# CSC Dictionary Entries

## Alternating Exposure Mode

In alternating exposure mode, sometimes also referred to as interleaved mode, there is a long (primary) and a short (secondary) frame time, and the observation alternates between them. The long frames are joined into one event file and the short frames are joined into another event file. This mode is used to look for variability on different timescales or when observing a source that is expected to be piled. The observer gets the long-frame "piled" information, as well as the short-frame unpiled data.

## Aperture Model Energy Fluxes

The conversion from aperture source count rates in each science energy band to aperture model energy fluxes is performed by scaling from a model spectrum folded through the calibrated response, as follows:

For a source model F(E) whose integral over the science band is F(band), calculate the corresponding band count rate C'(band) in counts s-1, given the effective area calibration A(E) [and, if available, the RMF, R(E, band)] appropriate to the observation; this is the integral of F(E)A(E)R(E, band) over all energies, or if a diagonal RMF is assumed, the integral of F(E)A(E) over the band.

Infer the aperture model energy flux from the measured aperture source count rate C(band) as F(band) = F(band)C(band)/C'(band). The power law spectral model has a fixed photon index, α, defined as FE~E- α, equal to 1.7, and a fixed total neutral Hydrogen absorbing column NH = NH(Gal) cm-2. The black body spectral model has a fixed temperature kT = 1.0 keV, and a fixed total neutral Hydrogen absorbing column NH = 3 x 1020 cm-2.

## Aperture Total Counts

"Aperture total counts" refer to the total number of source plus background counts measured in the modified source region, the modified background region, the modified elliptical aperture, and the modified elliptical background aperture, uncorrected by the PSF aperture fraction.

## ARF: Auxiliary Response Function (or File)

An ARF contains the combined telescope/filter/detector areas ("effective area") and the quantum efficiency (QE) as a function of energy, averaged over time (and therefore, aspect). The effective area is [cm²] and the QE is [counts/photon]; they are multiplied together to create the ARF, resulting in [cm² counts/photon].

When the input spectrum is multiplied by the ARF, the result is the distribution of counts that would be seen by a detector with perfect (i.e. infinite) energy resolution. The RMF is then needed to produce the final observed spectrum.

## Aspect Histogram

The aspect solution is given every 0.256 seconds during a Chandra observation. The aspect solution can be put into a very compressed form by making a histogram of the pointing vs. x-offset, y-offset, and roll-offset. The value in each bin is the time the pointing was within that offset bin during the observation, as modified by the good time interval and deadtime correction factor. (The information which is lost is the absolute time at which the pointing was at each offset.) The histogram is primarily used to shorten the time required to compute the response averaged over the observation. Another use for the histogram is to provide all the observational configuration details (detector, date) via the file's header.

## Black Body Model Spectral Fit

If there are at least 150 net (background subtracted) counts in the energy range 0.5-7.0 keV present in the source region of an ACIS observation (or in at least one ACIS observation contributing to the Master Source observation), then black body model spectra are fitted to PI event data exracted from the source region. The black body spectral fit includes corrections for the PSF aperture fraction, livetime, and ARF applied when fitting the models. The free parameters to be fitted are the total integrated flux, total neutral Hydrogen absorbing column density, and the black body temperature. For more information on spectral model fits used in the Chandra Source Catalog, see the Catalog Descriptions page "Spectral Properties."

Note: Spectral fit parameters may be unreliable for sources at large off-axis angles, where background levels can be high. A background-fitting approach will be considered for future releases of the Catalog.

## Blocking Factor

Blocking, also know as "binning", takes a table (e.g. an event list) and applies a rule to create an image; for instance, "bin sky=4" combines every 4 sky pixels into a single image pixel. CIAO binning syntax may also be used to rebin an existing image to a different blocking factor.

When performing source detection, a recursive blocking scheme may be used to cover a larger spatial region than the maximum image size for the tool. In such a case, the inner 2048x2048 pixel region of the dataset is searched for sources, then the inner 4096x4096 pixel region is blocked by 2 and searched for sources, then the inner 8192x8192 is blocked by 4, and so on.

The blocking factors used for source detection in the CSC are:

• ACIS-S and ACIS-I: 1, 2, and 4
• HRC-I: 2, 5, and 12
• HRC-S: 2, 5, 12, and 25

These binning factors ensure that the image being searched is always 2048x2048.

