Event Horizon Telescope
Memo Series

EHT Memo 2017-CE-03

Calibration & Error Analysis WG
Submillimeter Telescope Calibration Memo
Updated absolute amplitude calibration procedure for 2017
S. Issaoun1 , T. W. Folkers2 , D. P. Marrone2 , J. Kim2 , R. Tilanus1 and H. Falcke1
Oct 1, 2017 – Version 3.0
12

Department of Astrophysics/IMAPP, Radboud University Nijmegen, 6500 GL Nijmegen, the Netherlands
Arizona Radio Observatory, Steward Observatory, University of Arizona, AZ 85721 Tucson, USA

Memo history:• May 26, 2017 – Version 2.0• Apr 20, 2016 – Version 1.0
Abstract
This document presents a step-by-step explanation of the Submillimeter Telescope (SMT) calibration procedure for the antenna-based a priori amplitude calibration as part of the Event Horizon Telescope (EHT). During the EHT+ALMA April 2017 observing run, a number of calibration observations and tests were done. The
measurement and reduction processes for each of the SMT a priori calibration deliverables are described in thismemo. Improvements to the calibration procedure include: a newly estimated and updated beam-width for thetelescope at 228.1 GHz enabling a more accurate estimation of the flux density scaling using planet calibrator
observations; a step-by-step outline of the system temperature calibration process using exact equations from thetelescope software scripts; tests of various parts of the signal chain for potential amplitude losses; and a newly
determined gain curve for 2017.
Note: This document can be used in conjunction with similar calibration outlines from other stations forprocedural comparisons. It contains the most accurate calibration information for the SMT to date and thus
renders results from previous memos obsolete.

Contents
1

Introduction

4

2

Telescope-specific parameters and efficiencies

4

3

The SMT DPFU
3.1 The SMT beam-width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 The aperture efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5
5
7

4

System noise temperature

105

Obtaining the flux density of a source

116

Sky opacity at the SMT

127

Signal loss in the VLBI backend

128Updated Gain Curve for 201713

9

SMT 1.3 mm Measurements

14

10 Conclusion

15

A Variables in LinuxPops

172

Relevant terminology

Efficiency and correction terms:• rsb : sideband ratio - since the SMT is a single-sideband
receiver, rsb = ggsi  1 since no signal comes from theimage band but some leakage can still be present

Relevant variables introduced in this document (brightness
temperatures approximated with the Rayleigh-Jeans approximation):• AM: amount of airmass in the line of sight of the receiver (elevation-dependent)• Chot : Counts measured when looking at the hot load
(vane)• τ0 : atmospheric opacity at the zenith• Ccold : Counts measured when looking at the cold load(liquid nitrogen)• Con : Counts measured observing a target• e−τ : atmospheric attenuation factor, which damps thesignal based on atmospheric opacity in the line of sightτ = τ0 × AM

• Csky : Counts measured when looking at blank sky• τ0,s : atmospheric opacity at the zenith in the signal
band (same principle for image band)• T cold : temperature of the cold load

• el: elevation of the antenna dish for a particular observation (in degrees)• T rx : receiver noise temperature• g(el): elevation-dependent gain curve correcting forchanging illumination of the main reflector and groundcontributions as the dish moves and tilts to different
elevations• T vane : temperature of the vane (blocker)• T amb : ambient temperature around the SMT, as measured by the weather station on Mount Graham (physical temperature)• ηl : forward efficiency representing the fraction of
power received through the forward atmosphere• ηtaper : efficiency loss due to non-uniform illuminationof the aperture plane by the tapered radiation pattern• T em : temperature of the atmospheric emission (alsointerchangeably called T sky in the literature, the brightness temperature of the sky)• ηblock : aperture blockage efficiency due to blocking of
the feed by the sub-reflector (including its support legs)• T em,i : temperature of the atmospheric emission in theimage band (sideband not used for observing)• ηspillover : feed spillover efficiency past the main reflector− it is the ratio of the power intercepted by the reflective elements to the total power• T em,s : temperature of the atmospheric emission in thesignal band

• ηRuze : surface error efficiency (or Ruze loss) calculated
from Ruze’s formula (Ruze 1952)• T cab : physical temperature of the receiver cabin (this is
assumed to be the same as the ambient temperature)• T cal : derived temperature to give a correct temperaturescale for the signal band

• ηA : aperture efficiency approximated for the SMT, acombination of various efficiencies ( = ηtaper × ηblock ×ηspillover × ηRuze )• T sys : system noise temperature of the system• Ageom : geometric area of the SMT dish∗
• T sys: effective system noise temperature (corrected foratmospheric attenuation and rearward losses)• Aeff : effective area of the SMT dish ( = ηA Ageom )3

1

Introduction

ponents, the effective system noise temperature and the antenna gain:∗
T sys
SEFD =(3)G
For the SMT, the SEFD at zenith is of order 13 000 Jansky.

A telescope’s system-equivalent flux density (SEFD) is simply the noise contribution of the system, given by the system
noise temperature, and all losses and gains, converted to aflux density scale. The SEFDs can be calculated using system
noise temperature T sys measurements and all efficiencies andcontributions to source attenuation and noise, and one candetermine the sensitivity of the telescope when comparedto other telescopes in the array. The higher a telescope’s
SEFD, the lower its sensitivity. Ultimately, the flux density ofa source is simply the telescope’s SEFD, which contains all
system and telescope parameters and efficiencies, multipliedby the ratio of signal to noise power (defined as rS/N ) of thesource detection. The equation for the SEFD can be subdivided into three main components, each with station-based
variations for how they are determined and measured. The
three components to the SEFD are:

In order to fully calibrate the SMT amplitude scale, we musthave a comprehensive understanding of the variables involvedin the calculation of source flux densities and how they aremeasured and obtained. The following equations and descriptions explain the entire procedure for the SMT. In order toeasily understand the variables within LinuxPops, the SMT
in-house calibration software, a compilation of each variable in the theoretical calibration below and its LinuxPops
counterpart can be found in Appendix A.

