A Conceptual Overview of Single-Dish Absolute Amplitude Calibration

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. Mart´ı-Vidal5 and H. Falcke1 Sept 15, 2017 – Version 1.0

1Department of Astrophysics/IMAPP, Radboud University Nijmegen, 6500 GL Nijmegen, the Netherlands 2Arizona Radio Observatory, Steward Observatory, University of Arizona, AZ 85721 Tucson, USA 3Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA 4Max Planck Institut fürRadioastronomie (MPIfR), Auf dem Hügel 69, 53121 Bonn, Germany 5Onsala Space Observatory, Chalmers University of Technology, Observatorievägen 90, 43992 Onsala, Sweden

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 Tsys 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 11 2.1 VLBI and the Event Horizon Telescope array ...... 11 2.1.1 Determining the antenna-based SEFD for VLBI ...... 11 2.1.2 A brief overview of a priori amplitude calibration ...... 11 2.1.3 Double-sideband (DSB) receivers ...... 11 2.2 Telescopes not using the chopper technique ...... 12 ∗ 2.3 Tsys or Tsys?...... 12

2 Relevant terminology Eﬃciency and correction terms:

Relevant variables introduced in this document (brightness • rsb: sideband ratio - since the SMT has a sideband- gi separating receiver, rsb = 1 since no signal temperatures approximated with the Rayleigh-Jeans approxi- gs mation): comes from the image band but some leakage can still be present

• AM: amount of airmass in the line of sight of the • Chot: Counts measured when looking at the hot load (vane) receiver (elevation-dependent)

• τ0: atmospheric opacity at the zenith • Ccold: Counts measured when looking at the cold load (liquid nitrogen) • e−τ: atmospheric attenuation factor, which damps the signal based on atmospheric opacity in the line of sight τ = τ0 × AM • Con: Counts measured observing a target • el: elevation of the antenna dish for a particular obser- vation (in degrees) • Csky: Counts measured when looking at blank sky • g(el): elevation-dependent gain curve correcting

• Tcold: temperature of the cold load for changing illumination of the main reﬂector and ground contributions as the dish moves and tilts to diﬀerent elevations

• Trx: receiver noise temperature • ηl: forward eﬃciency representing the fraction of power received through the forward atmosphere (ac- • Tamb: ambient temperature around the observatory, as counting for rearward losses) measured by a weather station (physical temperature)

• ηtaper: efficiency loss due to non-uniform illumination of the aperture plane by the tapered radiation pattern • Tsky: temperature of the atmospheric emission (the brightness temperature of the sky) • ηblock: aperture blockage efficiency due to blocking of the feed by the sub-reflector (including its support legs) • Tcab: physical temperature of the receiver cabin (this is 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 the • Tcal: derived temperature to give a correct temperature reflective elements to the total power scale for the signal band

• ηRuze: surface error efficiency (or Ruze loss) calculated from Ruze’s formula (Ruze 1952) • Tinject: injected known temperature (of calibrator or noise diode) in the signal chain • ηA: aperture efficiency approximated for the SMT, a combination of various efficiencies ( = ηtaper × ηblock × ηspillover × ηRuze ) • Tsys: system noise temperature of the system

• Ageom: geometric area of the SMT dish ∗ • Tsys: effective system noise temperature (corrected for atmospheric attenuation) • Aeff: effective area of the SMT dish ( = ηAAgeom )

3 1 Introduction to standard single- SEFD: τ Tsyse dish Tsys calibration techniques SEFD = (1) ηlG