## CTI: charge transfer inefficiency

CTI is the loss of charge in a CCD as it is shifted from one pixel to the next during readout. This is due to states in the lattice, called "traps," into which electrons can transition from the conduction band. Soft proton damage to the ACIS chips early in the Chandra mission greatly increased the numbers of these states, thereby increasing the parallel CTI and reducing the energy resolution. CTI effects can change the "grade" of events, which may result in some good events being rejected in processing.

There is CTI calibration data to compensate for the effect for the entire ACIS detector (chips I0-3 and S0-5) at the -120 C focal plane temperature.

The CIAO CTI why topic has further details.

## Deconvolved Source Extent

The deconvolved source extent is a parameterization of the extent of the PSF-convolved source; it consists of the 1σ radius along the major axis, the 1σ radius along the minor axis, and the position angle of the major axis of a rotated elliptical Gaussian source that is convolved with the ray-trace local PSF at the location of the source spatial event distribution. For more information on deconvolved source extent in the CSC, see the Catalog Descriptions page "Source Extent and Errors."

Note: In the first catalog release, the rotated elliptical Gaussian parameterization of the deconvolved source extent will be approximated by a circularly symmetric Gaussian parameterization. This limitation will be lifted in a future release.

## Degap Correction for HRC

The algorithm used to determine the centroid of the charge cloud exiting the rear microchannel plate of the HRC (and hence the x-ray event position) introduces systematic errors in the event positions, which are manifested by regularly-spaced gaps in both x and y in HRC images.

The degap correction is applied to compensate for this problem in HRC-I and HRC-S data.

The CIAO degap why topic has further details.

## Direct Flux Algorithm

To provide a model independent estimate of source fluxes, we calculate fluxes for source and background regions using the energies of each photon, estimated using their PI values and the effective areas appropriate for those photons given their detector position and energy.

## Dither

While observing a source, Chandra does not maintain a fixed pointing direction, but instead dithers in a Lissajous figure. Irrational periods are used in Y and Z directions to make a non-closed Lissajous pattern.

The CIAO dither why topic has details of the Chandra dither pattern.

The dead time correction factor represents the fraction of the total time per frame that contains useful X-ray data. For ACIS data, the equation is:

DTCOR = EXPTIME / (TIMEDEL + FLSHTIME)


The value is between 0 and 1, and depends on the operating mode used for the observation. The value for the standard, full-frame mode is 0.987337. If an observation uses alternating exposure mode, then there are separate values of DTCOR for the primary and secondary frames.

The observation times entry in the CIAO dictionary explains how the various Chandra exposure keywords are related.

The HRC dead time is an error-weighted mean over the good time of the observation. To learn how it is calculated, refer to the CIAO thread.

## Energy Bands

The energy bands used in the CSC are split into two categories: source detection energy bands and science energy bands. ACIS and HRC are each addressed in both categories. The Energy Bands page has further details.

## Event

X-ray astronomy instruments record a separate signal from every individual photon they detect. This is unlike typical optical CCDs (for example WF/PC2 on Hubble Space Telescope) which need to integrate the signal from a number of photons to generate a detectable signal. As a result, X-ray data is stored event by event, which retains more information and allows great flexibility of analysis.

Every X-ray "event" (a general term for a detection; may refer to a photon or a background cosmic ray) is characterized by: a "pulse height" (PHA) that encodes the energy of the incoming photon; a time; a grade, and typically two position coordinates. The large amount of information for each event allows rather complex and sophisticated analysis. For example, a user may wish to exclude events which occurred during a period of high background, and then display the events as a spectrum vs. time image. Such multi-dimensional analysis is common to X-ray astronomy.

Retaining the individual events also retains the Poisson ("counting statistics") nature of the data, and so allows the statistical significance of sources or features to be assessed more readily.

## Field of View

The field of view refers to the region of the sky that was imaged by the detector during the observation, including the effects of spacecraft dither.

## FITS Standard (ISO 8601) Format Time

The FITS standard (ISO 8601) formats time as "<YYYY>-<MM>-<DD>T<HH>:<MM>:<SS>", where <YYYY> is the 4 digit year, and <MM>, <DD>, <HH>, <MM>, and <SS> are the 2 digit month, day of month, hours, minutes, and seconds TT corresponding to the equivalent Mission Elapsed Time, truncated to integer seconds.