2

1. T sys : the total noise characterization of the system,
given by the system noise temperature

Telescope-specific parameters andefficiencies

The most basic parameter relevant for the sensitivity of aparabolic antenna is its geometrical collecting area, which
affects the amount of radiation reflected from the primary
onto the secondary reflector. The geometric area of the SMT
is calculated with a known diameter of the dish of 10 meters:τ

2. eηl : the correction terms for attenuation of the source
signal by the atmosphere and rearward losses

3. G: The antenna gain, including all the loss terms fromthe telescope and the conversion from a temperature
πD2
scale (K) to a flux density scale (Jansky), given by theAgeom =
≈ 78.54 m2(4)
4“degrees per flux density unit” factor (DPFU) in K/Jy
and the normalized elevation-dependent gain curveg(el):
The aperture efficiency represents the efficiency of the teleη A
DPFU = A 2kgeom , giving antenna gain G = DPFU × g(el) scope compared to a telescope with a perfect collecting area
(uniform illumination, no blockage or surface errors) and it isdetermined using observations of known calibrator sources,
This gives the following general equation for a telescope’s
usually planets. It corrects for the forward losses of the teleSEFD:
scope during observing. We can write the aperture efficiencyτT sys eSEFD =(1) as the combination of various efficiency terms:
ηlG
ηA = ηtaper × ηblock × ηspillover × ηRuze(5)
The flux density of a source detected with a given ratio of
signal to noise power rS/N is then:Each efficiency term corresponds to an aspect of the telescope
feed:2τ

S source = SEFD × rS/N =rS/N × T sys eηlG(2)• ηtaper is the efficiency loss due to non-uniform illumination of the aperture plane by the tapered radiation
pattern/feed function (also formally known as the illumination efficiency). It is the most important contributor to the aperture efficiency.

For mm-observatories, which measure the effective systemτ∗temperature T sys= T sys eηl directly using the chopper technique1 , the SEFD equation can be rewritten in only two com-

1 See the complementary A priori Calibration Memo by Issaoun et al. (2017) for details
2 See Baars, J., The paraboloidal reflector antenna in radio astronomy, Springer, 2007 for more detail on the measurement of the different losses.

4• ηblock is the aperture blockage efficiency due to blocking of the feed by the sub-reflector (including its support legs)

fJy/K =

1
2k=
[Jy/K]DPFU Ageom ηA(9)• ηspillover is the feed spillover efficiency past the mainreflector - it is the ratio of the power intercepted by
the reflective elements beyond the edge of the primary
and the sub-reflector to the total power. It is due partly
to cold sky and partly to a warm background, and iselevation-dependent.

The antenna gain will of course also be affected by the gain
curve, not included in fJy/K , depending on the elevation of theobserved source. A time-dependent DPFU can also be estimated using flux measurements of planets at different times
during the observing window.

• ηRuze is the surface error efficiency (also called “Ruzeloss” or scattering efficiency) calculated from Ruze’s
formula (Ruze 1952). It is due to randomly distributed
small-scale deviations of the reflector from the perfectparaboloidal shape. Ruze’s formula is presented below,
where σ is the surface rms (accounting for small-scaledeviations from a perfect surface through dish holography) and λ is the observing wavelength:

3.1

ηRuze (λ) = e−16π2 σ2λ2

The SMT beam-width

In order to confidently provide an aperture efficiency for thetelescope, a reliable beam-width must be used. The expected
flux densities from calibrators are dependent on the couplingof the source size to the telescope beam. Therefore an accurate estimation of the telescope beam-width is a necessary
input for the simulation programs giving expected calibrator
fluxes.

(6)Additionally, telescopes do not have perfect surfaces, andmust thus suffer some losses and gains of signal due to distorted illumination of the main reflector as they slowly moveto different elevations. This large-scale surface deformation
affects the received signal and is not taken into account in thegeneral aperture efficiency calculation. These losses however
can be determined by tracking sources through a wide rangeof elevations, and thus measure an elevation-dependent gain
curve for the telescope, where the maximum (g = 1) is set
where the telescope is expected to be the most efficient. This
normalized gain curve is usually written in the form of a second order polynomial (in the standard VLBA format), where‘el’ is the elevation in degrees:g(el) = a2 (el)2 + a1 (el) + a0

(7)

An updated gain curve for 2017 can be found in Section 8.

3

Figure 1: The 50 by 50 az-el continuum grid map of Jupiter taken
on 11 April, 2017. On the left side is the image of Jupiter mappedby the negative beam and on the right side is the image of Jupitermapped by the positive beam.

The SMT DPFU

After the EHT+ALMA run in April 2017, we made a continuum map of Jupiter to estimate the telescope beam-width at
the EHT observing frequency (228.1 GHz). The map is a 50
by 50 azimuth-elevation (az-el) grid map, with the telescopemoving horizontally in azimuth at each elevation incrementof 1000 . The map is shown in Fig. 1. The SMT has a chopping
beam with a separation of 40 between the two beams: during

We have previously defined the antenna gain as a combination of the telescope’s DPFU and its normalized elevationdependent gain curve. To obtain the SMT’s Jy/K factor fJy/K ,independent of elevation, we simply take the inverse of the
DPFU:
Ageom ηA
[K/Jy](8)
DPFU =
2k5

Figure 2: Left: Map of Jupiter by the positive beam of the SMT. Center: Best-fit 2D Gaussian model of the positive beam map. Right:
Residual map of the model-subtracted image.

