Event Horizon Telescope
Memo Series

EHT Memo 2017-CE-02

Calibration & Error Analysis WG
A conceptual overview of single-dish absolute
amplitude calibration
S. Issaoun1 , T. W. Folkers2 , L. Blackburn3 , D. P. Marrone2 , T. Krichbaum4 , M. Janssen1 ,I. Martı́-Vidal5 and H. Falcke1
Sept 15, 2017 – Version 1.0
1

Department of Astrophysics/IMAPP, Radboud University Nijmegen, 6500 GL Nijmegen, the Netherlands
Arizona Radio Observatory, Steward Observatory, University of Arizona, AZ 85721 Tucson, USA
3
Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA
4
Max Planck Institut für Radioastronomie (MPIfR), Auf dem Hügel 69, 53121 Bonn, Germany
5
Onsala Space Observatory, Chalmers University of Technology, Observatorievägen 90, 43992 Onsala, Sweden
2

Abstract
This document presents an outline of common single-dish calibration techniques and key differences between centimeter-wave and millimeter-wave observatories in naming schemes and measured quantities. It
serves as a conceptual overview of the complete single-dish amplitude calibration procedure for the Event
Horizon Telescope, using the Submillimeter Telescope (SMT) as the model station.

Note: This document is not meant to be used as a general telescope guide or manual from an engineering
perspective. It contains a number of common approximations used at observatories as an attempt to reason
through the methods used and the specific calibration information needed to calibrate VLBI amplitudes from
Event Horizon Telescope observing runs. This document can be used in conjunction with similar calibration
outlines from other stations for procedural comparisons.

Contents
1

Introduction to standard single-dish T sys calibration techniques
4
1.1 The antenna-based system-equivalent flux densities (SEFDs) . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.2 The receiver noise temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.2.1 Two-load (hot and cold) calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3 The effective system noise temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.3.1 Clarification of system temperature jargon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.3.2 Direct (switched noise diode) method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.3.3 Chopper (or single load) calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.4 Getting a flux density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.1 Determining a gain curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4.2 Determining the DPFU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.4.3 Other efficiencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10

2

Miscellaneous explanations
2.1 VLBI and the Event Horizon Telescope array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Determining the antenna-based SEFD for VLBI . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 A brief overview of a priori amplitude calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Double-sideband (DSB) receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Telescopes not using the chopper technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
∗2.3 T sys or T sys? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

11
11
11
11
11
12
12

Relevant terminology

Efficiency and correction terms:• rsb : sideband ratio - since the SMT has a sidebandseparating receiver, rsb = ggsi  1 since no signal
comes from the image 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 thereceiver (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)• e−τ : atmospheric attenuation factor, which damps thesignal based on atmospheric opacity in the line ofsight τ = τ0 × AM

• Con : Counts measured observing a target• el: elevation of the antenna dish for a particular observation (in degrees)• Csky : Counts measured when looking at blank sky• g(el): elevation-dependent gain curve correcting
for changing illumination of the main reflector andground contributions as the dish moves and tilts todifferent elevations• T cold : temperature of the cold load

• T rx : receiver noise temperature• ηl : forward efficiency representing the fraction of
power received through the forward atmosphere (accounting for rearward losses)• T amb : ambient temperature around the observatory, as
measured by a weather station (physical temperature)• ηtaper : efficiency loss due to non-uniform illuminationof the aperture plane by the tapered radiation pattern• T sky : temperature of the atmospheric emission (thebrightness temperature of the sky)• ηblock : aperture blockage efficiency due to blockingof the feed by the sub-reflector (including its supportlegs)• T cab : physical temperature of the receiver cabin (thisis assumed to be the same as the ambient temperature)• ηspillover : feed spillover efficiency past the main reflector − it is the ratio of the power intercepted by thereflective elements to the total power• T cal : derived temperature to give a correct temperaturescale for the signal band

• ηRuze : surface error efficiency (or Ruze loss) calculated from Ruze’s formula (Ruze 1952)• T inject : injected known temperature (of calibrator ornoise diode) in the signal chain• 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 for• Aeff : effective area of the SMT dish ( = ηA Ageom )atmospheric attenuation)
3

1

Introduction to standard singledish T sys calibration techniques

SEFD:
SEFD =T sys eτηl G

(1)