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 the signal to noise power rS/N is then: different quantities they output. The equations provided τ rS/N × Tsyse here contain various approximations commonly used but are S source = SEFD × rS/N = (2) not exact from an engineering perspective. They are only ηlG meant to serve as guidelines for a quick understanding of the outputs of the two different techniques. For mm-observatories, which measure the effective system ∗ eτ noise temperature T = Tsys directly using the chopper sys ηl 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) ∗ Tsys SEFD = (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 to a flux density scale. The SEFDs can be calculated using 1.2 The receiver noise temperature system noise temperature Tsys 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) calibration compared to other telescopes in the array. The higher a telescope’s SEFD, the lower its sensitivity. Ultimately, the flux density of a source is simply the telescope’s SEFD, During a two-load calibration (also called cold calibration), which contains all system and telescope parameters and ef- the Y-factor and the receiver noise temperature are mea- ficiencies, multiplied by the ratio of signal to noise power sured using voltage or counts measurements with a hot and (defined as r ) of the source detection. The equation for the a cold load. In principle the receiver noise temperature can S/N T SEFD can be subdivided into three main components, each be estimated from sys measurements at very low opacities τ T with station-based variations for how they are determined ( 1) by extrapolating a linear fit of airmass versus sys and measured. The three components to the SEFD are: to zero airmass. However, it is highly recommended to measure a receiver noise temperature at least once an observing night, as this yields more accurate Tsys measurements rather

1. Tsys: the total noise characterization of the system, than backtracking in post-processing. The Y-factor is calcu- given by the system noise temperature lated with the following: C τ hot 2. e : the correction terms for attenuation of the source Y = , (4) ηl Ccold signal by the atmosphere and rearward losses (ohmic where the numerator is C , the counts obtained from the losses, rearward spillover and scattering) of the tele- hot hot load and the denominator is C , the counts obtained scope cold from the cold load. The Y-factor also enables an easy di- agnostic of the sensitivity of the receiver. A high Y-factor 3. G: The antenna gain, including all the loss terms from means little receiver noise, and thus sensitive observations the telescope and the conversion from a temperature (of course what constitutes “high” depends on the type of scale (K) to a ﬂux density scale (Jansky), given by the receiver 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 fol- g(el): G = DPFU × g(el) lows: T − YT T = hot cold (5) This gives the following general equation for a telescope’s rx Y − 1

4 Here temperatures are used, where Thot is the temperature 1.3.2 Direct (switched noise diode) method of the hot load (for the SMT this is near room temperature ∼ 290K) and Tcold is the cold load temperature (for the SMT this is the temperature of liquid nitrogen ∼ 77K). This method is commonly used at cm-observatories, such 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 repre- A 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. Tinject is the section is an attempt to approximately explain two of the temperature of the diode or the brightness temperature of the source (known for common calibrators), which turns the common techniques for Tsys measurements (chopper wheel common for mm-telescopes, direct for cm-telescopes), what counts scale to a temperature scale. When the telescope is they output and what they mean for data processing. The pointed at blank sky in the calibration procedure, without following outlines are modelled for an SMT-like telescope, the source signal, the temperature contribution is entirely thus with a sideband-separating (SSB) receiver. noise from the receiver and the atmospheric emission, and thus is the system noise temperature Tsys:

T = T + T = T + T (1 − η e−τ) (9) 1.3.1 Clariﬁcation of system temperature jargon oﬀ,cal rx sky rx atm l

We define the system noise temperature as the contributions When the telescope is pointed at the calibrator (or diode) of by the receiver and the sky to a source measurement (assum- a known brightness (or physical) temperature, the source ing TCMB is negligible), where ηl is the forward efficiency, signal is added to the temperature contribution: accounting for rearward efficiency loss due to ohmic losses, rear spillover and scattering: Ton,cal = Trx + Tsky + Tinject (10)

−τ Tsys = Trx + Tsky = Trx + Tatm(1 − ηle ) (6) The system temperature is then determined in the following way:

Before entering the atmosphere, the source signal is defined Csky T Tsys = Tinject (11) as Sig = source. After attenuation by the atmosphere, the Con,cal − Csky −τ signal becomes Sig= ηle Tsource, where the exponential is Toff,cal the atmospheric attenuation factor (τ is the opacity in the = Tinject Ton,cal − Toff,cal line of sight) and ηl embodies rearward efficiency losses. Therefore, the ratio of signal to noise power of a telescope (Trx + Tsky) = Tinject must depend on this received signal (not taking into account (Trx + Tsky + Tinject) − (Trx + Tsky) ground and ambient contributions): (Trx + Tsky) = Tinject = Trx + Tsky (12) −τ Tinject ηle Tsource ηlTsource rS/N = = τ (7) Tsys e Tsys Since the brightness temperature of the source observed (or the diode temperature) is determined outside the atmo- We 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 atmo- τ τ τ spheric attenuation (eτ term). This is because the contribu- ∗ e e e Tsys = Tsys = Trx + Tatm( − 1) (8) tion of the source or diode is added to the signal chain (as ηl ηl ηl opposed 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 tempera- of the sensitivity of a telescope: the system sensitivity ture directly, only the receiver and sky contributions to the drops rapidly (exponentially) as opacity increases. noise (which cannot be disentangled from each other).