Figure 3: Left: Map of Jupiter by the negative beam of the SMT. Center: Best-fit 2D Gaussian model of the negative beam map. Right:
Residual map of the model-subtracted image.

the az-el grid mapping, both the negative and positive beam
slew through the map area, hence the two images of Jupiter.

is estimated as follows (accurate to about 2% for an extended
source). The diameter of Jupiter θJupiter = 41.1700 , as computed from the Planet program for 11 April 2017, was used
for the deconvolution. The beam-width is thus given by:

A first attempt at obtaining the beam-width of the telescope
was made by fitting a Gaussian to the flux density collectedby the telescope at the zero elevation change mark, along theazimuth change. The deconvolution of the beam-width θmb(the width of the main lobe of the telescope beam pattern)from the full width at half-maximum (FWHM) of a Gaussian

r
θmb =
6

FWHM2 −

ln(2) 2θ
2 Jupiter(10)This method gave wildly varying beam-widths between thepositive and negative beam. Therefore, a different analysis
was used for a more robust result: using a least-squares fit ofa 2D Gaussian model for each beam window. Since the noiselevel is near zero, the fits were done without subtraction of abackground.

shows. However, any uncertainty from the beam-width usedwill not be the dominating factor for the uncertainty in the
DPFU, making the average estimate an adequate value to use
for the calibration procedure.

A 2D Gaussian model was fitted to both the negative and positive beam images of Jupiter, and then the telescope beam size
was deconvolved from the resulting FWHM in the azimuthand elevation directions and the known diameter of Jupiter on
April 11, 2017, as previously outlined in Eq. 10. Fig. 2 andFig. 3 show the best-fit 2D Gaussian model for the Jupiterimages taken by the positive and negative beams of the SMT
respectively. Table 1 presents the dimensions of the telescope
beams estimated from the 2D Gaussian fits.
Table 1: Measured beam-widths from the Jupiter az-el grid map
using 2D Gaussian fits for the positive and negative beam.

Beam
Positive
Negative

Beam-width along∆Az [arcseconds]
36.1 ± 0.3
32.6 ± 0.3

Beam-width along∆El [arcseconds]
34.9 ± 0.3
35.4 ± 0.3
Figure 4: Map of Jupiter with a smaller flux density range enables
the viewing of the spillage from the secondary reflector support legs
for each beam.

Of course this method only gives a simple fit, but this should
be sufficient for the analysis of the telescope DPFU, in particular because other uncertainties dominate (pointing, focus).
The difference between the sizes of the positive and negative beams are not yet fully understood, but it is possibly
caused either by technical differences in the chopping of thesecondary between the two beams or by non-uniform illumination of the dish from distortions on the surface of themain reflector. The latter is also further shown by the nonsymmetrical distortions caused by the secondary support legs,
as shown in Fig. 4.

3.2

The aperture efficiency

With the newly updated telescope beam-width, the aperture
efficiency for the SMT can now be estimated from planet
observations. We have decided not to use the SMT in-house
Planet simulation software for the following calculations (aswas usually done in past memos) but to instead only use thebrightness temperatures and apparent sizes given for Saturn,
Jupiter and Mars and calculate every step of the calibration
independently. The newly estimated aperture efficiencies
with the method described here render measurements in past
memos outdated and innacurate.

This is also supported by the fact that the residuals of themodel-subtracted images from the two beams are not mirror
images of each other but are both biased to the left side: thisagain points toward asymmetry in dish illumination. Asymmetry between the two beams is also not an uncommon phenomenon for radio-telescopes with the same chopping setup.
Other possible effects could be atmospheric distortions, tracking errors or gridding problems during the mapping (the az-el
grid map mode is not used frequently at the SMT).

The observed planet fluxes from the SMT calibrator scans
are compared to expected planet flux densities in the telescope beam for a perfect telescope at the given frequency andbeam-width. The aperture efficiency ηA is found using the following equation, where T A∗ is the observed effective antenna
From Table 1 we can estimate an average beam-width fortemperature, g(el) is the telescope gain curve, k is the Boltzthe SMT at 228.1 GHz: θSMT = 34.8 ± 0.6 arcseconds. It ismann constant, Ageom is the geometric area of the telescope
important to note however that an approximation of a circular and S beam,sim is the expected flux density of the planet in the
beam might not be entirely accurate for the SMT, as Table 1 telescope beam estimated from planet information such as7

apparent size and brightness temperature:
T A∗
Ageom g(el)S beam,sim
2k

ηA =

Or similarly the DPFU is directly given by:
DPFU =T A∗
g(el)S beam,sim

where ν is the observing frequency in Hz, h is the Planck constant, c is the speed of light (in m/s), T B is the mean brightness temperature for the planet (assuming a disk of uniform(11) temperature), and Ωs is the solid angle of the source on thesky in steradians. Since we are dealing with very small objects, the latter can be approximated using the small angle approximation, where θs is the apparent diameter of the planet
observed, in radians:
(12)

Ωs '

For extended sources, it is important to calibrate the flux density observed in the beam because some emission might not
be picked up by the telescope. The aperture efficiency is only
concerned by the main beam flux density, and so the following equation is used to calibrate the simulated flux density inthe beam for an extended source, where S sim is the expected
total flux density of the source:
S beam,sim = S sim × K

πθs2
4

(19)

Of course this process heavily depends on assumptions madein the planet calibration, such as accurate predicted planet
fluxes, stable weather conditions and a well-calibrated instrument in terms of pointing and focus.
For the SMT, the Planet program provided brightness temperatures T B and apparent angular sizes of each planet at
a given date and frequency. The program gave, for eachplanet and each day, a major and minor axis estimation forthe planet disk. We approximated the diameter of the planetθs for the flux density calculation as the mean of the major
and minor diameters. Table 2 shows typical parameters ofplanet calibrators for EHT observing at the SMT.

(13)

K is determined by the following equation, where θmb is thehalf-power beam-width in arcseconds of the primary lobe of
the telescope beam pattern (telescope beam diameter) andθs is the diameter in arcseconds of the observed extended
source:
θs p
x=
ln(2)
(14) Table 2: Typical planet apparent sizes, brightness temperatures andK correction factors for EHT observing at the SMT over the past
θmbK=

1 − e−xx2

three years (for Mars, April 2016 only).