The following is an outline of the different calibration pro- The flux density of a source detected with a given ratio of
cedures for cm-wave and mm-wave observatories and thesignal to noise power rS/N is then:different quantities they output. The equations provided
rS/N × T sys eτhere contain various approximations commonly used but are
S source = SEFD × rS/N =(2)
ηl G
not exact from an engineering perspective. They are only
meant to serve as guidelines for a quick understanding ofFor mm-observatories, which measure the effective system
the outputs of the two different techniques.
τ∗
noise temperature T sys= T sys eηl directly using the chopper
technique (explained in the next section), the SEFD equation can be rewritten in only two components, the effective
1.1 The antenna-based system-equivalent flux system noise temperature and the antenna gain:

densities (SEFDs)
SEFD =

∗
T sys

(3)
G
A telescope’s system-equivalent flux density (SEFD) is sim- For the SMT, the SEFD at zenith is of order 13 000 Jansky.
ply the noise contribution of the system, given by the system
noise temperature, and all losses and gains, converted toa flux density scale. The SEFDs can be calculated using
1.2 The receiver noise temperaturesystem noise temperature T sys measurements and all efficiencies and contributions to source attenuation and noise,and one can determine the sensitivity of the telescope when 1.2.1 Two-load (hot and cold) calibrationcompared to other telescopes in the array. The higher atelescope’s SEFD, the lower its sensitivity. Ultimately, theDuring a two-load calibration (also called cold calibration),
flux density of a source is simply the telescope’s SEFD,
which contains all system and telescope parameters and ef- the Y-factor and the receiver noise temperature are measured using voltage or counts measurements with a hot andficiencies, multiplied by the ratio of signal to noise power(defined as rS/N ) of the source detection. The equation for the a cold load. In principle the receiver noise temperature can
SEFD can be subdivided into three main components, each be estimated from T sys measurements at very low opacities(τ  1) by extrapolating a linear fit of airmass versus T sys
with station-based variations for how they are determined
to zero airmass. However, it is highly recommended to meaand measured. The three components to the SEFD are:sure a receiver noise temperature at least once an observing
night, as this yields more accurate T sys measurements rather
than backtracking in post-processing. The Y-factor is calcu1. T sys : the total noise characterization of the system,
lated with the following:
given by the system noise temperature
Chot
τ
Y=
,
(4)
2. eηl : the correction terms for attenuation of the source
Ccold
signal by the atmosphere and rearward losses (ohmic
where the numerator is Chot , the counts obtained from thelosses, rearward spillover and scattering) of the telehot load and the denominator is Ccold , the counts obtained
scopefrom the cold load. The Y-factor also enables an easy diagnostic of the sensitivity of the receiver. A high Y-factor
3. G: The antenna gain, including all the loss terms frommeans little receiver noise, and thus sensitive observations
the telescope and the conversion from a temperature(of course what constitutes “high” depends on the type ofscale (K) to a flux density scale (Jansky), given by thereceiver and the observing frequency).
“degrees per flux density unit” factor (DPFU) in K/Jy
and the normalized elevation-dependent gain curve
Then the receiver noise temperature is determined as folg(el): G = DPFU × g(el)lows:
T hot − YT cold
T rx =(5)This gives the following general equation for a telescope’s
Y −14Here temperatures are used, where T hot is the temperature
1.3.2 Direct (switched noise diode) methodof the hot load (for the SMT this is near room temperature∼ 290K) and T cold is the cold load temperature (for the SMTThis method is commonly used at cm-observatories, such
this is the temperature of liquid nitrogen ∼ 77K).
as the VLBA. The system noise temperature is obtained using a known source or a switched noise diode with a known
temperature placed in the signal chain. The equation is the
1.3 The effective system noise temperature
following, where Csky represents the counts on blank sky,
so only receiver noise and sky contribute, and Con,cal repreA lot of confusion comes from mixtures of complicated cal- sents counts on the calibrator (or diode), such that the signal
ibration documents for different types of calibrations. This contains the source, the receiver and the sky. T inject is thetemperature of the diode or the brightness temperature ofsection is an attempt to approximately explain two of thecommon techniques for T sys measurements (chopper wheel the source (known for common calibrators), which turns thecommon for mm-telescopes, direct for cm-telescopes), what counts scale to a temperature scale. When the telescope ispointed at blank sky in the calibration procedure, without
they output and what they mean for data processing. The
following outlines are modelled for an SMT-like telescope, the source signal, the temperature contribution is entirely
noise from the receiver and the atmospheric emission, andthus with a sideband-separating (SSB) receiver.
thus is the system noise temperature T sys :1.3.1

T off,cal = T rx + T sky = T rx + T atm (1 − ηl e−τ )Clarification of system temperature jargon(9)

We define the system noise temperature as the contributions When the telescope is pointed at the calibrator (or diode) ofby the receiver and the sky to a source measurement (assum- a known brightness (or physical) temperature, the source
signal is added to the temperature contribution:
ing T CMB is negligible), where ηl is the forward efficiency,
accounting for rearward efficiency loss due to ohmic losses,
T on,cal = T rx + T sky + T inject(10)
rear spillover and scattering:
T sys = T rx + T sky = T rx + T atm (1 − ηl e−τ )(6) The system temperature is then determined in the following
way:

Before entering the atmosphere, the source signal is definedas Sig = T source . After attenuation by the atmosphere, thesignal becomes Sig= ηl e−τ T source , where the exponential isthe atmospheric attenuation factor (τ is the opacity in theline of sight) and ηl embodies rearward efficiency losses.
Therefore, the ratio of signal to noise power of a telescopemust depend on this received signal (not taking into accountground and ambient contributions):rS/N =