5 In order to obtain the effective system temperature, opac- counts measured when the blocker/chopper/vane is in place, ity measurements during observations must be obtained. and Csky is our usual blank sky counts. In terms of temper- This is done either by using water vapor radiometers (or atures, the temperature contribution when the blocker is in tipping radiometers) or by using the telescope as a tipper place Tblock is defined as: using sky tips. Tipping radiometers are notoriously unreli- able (although water vapor radiometers perform very well), Tblock = Trx + Thot, (16) and sky tips must be done very often (every 10 min) and where Thot is the temperature of the hot load itself. The take up valuable observing time. This makes this method load completely blocks the sky emission, which changes highly cumbersome for frequencies at which the atmosphere the calibration equations from the direct (or diode) calibra- cannot be neglected. tion method. As seen in the direct method, the blank sky contribution is simply the system noise temperature: The effective system temperature is thus: −τ Toff = Trx + Tsky = Trx + Tatm(1 − ηle ) (17) τ ∗ e Tsys = Tsys, (13) We can thus write the chopper equation (eq. 15) in terms of ηl temperatures: such that the effective (opacity-corrected) antenna tempera- C ture of a source (where C is the telescope signal on target) ∗ sky Toff on Tsys = Thot = Thot (18) can be given using ON-OFF measurements as: Chot − Csky Tblock − Toff Trx + Tsky C − C − τ = Thot (19) ∗ on sky ∗ Ton Toff ∗ e (Trx + Thot) − (Trx + Tsky) TA = Tsys = Tsys = TA (14) Csky Toff ηl We assume the hot load is at ambient temperature, and so T = T = T . This gives: 1.3.3 Chopper (or single load) calibration hot amb atm Trx + Tsky T ∗ = T (20) sys hot (T + T ) − (T + T ) The chopper (or single load) calibration technique is com- rx hot rx sky monly used by (sub)mm observatories. The system noise Trx + Tsky = Tatm (21) temperature is obtained by placing an ambient temperature (Trx + Tatm) − (Trx + Tsky) load Thot that has properties similar to a blackbody in front of the receiver, blocking everything but the receiver noise. −τ As we have defined Tsky = Tatm(1 − ηle ), we can simplify: As long as Tatm ∼ Thot, this method automatically compen- sates for rapid changes in mean atmospheric absorption. ∗ Trx + Tsky Tsys = Tatm (22) (Trx + Tatm) − (Trx + Tsky) For calibration of source measurements, we want to obtain −τ Trx + Tatm(1 − ηle ) the effective sensitivity of the system, not a comparison be- = Tatm −τ (23) (Trx + Tatm) − (Trx + Tatm(1 − ηle )) tween the receiver and sky contributions to noise. Therefore, T + T (1 − η e−τ) we want to obtain the effective system noise temperature rx atm l = Tatm −τ (24) ∗ Tatm − Tatm + ηle Tatm Tsys to calibrate source measurements. T + T (1 − η e−τ) = T rx atm l (25) ∗ atm η e−τT To first order, the chopper method directly measures Tsys. l atm −τ This is obtained via the following equation: Trx + Tatm(1 − ηle ) = −τ (26) τ τ ηle ∗ Csky e e Tsys = Thot = Trx + Tatm( − 1), (15) Chot − Csky ηl ηl Finally we obtain: where Csky is the voltage/count signal on blank sky and τ is the opacity in the line of sight. −τ Csky T + T (1 − η e ) How does the chopper technique directly provide T ∗ ? T ∗ = T = rx atm l (27) sys sys hot C − C e−τ This is shown simply by investigating the exact output by hot sky τ τ τ the chopper technique. The chopper system temperature e e e = Trx + Tatm( − 1) = (Trx + Tsky) (28) equation is given in telescope counts, where Chot are the ηl ηl ηl