2

(15)

PlanetThis K factor is the ratio of the beam-weighted source solid
angle and the solid angle of the source on the sky. It is infact the integral of the antenna pattern of the telescope (ap2
proximated as a normalized gaussian) P(θ, φ) = e− ln 2(2θ/θmb )and a disklike source with a uniform brightness distributionΨ(θ, φ) = 1 over the size of the extended source. This serves
very well for our a priori calibration purposes.
Z
Ωsum
1
K=
=
P(θ − θ0 , φ − φ0 )Ψ(θ0 , φ0 )dΩ0 (16)ΩsΩs source

K=

1
Ωs

Z

P(θ − θ0 , φ − φ0 )dΩ0

Mars
Jupiter
Saturn(17)

The expected total flux density is estimated by:2h ν3 Ωs,c2 e kThνB − 1

TB
[K]
208
170
150

K factor
0.96
0.63
0.93

Planet scans during three EHT observing windows, in 2015,2016 and 2017, were taken to estimate an aperture efficiency
per scan. The measurements with Saturn were discarded dueto the Saturn rings influencing the disk approximation for theplanet and the need for a more detailed brightness distribution model for accurate estimates of an aperture efficiency.
Furthermore, only antenna temperature measurements done
in upper sideband (USB) were kept for the analysis, as thisis the sideband used for EHT observing. Separate aperture
efficiency measurements were also determined per polarization (RCP and LCP). Fig. 5 shows the measurements used forthe analysis and the resulting mean aperture efficiency for the
SMT.

sourceS sim =

θs[arcseconds]
12 ± 142 ± 316 ± 2(18)

8

not trusted.

Figure 5: Aperture efficiency estimates for upper sideband measurements (USB) using Jupiter and Mars (2015-2017). The blue dashedline represents the mean aperture efficiency from the measurements.
Figure 7: Aperture efficiency measurements separated by polarizaThe error bars are estimated from the standard deviation from themean of the measurements approximated as a Gaussian distribution. tion: blue is LCP, red is RCP. Once again, the squares are Jupiterand circles are Mars measurements.

The lack of a trusted physical trend for the time-dependenceof the aperture efficiency thus led to the conclusion that asingle average aperture efficiency must be adopted for the
SMT until this behavior can be better understood or a better sampling of planet measurements (not just during EHT
weeks in April) can be obtained. The telescope is not particularly known to be under-performing or having difficulties
observing in daytime conditions.
The aperture efficiency of the SMT is determined by taking
the mean of all aperture efficiency measurements from all
three EHT observing windows, and the error on the measurement is the standard deviation determined from the scatter of
Figure 6: Aperture efficiency estimates using Jupiter (squares) andthe measurements around the mean value. It is likely that theMars (circles). 2015 points in blue, 2016 in green and 2017 in red.
error on the aperture efficiency is overestimated, as the scatterof the measurements can be caused by various other effects
such as the unsolved time-dependent trend, uncertainties
An attempt was also made to determine a time-dependentand fluctuations in the measurement of the effective system∗variation of the aperture efficiency as a function of UT time, temperature T sys(used to calibrate counts on source to T A∗
as shown in Fig. 6. A time-dependence of the aperture effiscale), changes in pointing and focus accuracy or a variable
ciency can be caused by the dish being affected by the Sun
atmosphere due to sparse cloud cover and high winds. In par∗and heating up during daytime hours, thus reducing telescope ticular, the uncertainty in the T sys
measurement is also present
sensitivity and efficiency. However, a very peculiar physical in the SEFD calculation for the telescope, thus it is likely thattrend was observed at the SMT as a function of UT time (lo- its error contribution is double-counted: once for the actual∗
cal time is in MST = UT − 7). It appears that the telescope
T sysvalues needed to calculate the SEFDs; and once again for
is at its most efficient during mid-day and reaches a minitheir effect on the ηA measurements.
mum in the night. The current measurements, which appear
to show an increase in efficiency during daytime hours, are
The mean aperture efficiency and DPFU (or Kelvin-to-Jansky
3 More digits for the measurements can be obtained upon request.

9

4

factor fJy/K ) for the SMT are thus3 :ηA = 0.59 ± 0.03

System noise temperature(20)

ηA Ageom= 0.0168 ± 0.0008 [K/Jy]
2k
2k
fJy/K =
= 59 ± 3 [Jy/K]ηA Ageom

The first step to calibrating the flux density measurements(21) of various sources is to determine the noise temperature of
the system when the telescope is pointed off-source (blank(22) sky). At the SMT, this is done with the chopper method, by
getting a system-to-background ratio of the signal, comparing the power obtained when the telescope is pointed to the
Furthermore, the difference between RCP and LCP aperture sky directly (off-source counts), and when the receiver isefficiency (or DPFU) was not found to be significant. This
blocked by an absorber (chopper counts), and then calibrated
can be seen in Fig. 7 for individual measurements and in
to a temperature scale. The following is an account of theFigs. 8 and 9 for the mean distributions of aperture efficiency exact equations used in the telescope software to determine
measurements for RCP and LCP respectively.
the effective system noise temperature and other calibration
information.4 A more general approximate procedure can befound in the complementary A priori Calibration Memo by
Issaoun et al. (2017).
DPFU =

During a cold load calibration with a liquid nitrogen bath(hereby referred to as “cold cal”), the Y-factor and the receiver noise temperature are computed. This is done once
every few hours by the operator to refresh receiver noise temperature values. They do not vary much between cold cals:
Y=

Chot
,
Ccold

(23)

where the numerator is Chot , the counts obtained from thevane (chopper), and the denominator is Ccold , the counts obFigure 8: Aperture efficiency distribution for RCP approximated by
a Gaussian fit of a mean ηA,RCP = 0.5911 and a standard deviation of tained from the cold load (liquid nitrogen bath). The Y-factor
also enables an easy diagnostic of the sensitivity of the reσRCP = 0.0285.
ceiver. A high Y-factor means little receiver noise, and thus
sensitive observations (of course what constitutes “high” depends on the type of receiver and the observing frequency).
For the SMT, the Y-factor is typically ∼ 2.5 for EHT observing. Then the cold cal routine calculates the receiver noise
temperature as follows:
T rx =

T vane − YT cold
Y −1

(24)

Here temperatures are used, where T vane is the temperatureof the vane blocker (chopper; at room temperature ∼ 290 K)
and T cold is the cold load temperature (for the SMT it is thetemperature of liquid nitrogen ∼ 77 K).
Figure 9: Aperture efficiency distribution for LCP approximated by
a Gaussian fit of a mean ηA,LCP = 0.5904 and a standard deviation of
σLCP = 0.0281.