## Flux Significance

Flux-based signal-to-noise ratio of source in source region. It is defined as the photflux_aper value divided by the standard deviation of a 1-D Gaussian distribution that has the same FWHM as the probability density function from which the upper and lower confidence limits are determined. For more information on source significance in the Chandra Source Catalog, see the Column Descriptions page "Source Significance."

## Gain

The gainfile is used to compute the ENERGY and PI of an event from the PHA value; see the PI dictionary entry for more information. Each of the values is stored in a column of the same name in the event file. Using the correct gainmap is especially important if one is interested in ENERGY, PI, or the data is for a grating observation.

The grade is a number assigned to every event based on which pixels in its 3x3 island are above their threshold value. The initial grade is assigned by on-board processing, which first finds local maxima, then analyzes the values of the surrounding 3x3 neighboring pixels. Based on this pattern, the event is assigned a grade. For example, a single-pixel event has a grade of 0.

## GTI: Good Time Intervals

The range of valid data included for an observation is also known as a "good time interval" or GTI. The GTIs give the time periods when the mission time line parameters fell within acceptable ranges.

The GTI calculations for the CSC are different from those created by Chandra Standard Data Processing in that they exclude periods of high background (e.g. flares).

Criteria used to determine the GTI may include (but are not limited to):

• the first one or two frames of an observation may not contain useful science data
• the quality of the aspect solution
• whether there were any gaps in the aspect solution
• the motion of the optical axis
• the angles between the optical axis and the limb of the Sun, the limb of the Earth, the limb of the Moon, and the direction of motion of the spacecraft
• the temperatures of various portions of the telescope

## Hardness Ratio

Source hardness ratios are an approximate measure of the source spectral shape based on count ratios. The hardness ratio for a pair of bands x, y is defined as

where F(x) is the aperture source photon flux (field photflux_aper in MSC) in band x, and F(s), F(m), and F(h) are the aperture source photon fluxes in the soft, medium, and hard bands. For example, hard_ms is the medium-to-soft band hardness ratio, defined as

As the aperture photon fluxes are evaluated assuming a monochromatic energy exposure map in each energy band, the sum of the photon fluxes from individual bands can be somewhat different than the value of the photon flux calculated for the broad band, F(b).

The catalog includes only the ratios hard_ms, hard_hs, and hard_hm. Hardness ratios using the broad, ultrasoft, and HRC bands are not included in the catalog. For more information on spectral properties in the catalog, see the Column Descriptions page "Spectral Properties."

## ICRS Right Ascension and Declination

All right ascension and declination values in the CSC are given in ICRS, the International Celestial Reference System. ICRS is the current IAU-recommended system which decouples the definition of RA and Dec from the changing orientation of the Earth, including precession; ICRS positions are within 0.1 arcsecond of the older and more familiar FK5(J2000) system, so most Chandra users can think of them as being the same as J2000.

The ICRS origin is the solar system barycenter, but the Chandra data are, strictly speaking, tagged with coordinates that represent the ICRS with its origin translated to the spacecraft. Since we use ICRS positions for the aspect camera guide stars when solving for the Chandra pointing direction, no explicit correction for the motion of the Earth around the Sun needs to be made for objects in the far field. However, for near-field (solar system) objects, more care must be taken: see the CIAO thread on reprojecting coordinates of a solar system object for use of the Chandra orbit ephemeris data and ephemerides for solar system objects observed by Chandra.

See the USNO ICRS webpage for a detailed technical description of ICRS and related references.

## Inter-observation

An "inter-observation" source property is one whose derivation is based on data from all individual observations contributing to a set.

## Intra-observation

An "intra-observation" source property is one derived from an individual observation contributing to a set.

## Level 3

The CSC is created by processing each Chandra dataset with a series of automated data analysis pipelines. Collectively, the pipelines are known as "Level 3 Processing" and the data products are named as such, e.g. the event file is evt3.fits. The nomenclature follows Chandra Standard Data Processing, which produces the Level 1 and 2 Chandra data products.

For details on Level 3 processing, read the Catalog Processing section.

## Limiting Sensitivity

In the Chandra Source Catalog, the limiting sensitivity is defined to be the flux of a point source that meets but does not exceed the flux significance threshold for inclusion in the Catalog. It is a function of source position, background, and the algorithm used to calculate flux and flux significance. For details, see the memo "Computing L3 Limiting Sensitivity Maps."