6 If we compare the chopper effective system noise tempera- 1. The ratio of signal to noise power of the observed ture to the system temperature from the direct method: source as measured by the telescope (thus attenuated by the atmosphere): ∗ Tsys − T = (29) Ton Toff sys −τ rS/N = ηle Toff To first order, the chopper calibration (or alternatively 2. The total noise characterization of the system, includ- named the single-load calibration) corrects for atmospheric ing the correction term for atmospheric absorption, given by the effective system noise temperature: attenuation of an observed source and rearward losses of the τ ∗ ∗ e telescope by directly measuring T . It is also worth noting T = Tsys sys sys ηl that during VLBI observing, the quarter wave-plate is added to the signal chain to convert linear to circular polarization: 3. The antenna gain G, including all the loss terms from any losses associated with the addition of the wave-plate the telescope and the conversion from a temperature ∗ scale (K) to a flux density scale (Jansky), given by the will be automatically calibrated and included in the Tsys measurement from the chopper technique in the same way “degrees per flux density unit” factor (DPFU) and the as the atmospheric and rearward losses. gain curve: ηAAgeom G × g DPFU = 2k giving = DPFU (el)

1.4 Getting a flux density We can thus simplify the flux density equation using the three main terms actually measured by the SMT: We have defined the antenna temperature (modified for mea- τ ∗ rS/N × Tsyse rS/N × Tsys sured quantities at a telescope) as: S = = (35) ηlG G

∗ Con − Csky ∗ TA = Tsys (30) Csky 1.4.1 Determining a gain curve 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 ele- As previously mentioned, the characterization of the an- vation of the telescope and convert from a temperature scale tenna gain G is subdivided into two quantities that must to a flux density scale (where k is the Boltzmann constant), be separately provided for the calculation of the SEFDs: the gain curve g(el) and the DPFU (explained in the next dependent on the geometric area of the dish Ageom: section). The characterization of the telescope’s geometric ∗ TA 2k (opacity-free) gain curve is an important part of the flux S = (31) density calibration, and is particularly crucial for the EHT a ηAg(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 a northern hemisphere stations. flux density for a given source. If we expand using all the terms we’ve discussed, we get the following: Telescopes do not have perfect surfaces, and must thus suf- ∗ fer some losses of signal due to distorted illumination of the TA 2k 1 Con − Csky ∗ 2k S = = Tsys (32) main reflector as they slowly move to different elevations. ηAg(el) Ageom ηAg(el) Csky Ageom This large-scale surface deformation affects the received 1 Con − Csky Tsys 2k signal and is not taken into account in the measurements = −τ (33) ηAg(el) Csky ηle Ageom leading to the efficiency and DPFU characterization. These

Ton − Toff Tsys 2k losses can be determined by tracking sources through a = −τ (34) wide range of elevations, and thus measure an elevation- Toff ηAg(el)ηle 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 most noise 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, in- We can subdivide the flux density equation into three major cluding telescope efficiency (through the DPFU) and the ∗ parts: atmospheric attenuation of the signal (through Tsys). At the

7 SMT, this is done by observing two sources (usually K3-50 measurements to a flux density scale and is obtained us- and W75N, a planetary nebula and a star-forming region, ing known flux calibrators, particularly planets, or by boot- due to their similar up-time plots and wide range of eleva- strapping near-in-time observations of non-planet sources tion) contiguously, tracked as they increase and decrease in from telescopes with well-defined and accurate flux density elevation from the tree-line to transit and vice-versa. measurements. This enables to check the flux density scale obtained by the telescope by directly measuring an aperture The gain curve is estimated by fitting a polynomial (usually efficiency. second-order for standard radio-dishes). If more than one source is used, this is done once the flux density measure- The DPFU is estimated with the following equation, where ments are normalized around a plateau (to a relative gain k = 1.38 × 10−23J/K = 1.38 × 103 Jy/K: scale). This normalized gain curve must be written in the form of a second order polynomial (in the standard VLBA format for ANTAB), where ‘el’ is the elevation in degrees: ηAAgeom DPFU = [K/Jy] (37) 2k