Then the routine computes an estimate for T em = T sky , theatmospheric emission temperature (or equivalently the sky

4 Note: The SMT calibration script written by Thomas Folkers uses the ATM program, a sky emissivity correlation model of sky temperature part of the
ASTRO package of the GILDAS software by IRAM. For more detail on this procedure, see Calibration of spectral line data at the IRAM 30m radio telescopeby C. Kramer.

10brightness temperature), which is later corrected in an itera∗tive process with the ATM program to determine T sys:

they do not deviate from the standard AM= 1/ sin(el) equation until below 15◦ (at 15◦ it deviates by less than 1%).

Chot − Csky
(T vane − T cold )Chot − Ccold(25) The calibration temperature represents the difference between the temperature of the sky and the temperature of thevane, corrected for atmospheric losses, determined by theHere Csky corresponds to the counts obtained by looking at
ATM program. In the standard chopper calibration technique,blank sky. When the calibration routine is not a cold cal (butT cal ∼ T hot . It is important to note that the exponential term∗
is instead a T sys
routine before a scan/pointing/focus), thehere is highly misleading. The presence of this term in T cal
program uses the following to estimate T em :serves to stabilize fluctuations in T cal coming from changes in
the atmospheric emission temperature with opacity and elevaCsky
T em = (T rx + T vane )− T rx(26) tion. It is NOT the cause of the atmospheric correction factor
Chotfrom the chopper technique resulting in the measurement of∗
T sys. Actual values for T cal are very stable, fluctuate only by
The program then iteratively corrects T em using the ATM
1-2%, and do not change as a function of airmass.
program to determine the variables in the separate sidebands(signal and image). The atmospheric model (ATM) is used to Then the program calculates the effective system temperature∗
fit the emission of both sidebands to the sky temperature T em (including the atmospheric attenuation correction) T sys
in theby varying the amount of water vapor. The model uses a stan- following way:
dard atmosphere and radiative transfer to compute the total
Csky
absorption and thermal emission by water vapor and oxygen
∗
T sys
=
T cal
(29)
through the atmosphere. After running the ATM transmission
Chot − Csky
routines for the two sidebands, the atmospheric emission temperature for the image band is computed and simplified as
The chopper technique determines a system temperature infollows:
cluding all noise contributions from the receiver until the top
T em,s + rsb × T em,iT em,i =+ T cab = T em,s(27) of the atmosphere. Thus, the temperature it outputs is the ef1 + rsb∗
fective system noise temperature T sys, which, when appliedtosourcemeasurements,
wouldalready
correct for the source
With the assumption on the sideband ratio (rsb = 0 for
signalattenuation
bytheatmosphere.
This
is measured before
single-sideband (SSB) receiver) and the fact that the cabin
every
scan
on
a
target.
In
VLBI
mode,
the
quarter wave-plate
temperature is computed as T cab = 0.8T vane + 0.2T amb =
∗
is
added
to
the
signalchain,
but
since
T
is
measured before
0.8T amb + 0.2T amb = T amb , the atmospheric emission tempersys
5
every
scan
in
VLBI
mode,
the
choppercalibration
is doneatures in the image band and signal band are equal . This ofwiththequarterwave-plateinplace.
Thus
anylossesinducedcourse depends on the atmospheric absorption spectrum butbythequarterwave-plate
wouldbecorrected
via
thechopperremains a fair assumption for the Event Horizon Telescope∗calibration and will be offset within the T sys
measurement.
observing frequencies.
T em = T vane −Then the program proceeds to compute the calibration temperature T cal , which gives the temperature scale for the signal
band:

5

T cal = (1.0 + rsb )(T vane − T em,i )eτ0,s ×AM= (T vane − T em,i )eτ0,s ×AM

Obtaining the flux density of asource(28)

The opacity at the zenith in the signal band τ0,s comes fromthe ATM program, as does the atmospheric emission temperature in the image band T em,i . The program uses elaborate
equations to compute the airmass, designed for low-elevation
observing. These equations will not be explained here, but

We define the antenna temperature for each source measurement, using the signal-to-noise ratio of each measurement,as:

T A∗ =Con − Csky ∗
T sys
Csky

(30)

5 The assumption that the vane temperature is the same as the ambient/outdoor temperature is no longer trusted. The vane was moved from the receiver∗ calibration.
cabin to a room in the telescope building, and so the vane temperature is likely closer to room temperature. This is an outstanding issue of the T sys

11

Notice the (*) symbol: this is because this antenna temperature is corrected for atmospheric absorption, implicitly in∗cluded in T sys.

to determine an opacity, using weather conditions obtained
from the Mount Graham weather station, and can deem atipper reading as “unreliable” if the measured opacity does
not match that outputted by the ATM program. That value isthen not used for the calculation of the effective system noise
temperature, and it instead uses the last good tipper value tocompute this. The ATM program then computes the opacity
in the signal band in the line of sight at the exact observing
frequency using an iterative process, with the tipper opacityand other parameters from the Mount Graham weather station
as inputs.

Now correcting for the aperture efficiency and the gain curve
as a function of elevation:
T A∗
TR =
(31)
ηA g(el)T R is the final antenna temperature: it contains all corrections
for efficiency and telescope parameter terms, and is thought
to be the equivalent of the temperature of a resistor held directly in front of the receiver. Thus, it should account forevery telescope-specific and source-specific variables, apartfrom the conversion to a flux density.

