# fit lband gain(az,za) and compare pitch, roll, focus losses

The lband gain measurements (sefd,tsys,gain jul/aug00) were fit to a function of azimuth and zenith angle.  The fit  was:
g(az,za)=g0 + g1*za+g2*(za-14)^2 + g3*(za-14)^3 + g4*cos(az)   + g5*sin(az)
g6*cos(2az) + g7*sin(2az)     + g8*cos(3az) + g9*sin(3az)
... the (za-14) terms are only used for za > 14 degrees.
The fit can be used for:
•  exploring why the telescope gain is lower than the theoretical value (11-12 K/Jy) at lband.
•  calibrating lband telescope data.

• The degradation in gain from the theoretical value  can come from various sources:
1. misalignment of the mirrors and feed. These can be broken down into parts that are dependent and independent of az,za.
• Az,za independent: mounting of the dome on the azimuth arm, mounting of the secondary, tertiary, and feed relative to each other (assuming gravity changes with za are small).
• Az,za dependent    : motion of the dome relative to the primary. Affected by the azimuth and elevation rail shimming, and horizontal motion of the platform do to the platform support cables and the unbalanced load.
2. errors caused by irregularities in the mirrors (primarily the main dish).
3. pointing errors.
4. blockage of the structure.
The maxim measured gain of 9 K/Jy is about 20% below the expected value. Looking at the above list:
• maximum pointing errors of say  15 asecs at lband is a 2% loss.
• irregularities in the secondary, tertiary are measured to be small.
• blockage .. this one i'm not sure of.. my guess is the 11-12 K/Jy includes the blockage.
• That leaves us with the primary surface irregularities and the motion of the feed relative to the paraxial surface. The feed motion was measured using a theodolite and tilt sensors. The primary surface is in the process of being measured using photogrametry.

The symmetries of the platform includes a 3az term do to the 3 towers. The primary surface and its  support system has no obvious 3az term.  To check the validity of the gain measurements, pitch/roll/focus measurements, and the gain loss computation from the aoant program I compared the 3az term from the fit to the 3az term from the measured gain. These should have the same phase and amplitude (unless we've screwed up or left something out).

#### computing the pitch,roll, focus losses:

A fit in (az,za) was made to the pitch, roll, focus measurements. The pitch and roll errors were computed for the source tracks and added in quadtrature. This value was used in the aoant program to compute the pr loss. The focus loss along the source tracks was computed using a 24" full width half maximum gaussian. This was scaled from an sband focus curve that had:
(12.5*2)/1.74 * 21/12.6 = 24 inches fwhm focus for lband
12.5*2  fwhm sband in tiedown inches, 1.74  tiedown inches/platform inch
21/12.6 = the ratio of the wavelengths (cos of za was ignored).

The focus loss was then multiplied  by the pr loss. This gave a fractional loss from prf. To scale this to K/Jy, the prf fractional loss was multiplied by an idealized curve that included spill over gain:
gainIdeal=8.5 K/Jy  (0  to 15 deg za), then linear ramp 8.5 -> 8.5*.8 (15 to 20 deg za)
gainPRF=fractionalGainPRF * gainIdeal:
This computed gain was then fit using the same  formula as the measured data above. The 1az, 2az, and 3az components were plotted. You would expect the 3az loss to come from only the pitch,roll, focus losses (if we did the  pitch, roll, focus  measurement and computation correctly).

#### The fit results:

The figures are explained below:
1. Fig 1 shows the average gain (polA+polB)/2 plotted by source for the 4 frequencies :1405,1460,1290, and 1370 Mhz. To compute the gain, the source flux and the cal values are needed (as well as the measurement).  The first 14 and last 2 sources have large gain variations with frequency. These source used the lband narrow receiver where there is only a single cal measurement at 1420 Mhz. Using this same value for 1460,1290, and 1370 causes large errors. The 5 sources towards the end with little gain dispersion by frequency were taken with the lband wide receiver where the cal values have been measured at these frequencies.
2. Fig 2 plots the average gain versus za. The + are for the azimuth (feed) on the north part of the dish and * is the azimuth (feed) in the south part of the dish. The gain in the north part of the dish is better than the south.
3. Fig 3 is the average gain by sample for the 1405 Mhz data vs za. The fit to the  gain(az,za) is over plotted as a solid line. The rms residuals are .25 K/Jy.
4. Fig 4 is the measured gain minus the fit vs za.  The symbols +/* mark the feed in the north/south portion of the dish.
5. Fig 5 has the data-fit to az,za versus azimuth in black. The data - fit to only za is over plotted in red. The rms of the fit degrades from .25 K/Jy to .44 K/Jy when the azimuth dependence is left out. You can also see a gain difference in the northern (az=360) and southern (az=180) portions of the dish.
6. Fig 6 shows the fractional gain do to pitch, roll, focus errors vs za. These values were measured with the theodolite/tilt sensors and then computed using the aoant program.  The loss approaches 9% at high za.
7. Fig 7 shows the 1az, 2az, 3az terms for the fit to the data (solid line) and the 1az,2az,3az terms for the pitch,roll,focus gain (dashed line). The values for the complete data fit is at the top of the page and the fit to the pitch,roll,focus is at the bottom.