## Livetime

The livetime is the effective exposure time of an observation after applying the good time intervals (GTIs) and the deadtime correction factor.

## Master Sources Table

Each distinct X-ray source on the sky identified in the Chandra Source Catalog is represented by a single master source table entry. The master table entry records the best-estimates of the tabulated properties for a source, based on the data extracted from the set of source observations in which the source was detected.

The Catalog Organization page has further details on how the Master Sources Table and Source Observations Table are related.

## Mission Elapsed Time

Mission Elapsed Time (MET) is the standard time system for Chandra data. It counts the number of seconds TT since 1998-01-01T00:00:00 TT. Zero seconds MET corresponds to MJD 50814.0 TT.

## Modified Source Region

The modified source region and modified background region for each source are defined as the intersection of the source region and background region for that source with the field of view, excluding any overlapping source regions.

## OBI: Observation Interval

An observation interval (OBI) is an uninterrupted observing interval for a single ObsId. In the case of alternating exposure data, the OBI is also restricted to a single frame time.

## ObsID: Observation Id

Every Chandra observation is assigned an observation identification (ObsID), which indicates that it is an observation of a specific target for a specific proposal with a specific observational configuration. Each ObsId contains one or more observation intervals.

## PHA: Pulse Height Amplitude

The PHA value in Chandra event files is the total pulse height of an event; see also PI. For a given location, a gain table is used to map the PHA of an event to the energy value.

## PI: Pulse Invariant

In ACIS observations, the value of PI for an event is given by the equation:

  PI = [ (energy/14.6 eV) + 1],


where energy is a column in the event file. The energy is calculated from the event's PHA value, using the appropriate gain table. PI is an integer, not a real and the decimal portion of the computation is discarded; that is, the value is not rounded, but truncated. For example:

 energy (eV) (energy/14.6 eV) + 1 PI 992.9 69.01 69 1007.3 69.99 69

The energy and PI values are good enough to filter the event data into different bands, but not good enough for spectral analysis (since there is an implicit assumption that the RMF is infinitely good).

The calculation of PI for HRC data is more complicated. It is explained in the "Gain response of the HRC" Calibration pages; specifically, refer to the "HRC-I Gain Correction" memo (Posson-Brown & Kashyap, 2007).

## Position Angle

Position angles are expressed in degrees, and in the first catalog release, the 0 deg position angle is *not* equal to North for each source, but rather parallel to North at the location of the tangent plane reference for the observation (refer to the tangent plane reference right ascension (ra_nom), declination (dec_nom), and roll angle (roll_nom)).

## Power Law Model Spectral Fit

If there are at least 150 net (background subtracted) counts in the energy range 0.5-7.0 keV present in the source region of an ACIS observation (or in at least one ACIS observation contributing to the Master Source observation), then power law model spectra are fitted to PI event data exracted from the source region. The power law model spectral fit includes corrections for the PSF aperture fraction, livetime, and ARF applied when fitting the models. The free parameters to be fitted are the total integrated flux, total neutral Hydrogen absorbing column density, and power law photon index. For more information on spectral model fits used in the Chandra Source Catalog, see the Catalog Descriptions page "Spectral Properties."

Note: Spectral fit parameters may be unreliable for sources at large off-axis angles, where background levels can be high. A background-fitting approach will be considered for future releases of the Catalog.

## PSF Aperture Fraction

The "PSF aperture fraction" refers to the fraction of the PSF that is included in a specified region, e.g. the modified source region, the modified background region, the modified elliptical aperture, or the modified elliptical background aperture.

The PSF describes the shape of the image produced by a delta function (point) source on the detector.

Chandra produces sharper images than any other X-ray telescope to date and therefore provides an opportunity for high-angular and spectral resolution studies of X-ray sources. Crucial to these studies is the knowledge of the characteristics of the PSF. The blurring of the Chandra PSF is introduced by the HRMA PSF, the aspect, the limited size of detector pixels, and detector effects.