7

Signal loss in the VLBI backend

The final step is to convert the temperature scale into a flux
density scale in Jansky, where k = 1.38 × 10−23 J/K =1.38 × 103 Jy/K is the Boltzmann constant:

S =

6

2k
Ageom

TR =

T A∗
8k
πD2 ηA g(el)When calibrating VLBI data using individual telescope parameters and calibration information, there may be discrepancies in the instrumental setups for the observing run and thecalibration procedures. Since calibration information is usually obtained via the telescope’s in-house system, there may(32) be some additional unaccounted effects in the recorded VLBI
data as it is obtained through the separate VLBI backend(R2DBEs, Mark6 recorders and modules).

Sky opacity at the SMTIn April 2017, we attempted to constrain possible losses
through the VLBI backend using a Y-factor test. This wasdone by measuring and comparing the Y-factor (ratio between
a hot/ambient load and a cold load, see Eq. 23) obtained at
the receiver total-power box (part of the SMT setup) andinside the R2DBE itself (R2DBE output6 ).

The sky opacity for a particular air mass (AM) changes depending on the elevation of the dish as shown below, whereτ0 is the sky opacity at the zenith:τ = τ0 × AM

(33)

A spectral analyzer was used for the total-power box andR2DBE input to compare the results directly with the YThe opacity at the SMT is measured by a tipping radiomefactor measurements outputted by the R2DBE. It was foundter, which is placed about 100◦ away from the dish, on thethat the Y-factors at the receiver total-power box and at thetelescope building itself. The SMT building is on a rotating
VLBI backend, shown for RCP and LCP for both the highplatform, therefore the opacity is not measured through a con- and low bands in Fig. 10, are compatible. It is therefore safe
stant water vapor column but rotates to measure 100◦ fromto assume that there is no obvious signal loss through to thethe target source Azimuth.
R2DBE backend that is unaccounted for in the standard amplitude calibration described in this memo for EHT data taken
In particular, this becomes problematic when the telescopeat the SMT from 2015 onward.
observes targets at 200-210◦ Azimuth, as the tipper is pointed
right at the Large Binocular Telescope (LBT). The warm air Before 2015, the SMT used the R1DBE backend, where no
and reflection from the LBT building causes abnormal spikes Y-factor test was done, so it remains difficult to account forin the tipper opacity readings, and these readings become
possible signal losses in older datasets.
unreliable and not representative of observing conditions.
Furthermore, in order to correct T cal for atmospheric attenuation, the ATM program uses an iterative linear fit process
6 This was done using the LMT scripts collect mark6 data.py and genYSFactor.py written by L. Blackburn and K. Bouman.

12Figure 10: Spectral Y-factor at the receiver total-power box (in blue) and at the backend (input to the R2DBE; in orange), measured with thespectrum analyzer. Top left: low band LCP Y-factor. Top right: low band RCP Y-factor. Bottom left: high band LCP Y-factor. Bottom right:
high band RCP Y-factor.

8

Updated Gain Curve for 2017

If more than one source is used, this is done once the flux
density measurements are normalized around a plateau (to arelative gain scale). The source measurements used to obtain
Telescopes are not perfectly rigid paraboloids, and must thus a gain curve must of course be calibrated for all other effects,
including telescope efficiency (through the DPFU) and thesuffer some losses of signal due to distorted illumination of∗
the main reflector as they slowly move to different elevations. atmospheric attenuation of the signal (through T sys ).
This large-scale surface deformation affects the received signal and is not taken into account in the measurements leading Analogously to the previous runs in 2015 and 2016, continuum measurements were made at 1.3 mm of various nonto the efficiency and DPFU characterization. These losses
variablecalibrator sources. The elevation dependence wascan be determined by tracking sources over a wide elevation
investigatedusing two non-variable sources tracked acrossrange, preferably as they increase and decrease in elevation
the
sky
for
a
short period of time (10 hours over one night)
from the tree-line to transit and vice-versa. An elevationand
a
wide
range
of elevations: K3-50, a planetary nebula;dependent gain curve for the telescope is measured, where
and
W75N,
a
star
formation region. This tracking was thethe maximum (g = 1) is set where the telescope is expected
best
methodtoisolatethe elevation-dependent component in
to be most efficient. The gain curve is estimated by fitting atheSMT
output.
The
sourcesreached higher elevations but
polynomial (usually second-order for standard radio-dishes).
13

Table 3: Gain fit parameters for the 2017 elevation curve.

disappeared at very low elevations behind the solar avoidancezone. The relative gain for the source measurements weretaken using the ratio of the measurement against an averagevalue of the upper envelope of data points in a mid-to-highelevation range where the measurements were constant − fora radio telescope, this typically occurs between 40 and 60o .
The effect of elevation on the relative gain can be clearly seenin Fig. 11.
The elevation-dependent geometric (opacity-free) gain curve
can be constrained with little-to-no elevation effect from the∗atmosphere, an effect corrected by the T sys
measurementsfrom the chopper calibration. It is best approximated as thesecond-order polynomial shown in Fig. 11. Relative error
bars are obtained from the standard deviation of the noncorrected gain distribution around unity.

parameter
a0
a1
a2

value
0.73 ± 0.04
(0.9 ± 0.1) × 10−2(−0.8 ± 0.1) × 10−4

Following the constraint of the elevation gain curve g(el),LinuxPops can contain the gain curve terms and incorporate
them in the SMT calibration procedure in the following way,where the elevation is in degrees:g = −0.8 × 10−4 (el)2 + 0.9 × 10−2 (el) + 0.73

(34)

The final flux density is then given by:

1
2k ∗
S =
T
(35)
Observing was done in a similar manner to the 2016 gainηA g(el) Ageom A◦
curve, constraining the measurements to elevations above 20 .
The gain curve for the SMT for 2017 is once again a typical
Table 4: Mean flux densities of most EHT observing targets (sciradio-telescope elevation gain curve, with a plateau aroundence targets and calibrators) at 228.1 GHz (1.3 mm) during the40-60◦ .
EHT+ALMA observing run in April 2017.