ηl e T source ηl T source
= τ
T sys
e T sysCsky
T inject
Con,cal − Csky
T off,cal
T inject=
T on,cal − T off,cal(T rx + T sky )=T inject(T rx + T sky + T inject ) − (T rx + T sky )(T rx + T sky )=T inject = T rx + T sky
T injectT sys =

−τ

(11)

(12)

(7)

Since the brightness temperature of the source observed(or the diode temperature) is determined outside the atmoWe thus define the effective system noise temperature
sphere, the system noise temperature calculated with this
∗
T sys
:
method does not include effects on sensitivity due to atmoτττ
spheric attenuation (eτ term). This is because the contribue
e
e
∗
T sys
= T sys = T rx + T atm ( − 1)(8) tion of the source or diode is added to the signal chain (asηlηlηlopposed to the chopper technique that blocks everything but
the receiver noise, explained and derived in the next section).
The effective system temperature is the best description
This method does not provide an effective system temperaof the sensitivity of a telescope: the system sensitivity
ture directly, only the receiver and sky contributions to thedrops rapidly (exponentially) as opacity increases.
noise (which cannot be disentangled from each other).
5In order to obtain the effective system temperature, opacity measurements during observations must be obtained.
This is done either by using water vapor radiometers (ortipping radiometers) or by using the telescope as a tipper
using sky tips. Tipping radiometers are notoriously unreliable (although water vapor radiometers perform very well),and sky tips must be done very often (every 10 min) andtake up valuable observing time. This makes this method
highly cumbersome for frequencies at which the atmospherecannot be neglected.

counts measured when the blocker/chopper/vane is in place,and Csky is our usual blank sky counts. In terms of temperatures, the temperature contribution when the blocker is inplace T block is defined as:
T block = T rx + T hot ,where T hot is the temperature of the hot load itself. The
load completely blocks the sky emission, which changes
the calibration equations from the direct (or diode) calibration method. As seen in the direct method, the blank sky
contribution is simply the system noise temperature:

The effective system temperature is thus:
e
∗
T sys
=

τ

(16)

T off = T rx + T sky = T rx + T atm (1 − ηl e−τ )(17)

T sys ,

(13) We can thus write the chopper equation (eq. 15) in terms oftemperatures:such that the effective (opacity-corrected) antenna temperaCsky
T off∗ture of a source (where Con is the telescope signal on target)
T sys= T hot= T hot
(18)
C
−
C
T
− T off
hot
sky
block
can be given using ON-OFF measurements as:
T rx + T sky= T hot
(19)
τ
C
−
C
T
−
T
e
on
sky ∗
on
off ∗
(T rx + T hot ) − (T rx + T sky )T A∗ =T sys =T sys =T A (14)
Csky
T off
ηl
ηl

1.3.3

We assume the hot load is at ambient temperature, and so
T hot = T amb = T atm . This gives:

Chopper (or single load) calibration

T rx + T sky(T rx + T hot ) − (T rx + T sky )T rx + T sky= T atm(T rx + T atm ) − (T rx + T sky )∗T sys= T hot

The chopper (or single load) calibration technique is commonly used by (sub)mm observatories. The system noise
temperature is obtained by placing an ambient temperatureload T hot that has properties similar to a blackbody in frontof the receiver, blocking everything but the receiver noise.
As long as T atm ∼ T hot , this method automatically compensates for rapid changes in mean atmospheric absorption.

(21)

As we have defined T sky = T atm (1 − ηl e−τ ), we can simplify:
T rx + T sky(T rx + T atm ) − (T rx + T sky )T rx + T atm (1 − ηl e−τ )= T atm(T rx + T atm ) − (T rx + T atm (1 − ηl e−τ ))T rx + T atm (1 − ηl e−τ )= T atm
T atm − T atm + ηl e−τ T atm
T rx + T atm (1 − ηl e−τ )= T atmηl e−τ T atm
T rx + T atm (1 − ηl e−τ )=ηl e−τ

∗T sys= T atm

For calibration of source measurements, we want to obtain
the effective sensitivity of the system, not a comparison between the receiver and sky contributions to noise. Therefore,
we want to obtain the effective system noise temperature∗
T sys
to calibrate source measurements.
∗
To first order, the chopper method directly measures T sys.
This is obtained via the following equation:

Csky
eτ
eτ
∗
T sys
= T hot= T rx + T atm ( − 1),
Chot − Csky
ηl
ηl

(20)

(22)
(23)
(24)
(25)
(26)

(15)
Finally we obtain:

where Csky is the voltage/count signal on blank sky and τ isthe opacity in the line of sight.

