58.85 +0.22 14.67 +0.27 1213.39 436.00 322 3.9 5.4 0.062  13 44 16.195 -0.19 +47 52 49.66 +0.09 13.02 +0.22 1216.87 459.74 1468 3.9 11.7 0.066  13 44 19.898 -0.22 +48 10 39.63 -0.06 13.85 -0.05 298.79 313.20 682 4.0 7.9 0.068  13 44 20.062 +0.02 +48 04 53.84 -0.10 14.83 +0.23 593.17 372.12 279 4.1 4.9 0.069  13 44 22.521 -0.48 +48 04 35.57 -0.46 16.41 +0.31 604.63 396.15 65 3.6 2.5 0.101  13 44 22.639 -0.63 +48 01 43.30 -0.71 10.78 -0.12 751.23 425.86 11558 3.9 32.9 0.065  13 44 27.332 -0.14 +48 12 55.71 +0.57 14.13 +0.23 170.51 353.86 528 3.9 6.8 0.070  13 44 28.405 +0.52 +47 57 14.49 -0.65 13.30 -0.20 970.63 519.99 1135 3.9 10.2 0.062  13 44 29.486 -0.19 +48 03 15.08 -0.11 15.31 +0.01 661.57 469.01 178 3.8 4.0 0.064  13 44 30.369 +0.39 +48 09 30.81 -0.19 14.46 +0.06 340.01 413.86 391 3.9 6.1 0.066  13 44 30.791 -0.01 +47 51 56.00 +0.43 15.46 -0.24 1238.03 593.68 155 4.8 2.7 0.061  13 44 34.675 +0.40 +48 14 23.95 -0.21 14.68 +0.08 83.15 401.65 319 4.0 5.3 0.069  13 44 34.691 -0.07 +48 09 51.73 +0.62 11.96 +0.06 314.97 447.19 3920 3.9 19.2 0.064

10. Mar 12, 2010 Designing an Experiment, cont’d Second example: Measuring the Position of a Star, cont’d.  RA dRA Dec. dDec R dR x y Flux FWHM Peak Fit  h m s " ° ' " " mag mag ADU " SNR RMS  ------------------------------------------------------------------------------------------------------------  13 44 35.727 -0.23 +48 08 34.25 +0.19 12.69 -0.01 379.24 468.95 1993 4.0 13.6 0.061  13 44 36.443 -0.91 +48 03 52.65 -0.30 12.05 -0.05 617.91 522.11 3597 3.9 18.4 0.064  13 44 37.004 +0.12 +48 15 50.51 -0.93 16.48 +0.18 5.56 407.03 61 3.6 2.2 0.173  13 44 43.049 +0.45 +48 00 14.45 +0.13 12.40 -0.20 792.69 615.05 2605 3.9 15.5 0.062  13 44 43.190 +0.72 +48 00 51.46 -0.37 15.24 -0.16 760.93 610.05 190 4.0 4.0 0.066  13 44 43.975 -0.08 +48 10 58.22 -0.01 15.04 +0.14 242.86 515.18 228 4.0 4.4 0.063  13 44 44.397 +0.70 +48 11 47.11 -0.48 15.30 +0.10 200.51 510.60 181 4.1 3.7 0.079  13 44 46.540 +0.56 +48 12 34.05 +0.11 16.64 +0.14 156.97 521.00 53 3.7 2.0 0.101  13 44 47.377 +0.46 +47 58 26.97 +0.13 16.06 +0.16 876.96 670.07 89 3.8 2.8 0.075  13 44 48.015 -0.11 +48 02 10.59 -0.07 15.65 -0.25 685.42 638.01 130 4.8 2.5 0.059  13 44 49.530 -0.29 +48 09 37.15 +0.30 15.42 -0.48 302.60 576.08 161 4.9 2.8 0.046  13 44 50.175 -0.42 +48 12 51.77 +0.52 15.91 +0.01 135.81 548.99 103 4.0 2.8 0.073  13 44 52.062 +0.12 +48 09 28.66 -0.01 14.90 +0.10 305.58 599.09 260 4.0 4.7 0.068  13 44 55.619 (+1.81) +47 57 47.62 (-1.72) 14.33 -0.27 896.55 747.17 440 4.8 4.9 0.031  13 45 00.635 -0.21 +47 51 55.78 -0.09 16.42 +0.02 1187.69 849.38 64 4.3 2.0 0.111  13 45 02.140 (-1.09) +47 57 57.86 (-0.38) 15.77 -0.53 876.80 801.23 116 4.3 2.6 0.045  13 45 03.721 -0.20 +48 11 41.32 +0.16 16.88 -0.12 173.10 676.18 42 3.9 1.8 0.138  13 45 04.492 (-0.14) +48 05 18.50 (-1.43) 14.95 +0.25 497.67 747.13 249 3.8 4.8 0.067  13 45 08.319 -0.88 +48 07 27.77 +0.60 13.70 -0.10 381.19 758.03 788 3.9 8.6 0.058  13 45 08.982 +0.27 +48 11 54.47 +0.64 16.28 -0.02 153.08 718.80 73 4.7 2.0 0.098  13 45 10.493 +0.05 +48 13 50.93 -0.29 16.45 +0.25 51.44 712.09 62 4.0 2.0 0.122  13 45 13.251 -0.27 +47 52 56.08 -0.16 17.13 +0.13 1114.89 947.29 33 3.2 1.8 0.137  13 45 21.844 -0.44 +47 58 24.47 -0.04 11.61 -0.29 820.72 965.30 5409 3.9 22.1 0.066  13 45 21.983 -0.44 +48 14 24.26 -0.63 15.69 -0.31 3.78 804.32 125 4.7 2.4 0.069