Source
name
3C84
3C273
3C279
3C345
0510+180
BLLAC
CENA
M87
OJ287
SGRA
Figure 11: The best-fit second order polynomial to the relative
gains of the two calibrator sources, K3-50 and W75N is used as theelevation-dependent gain curve.

9

Mean flux
density [Jy]
11.4
7.2
8.5
2.3
2.1
1.7
8.5
2.1
3.9
9.0

± Statisticaluncertainty [Jy]
0.3
0.2
0.2
0.1
0.3
0.1
0.3
0.3
0.3
0.6

SMT 1.3 mm Measurements

During the EHT+ALMA observing run in April 2017, flux
density measurements were obtained for a number of EHT
The best fit parameters of the elevation-dependent gain curve science targets and calibrator sources, by averaging multiple(of the form a2 (el)2 + a1 (el) + a0 ) are shown in Table 3 below7 . 2.7 min continuum ON-OFF scans in single-dish mode over
These are consistent within the error margins with the coeffi- the EHT observing window (April 4-12, 2017).
cients of the gain curve for 2016 determined in Memo 2. This
result is reassuring, as no work has been done on the teleThe cadence of single-dish observations was not constantscope in the past year and so the gain curve was not expected or identical between all the sources: some sources were obto change.
served more frequently due to their availability during SMT
7 More digits and the covariance matrix for the fit parameters can be obtained upon request.

14

off-time from the VLBI schedules. Table 4 below presents
the mean flux densities for these observed sources during
the EHT + ALMA observing run in April 2017. Not all
EHT sources are present, some were not visible for longenough during the run to observe them separately with the
SMT alone.

10

The gain curve was also determined once more in 2017. The
resulting fit parameters are consistent, within error estimates,with the curve measured in 2016 (Memo 2). This is an expected result as no work has been done on the telescope since,
and it confirms the robustness of the method used to determine the gain curve.

Conclusion

The April 2017 run gave the opportunity to re-evaluate every step of the calibration procedure for amplitude scaling
for the SMT. A more detailed understanding of the calibration procedure for the telescope was obtained from previous
memos, and enabled a more critical evaluation of the strategy
currently used. Some limitations were found concerning thetelescope DPFU and aperture efficiency: one of the key results of this memo is an updated estimate for the beam-widthof 34.8 ± 0.6 arcseconds at 228 GHz, for a more accurate
coupling of the planet calibrators to the telescope beam thanin previous years. Furthermore, a more realistic estimation of
the aperture efficiency was obtained, no differences in DPFU
were found between the two polarizations, and measurements
were quite consistent over the years, with no significant timedependence detected.

The complete calibration procedure, when taking into accountelevation-dependent gain from the gain curve and the newly
determined DPFU, yields all the necessary calibration information for adequate antenna-based scaling of the visibility
amplitudes from the SMT for EHT observations. For example, the calibration information is currently consistent with
the necessary scaling done by Dr. Rusen Lu and Dr. Thomas
Krichbaum for the amplitude calibration of the SMT baselines in EHT 2013 data.

There remains, however, a number of limitations in the calibration procedure that should be investigated further in the
future. The beam-width estimation, although adequate considering the uncertainty in pointing and focus dominates the
DPFU analysis, is still not quite accurate. We have assumed
that the SMT has a circular beam, but from the Gaussian fits
it appears to be elliptical instead. A more accurate fit of theJupiter maps, such as fitting both beams simultaneously or
fitting a convolution of the planet disk and a Gaussian insteadof a simple Gaussian would yield better results. Anotherlimitation in the calibration is the chopper procedure: the assumption that the vane temperature is the same as the ambient
temperature is likely incorrect, as the vane was moved fromA test of the signal loss in the VLBI backend was also donein 2017, to investigate the hypothesis that a possible discrep- the receiver cabin (which is at around ambient temperature)ancy between the single-dish calibration information and the to a room in the telescope building (which is closer to room
scaling needed for EHT data was due to attenuation of signal temperature). Some additional assumptions, such as whether
in the VLBI equipment. It was found, using the Y-factor test the sideband ratio is indeed negligible, must also be lookedat more carefully. The chopper technique itself also has its
at the SMT total-power box, at the input to the R2DBE andown limitations outside of telescope-specific assumptions.
within the R2DBE itself, that no signal loss was occuring
and the Y-factor results were consistent with each other. It is Nevertheless, the current method should yield suitable results
therefore safe to assume that the single-dish calibration proce- for the calibration of EHT observations.
dure should be suitable to calibrate SMT visibility amplitudes
in EHT data in its entirety.
This memo also described the full procedure for the systemtemperature estimation via the chopper technique, using exact
equations from the telescope software written by Tom Folkers. We confirmed that the output of the chopper technique is∗the effective system temperature T sys, which already corrects
for atmospheric attenuation of the signal and rearward losses.
∗
The T sysmeasurements are done before each VLBI scan, but
the SMT additionally records total-power data that can beused to track intra-scan system temperature trends. This totalpower data is available on the EHT wiki as part of the SMT
logs for April 2017 observing.

Acknowledgements
This work is supported by the ERC Synergy Grant BlackHoleCam: Imaging the Event Horizon of Black Holes (Grant
610058). D.P.M., J.K. and T.W.F. receive support via NSF
MSIP award AST-1440254.

15

Useful References

Kraus, J.D., Radio Astronomy, 1986, Cygnus-Quasar Books,
Powell OH.