Simulating the HRMA PSF using SAOTrace is the first step in obtaining a good model of the Chandra PSF for a given observation; SAOTrace is equivalent to the publicly-available Chandra Ray Tracer (ChaRT). The shape and size of the HRMA PSFs vary significantly with source location in the telescope field of view (FOV), as well as with the spectral energy distribution of the source. Because of the Wolter Type I design, the image quality is best in a small area centered about the optical axis. In fact, the Chandra mirrors were designed to produce images with better than one arcsecond resolution; in particular, to concentrate better than 85% of the energy at 0.277 keV within a 1 arcsecond diameter.

While ACIS reads out a frame, it is still taking data. Photons detected during the readout are clocked out in the wrong row and so have incorrect CHIPY values. For a bright source, you get a streak along the entire column of the source. Events above a source occur as a frame is being read out. Events below the source occur when the previous frame is being read out.

## RMF: Redistribution Matrix Function (or File)

An RMF is a probability that maps energy space into detector pulse height (or position) space. Since detectors are not perfect, this involves a spreading of the observed counts by the detector resolution, which is expressed as a matrix multiplication. In high resolution instruments (e.g. diffraction gratings, such as HETG and LETG) the matrix is almost diagonal. In proportional counters, the matrix elements are non-zero over a large area. CCD detectors, such as ACIS, are an intermediate case, with most of the response being almost diagonal, but escape peaks and low energy tails adding significant contributions.

## Science Energy Bands

Science energy bands are used for all science analyses and are chosen to maximize the scientific utility of the catalog.

The science energy bands for ACIS are:

• broad band (b): 0.5-7.0 keV
• ultrasoft (u): 0.2-0.5 keV
• soft (s): 0.5-1.2 keV
• medium (m): 1.2-2.0 keV
• hard (h): 2.0-7.0 keV

NOTE: Source properties in the catalog which have a value for each science energy band have the corresponding letters appended to their names, e.g. "flux_aper_b" and "flux_aper_h" represent the background-subtracted, aperture-corrected broad band and hard band energy flux, respectively.

Since the HRC does not have significant energy resolution, there is a single broad energy band for both source detection and science. The broad energy band corresponds approximately to photon energies 0.1-10 keV and is designated "w" to distinguish it from the ACIS broad band (b).

The Energy Bands page has further details.

## Source Detection Energy Bands

Source detection energy bands are used in processing to detect sources.

The ACIS bands are chosen to maximize the source detection efficiency for different kinds of sources observed using the back- and front-side illuminated CCDs.

• broad band (B): 0.5-7.0 keV
• soft (S): 0.5-1.2 keV
• medium (M): 1.2-2.0 keV
• hard (H): 2.0-7.0 keV

NOTE: Source properties in the catalog which have a value for each science energy band have the corresponding letters appended to their names, e.g. "flux_aper_b" and "flux_aper_h" represent the background-subtracted, aperture-corrected broad band and hard band energy flux, respectively.

Since the HRC does not have significant energy resolution, there is a single broad energy band for both source detection and science. The broad energy band corresponds approximately to photon energies 0.1-10 keV and is designated "w" to distinguish it from the ACIS broad band (b).

The Energy Bands page has further details.

## Source Detection Region

A source detection region is a region generated by wavdetect, an algorithm used to identify source candidates within a single observation in the Detect Pipeline of Catalog Processing. wavdetect detects sources within an image by repeatedly correlating it with "Mexican Hat" wavelet functions of different scale sizes, producing a set of source detection regions for each scale. The individual source detection regions produced by all wavelet scales for a source are scaled and merged within wavdetect to produce a single source region that effectively corresponds to the smallest scale that is bigger than the PSF. The resulting source region is what is used in all subsequent scientific analysis, as the source detection regions are needed only to create source regions in Catalog Processing.

## Source Extent

The source extent is a rotated elliptical Gaussian parameterization of the raw extent of a source. The parameterization is computed from a wavelet transform analysis of the counts in the "source region", and shall consist of the 1σ radius along the major axis, the 1σ radius along th minor axis, and the position angle of the major axis of the rotated elliptical Gaussian on the detector. For more information on source extent in the CSC, see the Catalog Descriptions page "Source Extent and Errors."

Note: In the first catalog release, the rotated elliptical Gaussian parameterization of the source extent may be approximated by a circularly symmetric Gaussian parameterization. This limitation will be lifted in a future release.

## Source Observations Table

Each distinct X-ray source on the sky identified in the Chandra Source Catalog is represented by an individual source entry for each observation in which the source was detected (one or more).