Csky
T rx + T atm (1 − ηl e−τ )=Chot − Csky
e−τ
τ
τ
τ
e
e
e
= T rx + T atm ( − 1) = (T rx + T sky )
ηl
ηl
ηl

∗
T sys
= T hot∗
How does the chopper technique directly provide T sys?
This is shown simply by investigating the exact output by
the chopper technique. The chopper system temperatureequation is given in telescope counts, where Chot are the

6

(27)
(28)

If we compare the chopper effective system noise temperature to the system temperature from the direct method:
∗
T sys
=

T sys
ηl e−τ

1. The ratio of signal to noise power of the observed
source as measured by the telescope (thus attenuatedby the atmosphere):
offrS/N = TonT−T
off

(29)

2. The total noise characterization of the system, including the correction term for atmospheric absorption,
given by the effective system noise temperature:

To first order, the chopper calibration (or alternatively
named the single-load calibration) corrects for atmosphericattenuation of an observed source and rearward losses of the∗telescope by directly measuring T sys. It is also worth notingthat during VLBI observing, the quarter wave-plate is added
to the signal chain to convert linear to circular polarization:
any losses associated with the addition of the wave-plate∗
will be automatically calibrated and included in the T sys
measurement from the chopper technique in the same way
as the atmospheric and rearward losses.

1.4

τ

∗
T sys
= eηl T sys

3. The antenna gain G, including all the loss terms fromthe telescope and the conversion from a temperaturescale (K) to a flux density scale (Jansky), given by the“degrees per flux density unit” factor (DPFU) and thegain curve:
ηA Ageom
DPFU =
giving G = DPFU × g(el)2kGetting a flux density

We can thus simplify the flux density equation using thethree main terms actually measured by the SMT:

We have defined the antenna temperature (modified for measured quantities at a telescope) as:
T A∗ =Con − Csky ∗
T sys
Csky

S =

∗
rS/N × T sys eτ rS/N × T sys
=
ηlG
G

(35)

(30)
1.4.1

To get a flux density, we must correct for the aperture efficiency ηA (determined through different loss terms or planet
flux measurements) and gain curve g(el) as a function of elevation of the telescope and convert from a temperature scale
to a flux density scale (where k is the Boltzmann constant),
dependent on the geometric area of the dish Ageom :Determining a gain curve

As previously mentioned, the characterization of the antenna gain G is subdivided into two quantities that must
be separately provided for the calculation of the SEFDs:
the gain curve g(el) and the DPFU (explained in the next
section). The characterization of the telescope’s geometric(opacity-free) gain curve is an important part of the flux
T A∗
2k
S =
(31) density calibration, and is particularly crucial for the EHT a
ηA g(el) Ageom
priori amplitude calibration due to the low-elevation observations of some science targets (including Sgr A*) for the
The equation above is then the final expression to obtain anorthern hemisphere stations.
flux density for a given source. If we expand using all theterms we’ve discussed, we get the following:
Telescopes do not have perfect surfaces, and must thus suffer some losses of signal due to distorted illumination of the∗TA
2k
1 Con − Csky ∗ 2k
S =
=
T sys
(32) main reflector as they slowly move to different elevations.
ηA g(el) Ageom ηA g(el) Csky
Ageom
This large-scale surface deformation affects the received
1 Con − Csky T sys 2k
signal and is not taken into account in the measurements=(33)
leading to the efficiency and DPFU characterization. TheseηA g(el) Csky
ηl e−τ Ageom
losses can be determined by tracking sources through a
T sys
T on − T off
2k=(34) wide range of elevations, and thus measure an elevation−τ
T offηA g(el)ηl e Ageom
dependent gain curve for the telescope, where the maximum
Now the flux density is rewritten also in terms of a system(g = 1) is set where the telescope is expected to be mostnoise temperature determined with the direct method.
efficient. The source measurements used to obtain a gain
curve must of course be calibrated for all other effects, inWe can subdivide the flux density equation into three major cluding telescope efficiency (through the DPFU) and the∗parts:
atmospheric attenuation of the signal (through T sys). At the7

SMT, this is done by observing two sources (usually K3-50
and W75N, a planetary nebula and a star-forming region,
due to their similar up-time plots and wide range of elevation) contiguously, tracked as they increase and decrease inelevation from the tree-line to transit and vice-versa.

measurements to a flux density scale and is obtained using known flux calibrators, particularly planets, or by bootstrapping near-in-time observations of non-planet sourcesfrom telescopes with well-defined and accurate flux density
measurements. This enables to check the flux density scaleobtained by the telescope by directly measuring an aperture
efficiency.

The gain curve is estimated by fitting a polynomial (usuallysecond-order for standard radio-dishes). If more than one
source is used, this is done once the flux density measureThe DPFU is estimated with the following equation, wherements are normalized around a plateau (to a relative gain
k = 1.38 × 10−23 J/K = 1.38 × 103 Jy/K:
scale). This normalized gain curve must be written in theform of a second order polynomial (in the standard VLBA
format for ANTAB), where ‘el’ is the elevation in degrees:ηA Ageom[K/Jy](37)
DPFU =
2kg(el) = a2 (el)2 + a1 (el) + a0(36) The geometric area Ageom is simply the area of the dish,where D is the dish diameter:

Each parameter must not be rounded to the uncertainties ofthe fit but instead many significant figures should be provided. Uncertainties for each parameter as outputted by thepolynomial fit must also be provided, along with the fullcovariance matrix of the fit parameters. This will help determine an error estimate for the gain curve and propagate to
the error estimation of the final SEFDs. Additionally, a plotof the relative gains (normalized fluxes) versus elevationand the fitted polynomial should be provided if possible, as
shown in Fig. 1.