### Discussion:

#### North/south gain difference , 1az term:

The north part of the dish has a higher gain than the south. Looking at g(az,za) plot the south west side is worse than the southeast (sband showed the southeast fill area to be the worst). The 1az term measures this north, south difference (fig 6.) An amplitude of .4 K/Jy is .4/9K/Jy =.04 degradation.

The prf gain does not show much 1az term.  prf 1az terms come from a residual platform tilt (the tiedown cables are not the correct length). A 4% degradation would require a .18 degree residual tilt. Using the tilt sensors to measure the 1az term we are pretty sure that this is not the case. So the 1 az term is probably coming from north/south differences in the dish surface irregularities or horizontal translations of the platform relative to the dish. If you assumed it was just a focus error of 2*4% (peak to peak), then at lband you would need to go 4 inches out of focus.

#### 2az term:

The prfgain has very little 2az gain. The measured 2az term could be interacting with the za dependence. For sources that come near 18 deg dec, they will have strong za dependence spaced by about 180 degrees. If the za part of the fit does not completely remove this dependence, then you could end up getting a 2az term.

#### 3az term:

The 3az term in the data is .3 K/Jy or a .3/9 = 3.3% loss at lband. The phase of the measured 3az term aligns with the prf (pitch/roll/focus) phase which  shows that we've got the phase of the prf  3az term correct.  The 3az prf amplitude  of .14 K/Jy is .14/.33 = 40%  of the measured value.  Assuming the measured 3az gain term is correct,  we have underestimated the prf 3az term.  The measurement of the 3az prf term is pretty straight forward since it comes from the 3az term of the theodolite/tilt sensor measurements.  The 3az prf error sits on top of any constant prf error. Since the error is nonlinear in prf, any errors in the estimation of the constant term could explain the difference in the amplitude.

#### g(az,za) fit:

The fit is for 1405 Mhz average gain (polA+polB)/2 , gain polA, and gain PolB in K/Jy. The za is in degrees and the (za-14) terms are only used for za > 14 degrees.

gainavg(az,za)    =  8.78131     -.11062 *za  + ( .00183)*(za-14)^2 +(-.00310)(za-14)^3
+ .37864*cos(1az) + ( .16833)*sin(1az)
-.10007*cos(2az) + ( .02403)*sin(2az)
-.26800 *cos(3az)- (.14245)*sin(3az)
The fits to gainPola and gainPolB in K/Jy were:

gainpolA(az,za)=  8.92204 -(0.11083)*za +( 0.00214)*(za-14)^2 +(-0.00342)(za-14)^3
+ .41611*cos(1az) + ( 0.18207)*sin(1az)
-0.10120*cos(2az) + ( 0.02740)*sin(2az)
-0.30386*cos(3az) + (-0.14610)*sin(3az)

gainpolb(az,za)= 8.64057 -(0.11041)*za +( 0.00152)*(za-14)^2 +(-0.00278)(za-14)^3
+0.34117*cos(1az) + ( 0.15459)*sin(1az)
-0.09894*cos(2az) + ( 0.02066)*sin(2az)
-0.23214*cos(3az) + (-0.13879)*sin(3az)

#### How good is the fit for calibrating data:

Errors in the gain measurement come from the measurement technique, the source fluxes, and the cal values used. The rms of the fit was .25 K/Jy (.25/9= 3%). This probably includes the errors in the source fluxes. The data was taken with the lband narrow and lband wide receivers which have different cals. Two sources were measured using both systems.  The ratio of the gains were gainlbw/gainlbn = 1.05. Assuming the gains are the same (even though the horn illumination is a bit different) then the cals differed by 5%. This is a systematic error. The lbw gain values were scaled to the lbn scale (decreased by 5% since there were many more lbn measurement than lbw). On the other hand, the lbw cal is 5 times larger than the lbn cal (10,1.9) so it is probably a better cal to use.

#### Converting the cals to Janskies:

A large uncertainty in the above measurement is the value of the cal in Kelvins. If the cals are stable over long periods of time, then we can use the size of the cal itself as the temperature unit and bypass the uncertainty in the kelvins/cal.  The conversion is:  (K/Cal)/(gainK/Jy) = Jy/Cal. So take the measured calVal and divide it by the gain in K/Jy to get  Jy per cal. Most of this data was taken with the lbn cal but a fraction of it was taken with the lbw cal and then scaled to the gains we got with lbn cal. You should probably only use these equations with the lbn cal.

#### Frequencies other than 1400 Mhz:

The large variation in gains for the 1290,1370, and 1460 Mhz makes it doubtful that the lbn cal is a constant over this frequency range so the gain in K/Jy is incorrect. It is still valid to compute the cal in Jy using the gain(az,za) equations for each of these frequencies and the 1420Mhz cal value (since that is what was used to compute the K/Jy gains). These gains (1290,1370,1460) will be provided separately.
`processing: x101/callb/Aug00/fig_fitazza.pro ..`
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