Altenhoff, W.J., The Solar System: (Sub)mm Continuum
Observations, Proceedings of the ESO-IRAM-Onsala Workshop on (Sub)Millimeter Astronomy, 1996.
Bach, U., Seminar: Pointing and Single Dish Amplitude Calibration Theory, Max Planck Institut für Radioastronomie.
Baars, J.W.M., The Measurement of Large Antennas with
Cosmic Radio Sources, IEEE Transactions on Antennas andPropagation, Vol. AP-21, No. 4, July 1973.
Baars, J.W.M., The paraboloidal reflector antenna in radio
astronomy, Springer, 2007.

Iguchi, S., Radio Interferometer Sensitivities for Three Typesof Receiving Systems: DSB, SSB and 2SB Systems, PASJ 57,
643-677, August 2005.
Issaoun .S, Folkers T.W., Blackburn L., et al. A conceptual
overview of single-dish absolute amplitude calibration, EHT
Memo 2017-CE-01, 2017.
Mangum, J.G., Equipment and Calibration Status for the
NRAO 12 Meter Telescope, National Radio Astronomy Observatory, September 1999.
Mangum, J.G., Main-Beam Efficiency Measurements of theCaltech Submillimeter Observatory, Publ. of the Astron.
Soc. of the Pacific 105, 116-122, January 1993.

Bensch F., Stutzki S., Heithausen A., Methods and constraints for the correction of error beam pick-up in single
dish radio observations, A&A 365, 285-293 (2001).

Martı́-Vidal I., Krichbaum T.P., Marscher A. et al. On thecalibration of full-polarization 86 GHz global VLBI observations, A&A, 542, A107 (2012).

Berdahl P., Fromberg R., The Thermal Radiance of Clear
Skies, Solar Energy, Vol. 29, No. 4, pp. 299-314, 1982.

O’Neil, K., Single Dish Calibration Techniques at Radio
Wavelengths, NAIC/NRAO School on Single Dish Radio
Astronomy, ASP Conference Series, 2001, Salter, et al.

Berdahl P., Martin M., Technical Note: Emissivity of clear
skies, Solar Energy, Vol. 32, No. 5, pp. 663-664, 1984.
Berger X., Buriot D., Garnier F., About the Equivalent Radiative Temperature for Clear skies, Solar Energy, Vol. 32,
No. 6, pp. 725-733, 1984.

Pardo J.R., Cernicharo J., Serabyn E., Atmospheric Transmission at Microwaves (ATM): An Improved Model for
Millimeter/Submillimeter Applications, IEEE Transactions
on Antennas and Propagation, Vol. 49, No. 12, December
2001.

Burke B.F., Graham-Smith F., An Introduction to Radio
Astronomy, 1997, Cambridge University Press.

Rohlfs, K., Tools of Radio Astronomy, 1986, SpringerVerlag: Berlin, Heidelberg.

Cappellen, W. van, Efficiency and sensitivity definitions forreflector antennas in radio-astronomy, ASTRON, SKADSMCCT Workshop, 26-30 November 2007.

Ruze, J., The Effect of Aperture Errors on the Antenna Radiation Pattern, Il Nuovo Cimento Volume 9, Supplement 3,
pp 364-380, March 1952.

Gordon M.A., Baars J.W.M., Cocke W.J., Observations ofradio lines from unresolved sources: telescope coupling,
Doppler effects, and cosmological corrections, A&A 264,
337-344 (1992).

Sandell, G., Secondary calibrators at submillimetre wavelengths, Mon. Not. R. Astron. Soc. 271, 75-80 (1994).

Greve A., Bremer M., Thermal Design and Thermal Behavior of Radio Telescopes and their Enclosures, Chapter 4,
Section 7, Springer, 2010.

Thompson A.R., Moran J.M., Swenson G.W., Interferometry and Synthesis in Radio Astronomy, 1986, John Wiley andSons; New York.

Kraus, A., Calibration of Single-Dish Telescopes, Max
Planck Institut für Radioastronomie, ERATec-Workshop- Bologna, 28 October 2013.

Tournaire, A., Controle thermique du telescope et du batiment, Themis, INSU/TECH/AT/CT/No86/8, France (1986).
P.R. Bevinson and D.K. Robinson.Data Reduction and Error Analysis. McGraw Hill, 2003.

Kramer, C., Calibration of spectral line data at the IRAM
30m radio telescope, Version 2.1, IRAM, January 24th
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Kramer, C., Millimeter Calibration, presentation at IRAM
Summer School 2013, IRAM, Granada, Spain.
16

A

Variables in LinuxPops

In this memo, the variables for the LinuxPops and telescope calibration techniques were introduced. However, in the programs, some variables are called differently. Below is a table with the different variables introduced and their respective
LinuxPops or atm.c (telescope calibration script) symbol.
Table 5: Calibration variables in LinuxPops and atm.c

variable nameelevation
hot load counts
blank sky counts (for system temp)
cold load counts
vane (hot load) temperaturecold load temperature
receiver noise temperature
ambient temperature
atmospheric emission temperature
atmospheric emission temperature in image band
atmospheric emission temperature in signal band
cabin temperature
forward efficiency
sideband ratio
counts on-source
counts off-source (for observing)
signal-to-noise ratio
calibration temperatureeffective system noise temperature
opacity corrected antenna temperature
sky opacity at the zenith
sky opacity at the zenith in signal band
atmospheric attenuation factor
elevation gain correction factor
efficiency correction factor
calibrated antenna (resistor) temperature
flux density

symbol in calibration procedure
El
Chot
Csky
Ccold
T vaneT cold
T rx
T amb
T em
T em,iT em,s
T cab
ηeff
rsb
Con
Csky

symbol in LinuxPops or atm.c
El
Thot
Tsky
Tcold
Tvane
TCOLD
Trx
Tamb
Tem
Temi
Tems
Tcab
be eff
sbr
Sig
Sky

S −N
N

S ig−S ky
S ky

T cal
∗
T sys
T A∗
τ0
τ0,s

Tcal
Tsys
Ta*
Tau
taus
TauEL
GainELFacTr
Jy’seτ0 ∗AM
1
G
1
ηA

TR
S

17