Ageom =

πD2
4

(38)

The aperture efficiency is the most difficult part of the estimation of the DPFU. It represents the efficiency of thetelescope compared to a telescope with a perfect collecting
area (uniform illumination, no blockage or surface errors)
and it is determined using observations of known calibrator
sources, usually planets. The observed planet fluxes are thencompared to expected planet brightness temperatures froma planet simulation software for a perfect telescope at thegiven frequency and beam width.
The aperture efficiency ηA is found using the following
equation, where T A∗ is the observed effective antenna temperature, g(el) is the telescope gain curve, k is the Boltzmann
constant, Ageom is the geometric area of the telescope andS beam,sim is the expected flux density of the planet in thetelescope beam from the simulation program used:ηA =Figure 1: Example of a normalized geometric gain curve plot.

T A∗
Ageom g(el) S beam,sim
2k

(39)

Or similarly the DPFU is directly given by:
1.4.2

Determining the DPFU

DPFU =The degrees per flux density unit (or DPFU) is the characterization of the temperature to flux density scale of atelescope. The DPFU is used to calibrate the telescope

T A∗
g(el) S beam,sim(40)

For extended sources, it is important to calibrate the flux
density observed in the beam because some emission might8not be picked up by the telescope. The aperture efficiency
can be approximated using the small angle approximation,is only concerned by the main beam flux density, and so the where θs is the apparent diameter in radians of the planet
following equation is used to calibrate the simulated flux
observed:
density in the beam for an extended source, where S sim isπθ2
the expected total flux density of the source:Ωs ' s(47)
4
S beam,sim = S sim × K(41)
Of course this process heavily depends on assumptions
made in the planet calibration, such as accurate predicted
planet brightness temperatures from available software, telescope beam width used, stable weather conditions and awell-calibrated instrument in terms of pointing and focus.

Here K is the following, where θmb is the half-power beamwidth in arcseconds of the primary lobe of the telescope
beam pattern (telescope beam diameter) and θs is the diameter in arcseconds of the observed extended source, usually
given by the simulation program:
x=

θs p
ln(2)
θmb

K=

1 − e−xx2

2

An average value for the aperture efficiency can be esti(42) mated from the individual measurements during a particular observing run, but it is preferable to keep the timedependence of the variable if a telescope’s efficiency is
expected to vary with temperature and sunlight, causing(43) systematic differences in the telescope performance between
day-time and night-time observing.

This 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 purposes1 .
Z
Ωsum
1
K=
=
P(θ − θ0 , φ − φ0 )Ψ(θ0 , φ0 )dΩ0 (44)ΩsΩs source

1
K=
Ωs

Z

P(θ − θ0 , φ − φ0 )dΩ0

Even more preferable, a plot of long-term trends of the aperture efficiency, using additional measurements outside EHT
observing or even from previous years, would greatly help
understand the time-dependent nature of the aperture efficiency of a particular telescope. As the scatter between
individual measurements can be caused by various factors,
such as unstable weather or changing pointing/focus accuracy, it is not always representative of the true aperture
efficiency change in the observations. A trend exhibited in
the long-term as a function of time would be more reliable
to estimate an aperture efficiency for a particular scan. Sucha plot is shown in Fig. 2, as an example from the JCMT.

(45)

sourceTo minimize the number of approximations used by different planet simulation softwares, the expected total flux
density can be estimated by:
S sim =

2h ν3 Ωs,c2 e kThνB − 1(46)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 temperature) from the simulation program, andΩs is the solid angle of the source on the sky in steradians. Since we are dealing with very small objects, the latter

Figure 2: Example of long-term trend for the time-dependent aperture efficiency ηA .

1 More detail on this method in Calibration of spectral line data at the IRAM 30m radio telescope by C. Kramer.

9If a UT time-dependence is found for a particular station,a fit for this dependence must be provided, as well as thecovariance matrix for the fit parameters, for error analysisof the a priori deliverables. A fit to the UT time-dependence
would be the most robust against various observing effects
from day to day and session to session and should be very
stable over the years, provided no major work has been
done on the telescope. For telescopes with no visible timedependence, a mean aperture efficiency (or DPFU) will
suffice, with the appropriate error estimate.

In summary, the aperture efficiency accounts for all forward
losses of the telescope, which come from different contributions. As previously mentioned in section 1.3.3, the chopper
technique itself account for the rearward losses of the telescope automatically. These losses are also outlined in thefollowing section.