2 The geometric area A is simply the area of the dish, g(el) = a2(el) + a1(el) + a0 (36) geom where D is the dish diameter:

Each parameter must not be rounded to the uncertainties of the fit but instead many significant figures should be pro- πD2 A = (38) vided. Uncertainties for each parameter as outputted by the geom 4 polynomial fit must also be provided, along with the full covariance matrix of the fit parameters. This will help deter- The aperture efficiency is the most difficult part of the es- mine an error estimate for the gain curve and propagate to timation of the DPFU. It represents the efficiency of the the error estimation of the final SEFDs. Additionally, a plot telescope compared to a telescope with a perfect collecting of the relative gains (normalized fluxes) versus elevation area (uniform illumination, no blockage or surface errors) and the fitted polynomial should be provided if possible, as and it is determined using observations of known calibrator shown in Fig. 1. sources, usually planets. The observed planet fluxes are then compared to expected planet brightness temperatures from a planet simulation software for a perfect telescope at the given frequency and beam width.

The aperture efficiency ηA is found using the following ∗ equation, where TA 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 and S beam,sim is the expected flux density of the planet in the telescope beam from the simulation program used:

∗ 2k TA ηA = (39) Figure 1: Example of a normalized geometric gain curve plot. Ageom g(el) S beam,sim

Or similarly the DPFU is directly given by: 1.4.2 Determining the DPFU T ∗ DPFU = A (40) g(el) S beam,sim The degrees per flux density unit (or DPFU) is the characterization of the temperature to flux density scale of a For extended sources, it is important to calibrate the flux telescope. The DPFU is used to calibrate the telescope density observed in the beam because some emission might

8 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 is πθ2 the expected total flux density of the source: Ω ' s (47) s 4 S beam,sim = S sim × K (41)

Of course this process heavily depends on assumptions Here K is the following, where θmb is the half-power beam- made in the planet calibration, such as accurate predicted width in arcseconds of the primary lobe of the telescope planet brightness temperatures from available software, tele- beam pattern (telescope beam diameter) and θs is the diam- scope beam width used, stable weather conditions and a eter in arcseconds of the observed extended source, usually well-calibrated instrument in terms of pointing and focus. given by the simulation program: An average value for the aperture efficiency can be esti- θ p x = s ln(2) (42) mated from the individual measurements during a partic- θmb ular observing run, but it is preferable to keep the time- dependence of the variable if a telescope’s efficiency is 1 − e−x2 expected to vary with temperature and sunlight, causing K = (43) x2 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 Even more preferable, a plot of long-term trends of the aper- angle and the solid angle of the source on the sky. It is in ture efficiency, using additional measurements outside EHT fact the integral of the antenna pattern of the telescope (ap- observing or even from previous years, would greatly help − 2 proximated as a normalized gaussian) P(θ, φ) = e ln 2(2θ/θmb) understand the time-dependent nature of the aperture ef- and a disklike source with a uniform brightness distribution ficiency of a particular telescope. As the scatter between Ψ(θ, φ) = 1 over the size of the extended source. This serves individual measurements can be caused by various factors, very well for our a priori calibration purposes1. such as unstable weather or changing pointing/focus ac- curacy, it is not always representative of the true aperture Ω 1 Z K = sum = P(θ − θ0, φ − φ0)Ψ(θ0, φ0)dΩ0 (44) efficiency change in the observations. A trend exhibited in Ω Ω s s source the long-term as a function of time would be more reliable to estimate an aperture efficiency for a particular scan. Such a plot is shown in Fig. 2, as an example from the JCMT. 1 Z K = P(θ − θ0, φ − φ0)dΩ0 (45) Ωs source

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

2h ν3Ω S = s , (46) sim 2 hν c kT e B − 1 where ν is the observing frequency in Hz, h is the Planck constant, c is the speed of light (in m/s), TB is the mean brightness temperature for the planet (assuming a disk of uniform temperature) from the simulation program, and Ωs is the solid angle of the source on the sky in steradi- Figure 2: Example of long-term trend for the time-dependent aper- ans. Since we are dealing with very small objects, the latter ture eﬃciency ηA. 1More 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, In summary, the aperture efficiency accounts for all forward a fit for this dependence must be provided, as well as the losses of the telescope, which come from different contribu- covariance matrix for the fit parameters, for error analysis tions. As previously mentioned in section 1.3.3, the chopper of the a priori deliverables. A fit to the UT time-dependence technique itself account for the rearward losses of the tele- would be the most robust against various observing effects scope automatically. These losses are also outlined in the from day to day and session to session and should be very following section. stable over the years, provided no major work has been done on the telescope. For telescopes with no visible time- dependence, a mean aperture efficiency (or DPFU) will 1.4.3 Other efficiencies suffice, with the appropriate error estimate.

