Requirements for Single-Dish Holography

1
Requirements for Single-Dish Holography
Parameter Specification
Goal Measurement error lt10 ?m rms lt5
?m rms Transverse resolution lt0.1 m
lt0.1 m Measurement time 60 min
30 min (per observing frequency) Primary
frequency, f1 80-120 GHz Secondary
freq, f2 f2/f1gt1.2 or lt0.8
(at least 20 separation) Tuning about
f1 and f2 Minimum step size lt5 MHz
lt1 MHz Range gt130 MHz
gt200 MHz Settling time lt60 sec
lt1 sec
2
Design Parameters
Primary frequency 104.0 GHz Secondary
frequency 78.9 GHz Tuning about each
frequency gt130 MHz range, lt1 MHz
steps Range 300 m Transmitter height above
ground 50 m Transmitter height
above elevation axis 43 m Nominal
polarization (xmtr and rcvrs) vertical Receiver
processing bandwidth 10 kHz Integrating
time per measurement 12...48 msec
(nominally 48 msec) Derived Scan angle for 0.1
m resolution 2.18 deg at 78.9 GHz
(?1.09) 1.65 deg at 104.1 GHz
(?0.83) Minimum scan angle 2.29 deg
due to near field geometry Transmitter beamwidth
at -3 dB 4.6 deg (twice antenna
angle_at_xmtr) Transmitter antenna gain 33
dB Transmitter EIRP gt20
?W Transmitter power to antenna gt10
nW Reference antenna beamwidth, -3 dB 4.6
deg (twice scan range) Main antenna feed
beamwidth, -3 dB 128 deg (-3 dB edge taper)
3
Error Budget
Thermal noise lt5 ?m
rms (xmtr pwr, integ time) Feed phase pattern
knowledge lt5 ?m pk (critical)
Reference antenna pattern knowledge lt10 ?m
(insensitive) Multipath interference
0.4 ?m (-20 dB, random phase)
Frequency error 0 Near
field correction error unknown RSS
(except near field correction) 7.2 microns
4
Transmitter Power Calculation, inputs
Assumed hardware parameters Receiver
pre-correlation bandwidth B 10 kHz System
temperature, each receiver T 3200 K Frequency
f 92 GHz (l3.26mm) Antenna
diameter D 12 m j3dB .01556o
Range R 300 m Transmitter EIRP P TBD
Measurement requirements Transverse
resolution D 0.1 m Surface displacement
accuracy dz 5 mm Total measuring time
(OTF scanning) t lt30 min Derived Paramters
Scan angle ? 1.867o (-0.934o) ?
(l/D)(D/D) Number of measurements K 1802 K
(sD/D)2, oversampling factor s1.5
Integrating time per measurement t 27.8 msec t
tK overhead, allowing 100 overhead.
Reference antenna diameter d 50 mm
-3dB beam 2q l/d
5
Transmitter Power Calculation, outputs
Reference antenna power rcvd Pr (1.736e-9 P)
Pr (1/16)(d/R)2 P Main antenna power rcvd
on boresight Ps(0) (1.000e-4 P) Ps
(1/16)(D/R)2 P Receiver noise
power kTB 4.42e-16 W On-boresight noise
?0 (1.59e-22W)(P)1/2
Off-boresight noise (Pr term)
?1 (2.76e-27W)(P)1/2 Generally,
?2 kTB Pr Ps(?) kT/? Ps(?)
Ps(0)J1(??D/?)/(??D/2?)2 where ? is scan
angle, -?/2 lt a lt ?/2. Can neglect kTB term for
any Pgt1 ?W. Noise dominated by Ps term until
power is down by 50 dB. That happens when
J1(2x)/x gt 3e-3 gt x gt 31.7 gt ? gt 20.2l/D.
Outside there, the Pr term dominates.
6
Average Noise Over Map Very conservative
estimate With sampling every 0.75l/D (s1.5),
there are about p 272 2270 samples where the Ps
term dominates. Thus, there are 1802-2270
30,130 samples where the Pr term dominates. Of
those where Ps dominates, take the inner 4 to be
s0, the next 25 to be 10x lower, and the rest to
be 100x lower, in accordance with the envelope of
J1(x)/x. savg2 (425/102241/100)s02 30130
s12 / K 9.084e-4 s02 (1.444e-25W) P
This is actually the noise from a single, real
correlator. For complex correlation, we need to
increase this by 2 times. From DAddario (1982)
eqn (30) dz .044 (l/sD)2 D2 K1/2 savg /
(l M0) .044 l/s2 K1/2 (savg / M0)
.044(3.26mm)/2.25 1802(1.444e-25W)/P(1/2)/4.167e
-7 (2.754e4 m) (2.888e-25W) / P(1/2) For
dz 5 mm, this implies P 8.76 mW.
7
Observing Strategy

• Ground-based single-dish holography
• Transmitter on tower
• OTF raster scanning
• max 0.5d/sec
• 12 min minimum scan time
• Multiple frequencies for multipath mitigation
  repeat raster at each
• Ground-based interferometric holography
• Transmitter on tower
• Astronomy receivers
• Test correlator
• Astronomical interferometric holography
• Astronomy receivers, test correlator
• Natural sources, primarily SiO masers