1.4.3

Other efficiencies

The main beam efficiency of a telescope is the fraction ofobserved power in the main lobe of the telescope beam pattern. Let the beam solid angle (the full antenna pattern) beΩA and the main beam solid angle (the main lobe) be Ωmb .
Themain beam efficiency is written as the ratio between the(48)
total beam and main beam solid angles:

We can also write the aperture efficiency as the combination of various individual forward efficiencies, each closely
approximated for the telescope via various measurements:
ηA = ηtaper × ηblock × ηspillover × ηRuzeηmb =Each efficiency term corresponds to an aspect of the telescope feed 2 :

Ωmb
ΩA

(50)

It is estimated with the following, where T mb is the main
beam temperature of a source that fills the main beam, as
estimated from the simulation program:• 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 theillumination efficiency). It is the most important contributor to the aperture efficiency.

ηmb =S beam,sim ηA Ageom
T mb
2k

(51)

• ηblock is the aperture blockage efficiency due to block- It should be noted that the main beam efficiency is noting of the feed by the sub-reflector (including its sup- the same as the aperture efficiency and should not be
port legs)
used to determine telescope DPFUs and SEFDs.
• 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 subreflector and primary to the total power. It is due
partly to cold sky and partly to a warm background,and is elevation-dependent.
• ηRuze is the surface error efficiency (also called “Ruzeloss” or scattering efficiency) calculated from Ruze’s
formula (Ruze 1952). It is due to small scale, randomly distributed deviations of the reflector fromthe perfect paraboloidal shape. Ruze’s formula ispresented below, where σ is the surface rms (accounting for small-scale deviations from a perfect surface
through dish holography) and λ is the observing wavelength:

ηRuze (λ) = e−16π2 σ2
λ2

The forward efficiency ηl represents the fraction of powerreceived through the forward atmosphere (in other terms itis the coupling of the receiver to the cold sky) and is written
as the ratio between the solid angle over the forward hemisphere of the telescope and the beam solid angle and it istypically close to unity (but drops with frequency due to lossof receiver sensitivity):
ηl =

Ω2π
ΩA

(52)

The only way to estimate it is via sky-dips, by measuring
the atmospheric emission with elevation:3
T A (el) = ηl T atm (1 − e−τ/ sin(El) ) + (1 − ηl )T amb(53)

It is important to note that sky-dips measure both the atmospheric opacity and the forward efficiency so they need to bedisentangled. Fortunately, this is not an issue for the EHT
because the chopper technique implicitly corrects for the(49) forward efficiency ηl (see Section 1.3.3).

2 More detail on the measurement of the different losses, see Baars, J., The paraboloidal reflector antenna in radio astronomy, Springer, 2007.
3 Overview in: Kramer, C., Millimeter Calibration, presentation at IRAM Summer School 2013, IRAM, Granada, Spain.

10

2

Miscellaneous explanations

2.1

VLBI and the Event Horizon Telescopearray

to a signal plane above the atmosphere, then the resulting visibility amplitudes will be corrected for atmosphericlosses.

2.1.3
2.1.1

Double-sideband (DSB) receivers

Determining the antenna-based SEFD for VLBI

It is worth noting that the equations presented in the previous sections for amplitude calibration are modeled after the
The SEFD needed for calibration of single-dish on-off obserSMT, which has a sideband-separating receiver. However,vations and that for VLBI are identical. The equation for thea few stations in the Event Horizon Telescope array haveantenna-based SEFD for VLBI observations is thus:
double-sideband (DSB) receivers, which lead to some modifications of the equations for amplitude calibration. The
most relevant difference between SSB and DSB receiver is
∗
∗
T sys
T sys
the handling of measured signals. For an SSB receiver, all
SEFD ==(54) the measured signal comes from only one sideband, but forG
DPFU × g(el)a DSB receiver it comes from two sidebands folded together
into one single larger band, usually used for spectral-line
It is important to note that the SEFD contains corrections
observing. However, for continuum VLBI with the EHT,
for system noise, atmospheric absorption, antenna gain
only one sideband of the DSB receiver systems is used as
terms and temperature-to-Jansky conversion.
the signal sideband and gets correlated, but the rest of thetelescope continues to operate as a DSB system. Therefore,
the sensitivity of the measurements during EHT observing
2.1.2 A brief overview of a priori amplitude calibration (through one sideband) is about a factor of two lower than
the normal operation of the telescope as a perfect DSB system. This rescaling of the telescope sensitivity from two
For VLBI observations, there are very few suitable calibra- sidebands to one is done by correcting T ∗ .
systors that do not become resolved on some baselines, thus wecannot use the primary calibrator scaling to calibrate VLBI For a measured effective system temperature from a perfectamplitudes. An alternative approach is to calibrate the VLBI DSB system T ∗sys,DSB , the actual effective system temperaamplitudes using the system temperatures and collecting
ture for VLBI observing with only one sideband is:areas of the individual antennas. The visibility amplitudes∗∗
can be calibrated in units of flux density by multiplying the
T sys= 2T sys,DSB(56)
normalized visibility amplitudes by the geometric mean of
For EHT observing we use half the number of sidebands,
the SEFDs of the two antennas concerned (TMS Section
thus the telescope sensitivity must drop by a factor of two,
10.1.). On a baseline between two telescopes, for example
the SMT and the LMT, which both use the chopper method, leading to the effective system temperature increasing by
the same factor. However, if the telescope does not have a
the amplitude calibration for the correlated source signal
perfect DSB system but one sideband has more gain thanrcorr,SMT−LMT (compensated for digitization and sampling
the other, then the equation becomes, more generally:
losses) on that baseline is given by:

S SMT−LMT =p
SEFDSMT × SEFDLMT rcorr,SMT−LMT ,

∗
∗
T sys
= (1 + rsb )T sys,DSB,(55)

where SEFDSMT and SEFDLMT are determined as shown
above and S SMT−LMT is then the source signal in Jansky onthat baseline.
Since the SEFDs for the telescopes are expected to include
the effective system noise temperature, which corresponds(57)

where the sideband ratio (rsb ) is the ratio of source signal
power in the remaining sideband to the signal power in the
sideband of interest (the sideband to be correlated). For aperfect DSB system, the gains of each sideband are equal,
giving rsb = 1, which gives back Eq. 56. For a perfect SSB
system, where all signal is in one sideband, rsb = 0 and this∗gives back simply T sys
needed for the EHT.
11

∗
Once this correction is applied to T sys, the rest of the amplitude calibration process remains the same. For planet scans
to determine the telescope’s DPFU, the signal is collectedby both sidebands in a DSB system, thus the effective antenna temperature is usually measured in DSB mode. Thisis sufficient to reflect the conversion from Kelvin to Jansky within the aperture efficiency and DPFU estimation. Itshould be noted that the correction from a DSB system to∗an SSB system for VLBI should only be done on T sys, otherwise the resulting SEFDs would be double-corrected for a
DSB system.

2.2

Telescopes not using the chopper techniqueat the observing frequency but instead is converted(sometimes not so accurately) from a different frequency
Another possible solution is to use the telescope itself as
a tipper: using the dish to observe blank sky through a big
elevation range in the direction of observing to determine
the relationship between elevation and sky temperature andget an estimate of the zenith opacity.
This tipping method solves the radiometer issues of getting an opacity in the direction of observing and at the rightobserving frequency. However, these tipping scans are required very frequently, every 10 minutes or so, and take up
valuable observing time just to get accurate opacities.
An alternative is then to obtain opacities using approximations in post-processing. The system noise temperature, as
measured in the direct method, is defined as seen previously.
For τ0  1, we can approximate:

As explained above, the result for telescopes like the SMT
and the LMT, which both use the chopper (or single-load)
technique, is very clean and simple. Now what happens
when there is a telescope in the array that does not use
the chopper technique but instead uses the direct (or noise
diode) method?4

T sys ≈ T rx + T atm (1 − e−τ ) ≈ T rx + T atm × τ0 × AM

In that case, on baselines with telescopes with the chopper method, there will be inconsistencies in the amplitude
calibration if the same corrections are applied in postprocessing to both stations on that particular baseline. This∗is precisely because the chopper technique gives T sys
andthe direct method only gives T sys .
Fortunately, as explained in the previous section, the rela∗tionship between the two is well-understood and T sys
caneasily be determined from the direct method using opacity
measurements. If the telescope has a tipping radiometer orwater vapor radiometer nearby measuring opacities, this can∗give a fairly good estimate for T sys= eτ T sys .

(58)

By fitting a least-squares (or as it is done for the GMVA,a linear fit to the lower envelope) of T sys as a function of
airmass, the extrapolation of the fit will give an approximation for the receiver noise temperature T rx . If the telescope
frequently does a dual-load (cold cal) calibration to refresh
values for the receiver temperature, these values are usuallymore accurate to use.
With this linear relationship (or measured T rx ) and every
variable but the sky opacity known, measured or approximated by the telescope, we can get the sky opacity at thezenith and thus correct the system noise temperature for theatmospheric attenuation:

However, some aspects of radiometers hinder this approach:• The radiometer does not always point in the same direction as the telescope, thus under a varying or partly
cloudy sky the opacities from the radiometer are notentirely accurate to the observations.

T sys − T rx
1
τ0 = −
ln 1 −
AM
T atm

!
(59)

∗
2.3 T sys or T sys?

• The radiometer can have something blocking and corrupting the measurements (as on Mt Graham due toA crucial part of the amplitude calibration process is to
the LBT)
determine which variables are actually provided by each
• The radiometer does not always measure an opacity
telescope in the context of the entire EHT array. Are all
4 As far as the EHT is concerned, there are no stations in the array at this time without the chopper technique. However, this information could be
potentially useful for the a priori calibration of GMVA or HSA observations related to EHT, which are a mixture of mm- and cm-observatories.