We can also write the aperture efficiency as the combina- The main beam efficiency of a telescope is the fraction of tion of various individual forward efficiencies, each closely observed power in the main lobe of the telescope beam pat- approximated for the telescope via various measurements: tern. Let the beam solid angle (the full antenna pattern) be ΩA and the main beam solid angle (the main lobe) be Ωmb. ηA = ηtaper × ηblock × ηspillover × ηRuze (48) The main beam efficiency is written as the ratio between the total beam and main beam solid angles:

Ωmb Each eﬃciency term corresponds to an aspect of the tele- ηmb = (50) Ω scope feed 2: A

It is estimated with the following, where Tmb is the main • ηtaper is the efficiency loss due to non-uniform illumi- beam temperature of a source that fills the main beam, as nation of the aperture plane by the tapered radiation estimated from the simulation program: pattern/feed function (also formally known as the S ηAAgeom illumination efficiency). It is the most important con- η = beam,sim (51) mb T 2k tributor to the aperture efficiency. mb

• ηblock is the aperture blockage efficiency due to block- It should be noted that the main beam efficiency is not ing 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 main The forward efficiency ηl represents the fraction of power reflector - it is the ratio of the power intercepted by received through the forward atmosphere (in other terms it the reflective elements beyond the edge of the sub- is the coupling of the receiver to the cold sky) and is written reflector and primary to the total power. It is due as the ratio between the solid angle over the forward hemi- partly to cold sky and partly to a warm background, sphere of the telescope and the beam solid angle and it is and is elevation-dependent. typically close to unity (but drops with frequency due to loss of receiver sensitivity): • ηRuze is the surface error efficiency (also called “Ruze loss” or scattering efficiency) calculated from Ruze’s Ω2π ηl = (52) formula (Ruze 1952). It is due to small scale, ran- ΩA domly distributed deviations of the reflector from The only way to estimate it is via sky-dips, by measuring the perfect paraboloidal shape. Ruze’s formula is the atmospheric emission with elevation:3 presented below, where σ is the surface rms (account- −τ/ sin(El) ing for small-scale deviations from a perfect surface TA(el) = ηlTatm(1 − e ) + (1 − ηl)Tamb (53) through dish holography) and λ is the observing wave- It is important to note that sky-dips measure both the atmo- length: spheric opacity and the forward efficiency so they need to be disentangled. Fortunately, this is not an issue for the EHT because the chopper technique implicitly corrects for the −16π2σ2 2 ηRuze(λ) = e λ (49) forward efficiency ηl (see Section 1.3.3). 2More detail on the measurement of the different losses, see Baars, J., The paraboloidal reflector antenna in radio astronomy, Springer, 2007. 3Overview in: Kramer, C., Millimeter Calibration, presentation at IRAM Summer School 2013, IRAM, Granada, Spain.