The individual source entries record all of the tabulated properties about a detection extracted from a single observation, as well as the observation-specific data products. Source detections in each observation are matched with source detections in observations that overlap the same region of the sky. Detections that can be matched uniquely are merged to construct the master table entry for that source.

The Catalog Organization page has further details on how the Master Sources Table and Source Observations Table are related.

## Source Position

Source positions in the Chandra Source Catalog are the best estimate of the ICRS celestial position of the source at the epoch of observation. For release 1, the positions are derived directly from the wavelet source detection algorithm with no further modelling. Refer to the Position and Position Errors page for an in-depth discussion of the error ellipses used to define source position uncertainty.

## Source Region

A source region in a single observation is determined by scaling and merging the individual source detection regions resulting from the wavdetect source detection algorithm. The parameter values that define the soure region for each source are the ICRS right ascension and declination of the center of the source region, and the major axis, minor axis, and position angle of the error ellipse defining the uncertainty of the source position. For more information on source regions, see the page "Position and Position Errors" classified under "Column Descriptions."

## Streak Events

There is a flaw in the serial readout of the ACIS chips, causing a significant amount of charge to be randomly deposited along pixel rows as they are read out. Chip S4 (ccd_id=8) is most affected by this problem.

On account of the serial readout problem, the event file shows a variable pattern of linear streaks. These events may be flagged in or removed from the file with the CIAO tool destreak.

## TGAIN: Time-dependent Gain Correction

The time-dependent gain adjustment is necessary because the "effective gains" of the detectors are drifting with time as the result of an increasing CTI.

There is TGAIN calibration data for the entire ACIS detector at the -120 C focal plane temperature. There is also TGAIN calibration data for the ACIS-S1 and S3 back-illuminated chips at the -110 C focal plane temperature.

The CIAO time-dependent gain why topic has further details.

## Two-sided Confidence Limits

In the Chandra Source Catalog, the upper and lower confidence limits associated with estimates of source fluxes (e.g. photflux_aper_hilim, photflux_aper_lolim) and spectral properties (e.g. flux_powlaw_hilim, flux_powlaw_lolim) are reported as follows, at a 68.2% (1-sigma) confidence level, unless specified otherwise:

• If both of the confidence limits are within the allowed range (e.g. flux greater than zero, or spectral fit parameters within specified limits), then the computed values are reported.

Example:
flux = 1E-15
flux_lolim = 0.8E-15
flux_hilim = 1.2E-15

• If one of the confidence limits is outside the allowed range, but the parameter value is not, then the computed values are reported with the appropriate confidence limit at the null limit.

Example:
flux = 1E-15
flux_lolim = 0
flux_hilim = 1.4E-15

• If the parameter value and one of the confidence limits are outside the allowed range, then the parameter value is reported at the null limit and the appropriate confidence limit is set to INDEF.

Example:
flux = 0
flux_lolim = INDEF
flux_hilim = 1.4E-15

## wavdetect

wavdetect is a source detection algorithm that operates on its input in two stages: first, it detects putative source pixels within a dataset by repeatedly correlating it with "Mexican Hat" wavelet functions with different scale sizes. At each scale, the original image is correlated with the wavelet. Pixels with sufficiently large positive correlation values are removed from the image (as assumed sources), and subsequent correlations are performed at the same scale. This procedure of extracting source pixels from an image is called "cleansing". Typically, when very few source candidates are being found, or when an iteration-count limit is reached, the cleansing process stops. At this point, the estimated background (estimated from the cleansed image) is used to set detection thresholds, which are applied to the initial correlation image array values to identify putative sources. A set of outputs for the given wavelet scale is generated: a table of candidate sources (identified by correlation maxima), an image of the correlation of the data with the wavelet function, an image of the normalized (or flat-field) background (the image minus the "source" pixels), and the normalized background error image. Then the tool moves on to the next scale and repeats the process with a fresh copy of the image.

The second stage generates a source list from information from the first stage at each wavelet scale [correlation maxima tables, normalized (flat-field) background images (with errors), and correlation images]. Each correlation maximum from stage 1 is tested to see if it represents an independent, new source, or a source seen at other scales. For each source, a cell is computed that contains the majority of the source flux, and within that cell, source properties (location, count flux, etc.) are computed.