12telescopes providing T sys like cm-observatories do? Or are∗some telescopes providing in fact T sys
but labeling it as T sys(as is commonly done by mm-observatories)? Discrepancies in notation and a heavy background knowledge in thecontext of cm-observatories can cause misunderstandingsof the calibration information provided by the telescopes.
However, there is a nice way to do a quick check in postprocessing5 .
In order to visually understand the difference between thetwo variables, simulated measurements of system noise temFigure 3: Simulated system temperatures from the chopper andperatures for the SMT are presented. Using the standard
chopper equation, the calibration temperature was approxi- direct method calibration techniques show divergence as a functionof airmass.
mated to T cal = T amb = 280K, the receiver noise temperature
was set to T rx = 60K, and we have used a constant zenith
opacity of τ0 = 0.2, common for the SMT, for consistency.
∗
Figure 3 shows the effective system noise temperature T sys
using the chopper technique equation and the direct method∗
system noise temperature T sys ≈ e−τ T sys
as a function of airmass. It is clear that both temperatures indeed do vary with∗airmass, but T sys
is a lot more sensitive because in addition
it corrects for the increasing attenuation of a signal fromoutside the atmosphere, automatically determined with the
chopper technique.
It is a misconception to assume that because T sys does notcontain that term, it does not vary with airmass. T sys is inherently dependent on airmass because the sky brightness
temperature T sky , representing atmospheric noise, increases
with airmass as the telescope looks through a larger layer∗of atmosphere. This effect is also present in T sys, which, in
addition, corrects for the increasing signal attenuation that is
also elevation (airmass) dependent.
In the previous section, we introduced a useful tool to tell
the two system noise temperatures apart. For the case of
T sys , as shown by eq. 59, it is possible to untangle a zenith∗
opacity from T sys and T rx measurements. In the case of T sys,
because the opacity relationship is much more complicated,∗
it would not be valid. If we were to apply eq. 59 using T sys
instead of T sys , the opacities at zenith obtained would behighly inaccurate when compared to, for example, radiometer measurements during the same observing window.

Figure 4: Zenith opacities obtained by assuming the telescope provides T sys are much larger than what is actually measured by the
SMT tipper.

We use the opacity equation:
T sys − T rx
1
ln 1 −
τ0 = −
AM
T atm

!
(60)

Recall that at this point, what is plugged in as T sys is what is∗measured by the chopper technique (in fact it is T sys
but this
was still unknown).

The zenith opacities obtained from that equation were thencompared to the measured zenith opacities by the tipping
radiometer on the telescope scan-by-scan. These results
are presented in Fig.4. It is clear that the zenith opacities
obtained from “T sys ” are completely different, incrediblyThis reasoning was thus applied to the SMT to see if whathigh and inconsistent with the tipper measurements. Thiswas previously called “T sys ” was really T sys . For example,∗, which diverges
we can take the system noise temperatures and receiver tem- is of course because “T sys ” is in fact T sys
perature measured during the gain curve measurements for and is increasingly larger than T sys as a function of airmass.
2017. These were measured in the lapse of a few hours, thus Thus the zenith opacity equation does not work for what
is outputted by the chopper technique at the SMT, and thisminimizing opacity fluctuations due to changing weather.
output is definitely not simply T sys but something much
5 This check only works if the telescopes provide an elevation for each “Tsys ”, or alternatively these can be extracted from the VLBI Monitor database13∗more sensitive to opacity: T sys. For a telescope that is genuinely providing T sys , the opacity equation would give results much more in-line with the measured opacities from its
tipper/radiometer.

Kramer, C., Calibration of spectral line data at the IRAM
30m radio telescope, Version 2.1, IRAM, January 24th
1997.
Kramer, C., Millimeter Calibration, presentation at IRAM
Summer School 2013, IRAM, Granada, Spain.

Useful References

Kraus, J.D., Radio Astronomy, 1986, Cygnus-Quasar
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Altenhoff, W.J., The Solar System: (Sub)mm Continuum
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Iguchi, S, Radio Interferometer Sensitivities for Three
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Bach, U., Seminar: Pointing and Single Dish Amplitude
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Mangum, J.G., Equipment and Calibration Status for the
NRAO 12 Meter Telescope, National Radio Astronomy
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Baars, J.W.M., The Measurement of Large Antennas with
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Mangum, J.G., Main-Beam Efficiency Measurements of theCaltech Submillimeter Observatory, Publ. of the Astron.
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Baars, J.W.M., The paraboloidal reflector antenna in radio
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Bensch F., Stutzki S., Heithausen A., Methods and constraints for the correction of error beam pick-up in single
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Berdahl P., Fromberg R., The Thermal Radiance of Clear
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Pardo J.R., Cernicharo J., Serabyn E., Atmospheric Transmission at Microwaves (ATM): An Improved Model for
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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,
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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
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Thompson A.R., Moran J.M., Swenson G.W., Interferometry and Synthesis in Radio Astronomy, 1986, John Wileyand Sons; New York.

Kraus, A., Calibration of Single-Dish Telescopes, Max
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14