10 2 Miscellaneous explanations to a signal plane above the atmosphere, then the resulting visibility amplitudes will be corrected for atmospheric losses. 2.1 VLBI and the Event Horizon Telescope array 2.1.3 Double-sideband (DSB) receivers 2.1.1 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 obser- SMT, which has a sideband-separating receiver. However, vations and that for VLBI are identical. The equation for the a few stations in the Event Horizon Telescope array have antenna-based SEFD for VLBI observations is thus: double-sideband (DSB) receivers, which lead to some mod- ifications of the equations for amplitude calibration. The most relevant difference between SSB and DSB receiver is ∗ ∗ Tsys Tsys the handling of measured signals. For an SSB receiver, all SEFD = = (54) G DPFU × g(el) the measured signal comes from only one sideband, but for 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 the telescope 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 Tsys. tors that do not become resolved on some baselines, thus we cannot use the primary calibrator scaling to calibrate VLBI For a measured effective system temperature from a perfect amplitudes. An alternative approach is to calibrate the VLBI ∗ DSB system Tsys,DSB, the actual effective system tempera- amplitudes 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 Tsys = 2Tsys,DSB (56) normalized visibility amplitudes by the geometric mean of the SEFDs of the two antennas concerned (TMS Section For EHT observing we use half the number of sidebands, 10.1.). On a baseline between two telescopes, for example thus the telescope sensitivity must drop by a factor of two, the SMT and the LMT, which both use the chopper method, leading to the effective system temperature increasing by the amplitude calibration for the correlated source signal the same factor. However, if the telescope does not have a rcorr,SMT−LMT (compensated for digitization and sampling perfect DSB system but one sideband has more gain than losses) on that baseline is given by: the other, then the equation becomes, more generally:

p ∗ ∗ S SMT−LMT = SEFDSMT × SEFDLMTrcorr,SMT−LMT, (55) Tsys = (1 + rsb)Tsys,DSB, (57)

where the sideband ratio (rsb) is the ratio of source signal where SEFDSMT and SEFDLMT are determined as shown power in the remaining sideband to the signal power in the above and S SMT−LMT is then the source signal in Jansky on sideband of interest (the sideband to be correlated). For a that baseline. perfect DSB system, the gains of each sideband are equal, giving rsb = 1, which gives back Eq. 56. For a perfect SSB Since the SEFDs for the telescopes are expected to include system, where all signal is in one sideband, rsb = 0 and this ∗ the eﬀective system noise temperature, which corresponds gives back simply Tsys needed for the EHT.

11 ∗ Once this correction is applied to Tsys, the rest of the ampli- at the observing frequency but instead is converted tude calibration process remains the same. For planet scans (sometimes not so accurately) from a different fre- to determine the telescope’s DPFU, the signal is collected quency by both sidebands in a DSB system, thus the effective antenna temperature is usually measured in DSB mode. This Another possible solution is to use the telescope itself as is sufficient to reflect the conversion from Kelvin to Jan- a tipper: using the dish to observe blank sky through a big sky within the aperture efficiency and DPFU estimation. It elevation range in the direction of observing to determine should be noted that the correction from a DSB system to ∗ the relationship between elevation and sky temperature and an SSB system for VLBI should only be done on Tsys, oth- get an estimate of the zenith opacity. erwise the resulting SEFDs would be double-corrected for a DSB system. This tipping method solves the radiometer issues of getting an opacity in the direction of observing and at the right observing frequency. However, these tipping scans are re- 2.2 Telescopes not using the chopper tech- quired very frequently, every 10 minutes or so, and take up nique valuable observing time just to get accurate opacities.

An alternative is then to obtain opacities using approxima- As explained above, the result for telescopes like the SMT tions in post-processing. The system noise temperature, as and the LMT, which both use the chopper (or single-load) measured in the direct method, is defined as seen previously. technique, is very clean and simple. Now what happens For τ0 1, we can approximate: when there is a telescope in the array that does not use the chopper technique but instead uses the direct (or noise 4 diode) method? −τ Tsys ≈ Trx + Tatm(1 − e ) ≈ Trx + Tatm × τ0 × AM (58) In that case, on baselines with telescopes with the chopper method, there will be inconsistencies in the amplitude By fitting a least-squares (or as it is done for the GMVA, calibration if the same corrections are applied in post- a linear fit to the lower envelope) of Tsys as a function of processing to both stations on that particular baseline. This airmass, the extrapolation of the fit will give an approxima- ∗ is precisely because the chopper technique gives Tsys and tion for the receiver noise temperature Trx. If the telescope the direct method only gives Tsys. frequently does a dual-load (cold cal) calibration to refresh values for the receiver temperature, these values are usually Fortunately, as explained in the previous section, the rela- more accurate to use. ∗ tionship between the two is well-understood and Tsys can easily be determined from the direct method using opacity With this linear relationship (or measured Trx) and every measurements. If the telescope has a tipping radiometer or variable but the sky opacity known, measured or approxi- water vapor radiometer nearby measuring opacities, this can mated by the telescope, we can get the sky opacity at the ∗ τ give a fairly good estimate for Tsys = e Tsys. zenith and thus correct the system noise temperature for the atmospheric attenuation: However, some aspects of radiometers hinder this approach:

! 1 Tsys − Trx • The radiometer does not always point in the same di- τ0 = − ln 1 − (59) rection as the telescope, thus under a varying or partly AM Tatm cloudy sky the opacities from the radiometer are not entirely accurate to the observations. ∗ 2.3 Tsys or Tsys? • The radiometer can have something blocking and cor- rupting the measurements (as on Mt Graham due to the LBT) A crucial part of the amplitude calibration process is to 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 4As 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 Tsys like cm-observatories do? Or are ∗ some telescopes providing in fact Tsys but labeling it as Tsys (as is commonly done by mm-observatories)? Discrepan- cies in notation and a heavy background knowledge in the context of cm-observatories can cause misunderstandings of the calibration information provided by the telescopes. However, there is a nice way to do a quick check in post- processing5.

In order to visually understand the diﬀerence between the two variables, simulated measurements of system noise temperatures for the SMT are presented. Using the standard Figure 3: Simulated system temperatures from the chopper and chopper equation, the calibration temperature was approxi- direct method calibration techniques show divergence as a function of airmass. mated to Tcal = Tamb = 280K, the receiver noise temperature was set to Trx = 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 Tsys using the chopper technique equation and the direct method −τ ∗ system noise temperature Tsys ≈ e Tsys as a function of airmass. It is clear that both temperatures indeed do vary with ∗ airmass, but Tsys is a lot more sensitive because in addition it corrects for the increasing attenuation of a signal from outside the atmosphere, automatically determined with the chopper technique.

It is a misconception to assume that because Tsys does not contain that term, it does not vary with airmass. Tsys is in- Figure 4: Zenith opacities obtained by assuming the telescope pro- herently dependent on airmass because the sky brightness vides Tsys are much larger than what is actually measured by the temperature Tsky, representing atmospheric noise, increases SMT tipper. with airmass as the telescope looks through a larger layer ∗ of atmosphere. This effect is also present in Tsys, which, in addition, corrects for the increasing signal attenuation that is We use the opacity equation: also elevation (airmass) dependent. ! 1 Tsys − Trx In the previous section, we introduced a useful tool to tell τ0 = − ln 1 − (60) AM Tatm the two system noise temperatures apart. For the case of Tsys, as shown by eq. 59, it is possible to untangle a zenith Recall that at this point, what is plugged in as Tsys is what is ∗ ∗ opacity from Tsys and Trx measurements. In the case of Tsys, measured by the chopper technique (in fact it is Tsys but this because the opacity relationship is much more complicated, was still unknown). ∗ it would not be valid. If we were to apply eq. 59 using Tsys instead of Tsys, the opacities at zenith obtained would be The zenith opacities obtained from that equation were then highly inaccurate when compared to, for example, radiome- compared to the measured zenith opacities by the tipping ter measurements during the same observing window. radiometer on the telescope scan-by-scan. These results are presented in Fig.4. It is clear that the zenith opacities This reasoning was thus applied to the SMT to see if what obtained from “Tsys” are completely different, incredibly was previously called “Tsys” was really Tsys. For example, high and inconsistent with the tipper measurements. This ∗ we can take the system noise temperatures and receiver tem- is of course because “Tsys” is in fact Tsys, which diverges perature measured during the gain curve measurements for and is increasingly larger than Tsys 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 minimizing opacity fluctuations due to changing weather. is outputted by the chopper technique at the SMT, and this output is definitely not simply Tsys 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 database

13 ∗ more sensitive to opacity: Tsys. For a telescope that is gen- Kramer, C., Calibration of spectral line data at the IRAM uinely providing Tsys, the opacity equation would give re- 30m radio telescope, Version 2.1, IRAM, January 24th sults much more in-line with the measured opacities from its 1997. tipper/radiometer. Kramer, C., Millimeter Calibration, presentation at IRAM Summer School 2013, IRAM, Granada, Spain.

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