This page displays the results of your object search. The format of the
display depends on your search type and the output you requested.
HTML-formatted text is designed to be easily read by eye on your computer
display, while ASCII and XML VOTable output is appropriate for computerized
applications. This help page discusses the output as formatted for the HTML
pages, but the comments about the data here also apply to the data displayed
in any of the output formats. All formats have headers briefly explaining
the data output.
If you searched
NED using "Near Name", "Near Position", or "IAU Format", the objects are
listed in order of increasing distance from the center of the field. If
you used a "By Name" (with the extended name search flag set), "By Refcode",
"By Classifications", or "By Parameters" search, the objects are listed
in order of increasing Right Ascension. You may choose a different sort
order on the search input pages in the "Output Parameter" section.
By default, if fewer than five objects are returned by a search, their Basic
Data will be included on the same page as the summary list. If more than
five objects are found, click on the sequential number at the beginning
of each line to see the Basic Data for that object. The number of objects
with detailed data that will be displayed on the same page as the summary
list may also be changed in the "Output Parameter" section of the search
NED's default is to use an ASCII table for the summary list; this displays
long lists more quickly. An option on the search input page allows you to
choose an HTML-formatted summary list; this is easier to read, but is slower
to display long lists.
If you wish to make a finding chart of the objects returned by your
search, click the highlighted "Skyplot". This will display a table
with the coordinate data for your objects. If you wish, you may
modify the table before plotting the data.
Summary list of objects found
The following information is shown on this page:
A list of your input search parameters including (when applicable to
The Source List is preceded by a header line, the number of objects
returned by your search, and a link to plot the search results on the sky.
The Source List itself has the following information for each NED object
meeting you input criteria:
- Position and redshift constraints,
- Flux and object type constraints,
- Cosmology parameters and redshift reference frame, and
- Search radius in arcminutes.
- A sequential object number applicable to this list only.
- One of NED's names for the object; other names can be seen in the
"Cross-Identification" list below. An asterisk
preceding the name indicates the presence of an "Essential Note";
these follow the Data Index for each object.
- Right Ascension and Declination for the object at your requested
output equinox (default is J2000.0).
- NED's "Preferred Object Type" (e.g. "G" = galaxy, "GPair" = double
galaxy, "RadioS" = radio source, etc.).
- If you used the "By Classifications, Types, Attributes" search, NED's
standardized classifications are shown if available; the Morphological
Type is first, the Spectral Type follows.
- The heliocentric redshift, if known, listed first as
V = cz for V < 30,000 km/sec, then as z.
- A qualifier on the redshift. These are often taken from the published
source (a colon ":" or a question mark "?" are common), but are sometimes
added by NED to indicate a non-spectroscopic origin for the redshift --
"PHOT" for photometrically-measured redshifts, or "EST" for
redshifts estimated by other methods.
- NED's Basic Data magnitude and filter, generally in an optical band.
- Distance in arcminutes from the search position or from the position
of the named object, if applicable.
- Number of literature references in NED for the object.
- Number of literature or catalog Notes stored by NED for the object.
- Number of photometric data points stored by NED for the object.
- Number of position data points stored by NED for the object.
- Number of redshift data points stored by NED for the object.
- Number of diameter data points stored by NED for the object.
- Number of NED associations for the object.
- Link to the images for the object in NED's image archive.
- Link to the spectra for the object in NED's spectral archive.
- Finally, the sequential object number from the first column is repeated
Because NED is frequently updated, any of these data may be changed or
supplemented when more or better data become available.
Also, because NED's images and spectra are currently stored in different
internal databases, we do not yet display the number of images or spectra
held for a given object. However, clicking on the "Retrieve" link in the
"Images" or "Spectra" columns will retrieve all images or spectra currently
Detailed information for each object
This section of the output page is preceded by an Index containing
links that will display all of NED's data for each object. Each
section is blocked off on the screen for readability.
The Index also has a cutout from the Digitized Sky Survey for the object,
and -- if there are any data points in NED's photometric database for the
object -- a postage stamp version of the Spectral Energy Distribution.
These link to the image database and the photometric database, respectively.
Note that the DSS preview images, generated "on the fly" during your object
search, are centered on NED's current position for the object.
Many DSS images -- specifically for large galaxies, and for groups and
clusters -- are included in NED's image database accessible through
the image link. These images are independently-generated cutouts centered
on earlier, possibly less-accurate positions. While we update these
"pre-cutouts" periodically if they differ by more than 10 arcseconds from
current positions, there remains the possibility that these "pre-cutout"
positions will be centered on slightly different positions than the
"on-the-fly" DSS images.
Finally, the Index has a list of links to NED's databases of spectra,
redshift-independent distances, literature references, redshifts,
diameters, positions, notes, associations, and major catalogs.
Click on any of these links to see the additional data.
Here is detailed information about each data block:
- Essential Notes calling attention to e.g. errors in positions
or identifications in published catalogs, confused radio sources,
- All the Cross-Identifications in NED for the object, originating
from a process using positions (with uncertainties), redshifts, and
morphological information, as available in the related catalogs and
journal articles. The name prefixes are linked to information about
the catalog or paper in which the name originates.
- Coordinates, uncertainty, and reference:
- The reference code for the position adopted by NED.
This is linked to NED's bibliographic database so that you
may easily see the full reference and (for papers later than
1988) the paper's abstract.
- Equatorial and ecliptic positions for B1950.0 and J2000.0.
Equatorial positions are given in sexagesimal units as well
as in decimal degrees.
- Galactic and supergalactic positions in decimal degrees.
We compute Galactic coordinates following standard IAU practice
with the north Galactic pole at RA = 12h 49m, Dec = +27d 24m and
the longitude origin at 17h 42.4m, -28d 55m (B1950.0).
We follow RC3 and calculate supergalactic coordinates with the
north supergalactic pole at lII = 47.37d, bII
= +6.32d, and the longitude origin at lII = 137.37d,
bII = 0.0d.
- Position uncertainties in arcseconds given as the 95% confidence
ellipse semi-axes and position angle; or, for galaxy pairs,
triples, and other multiplets, the radius of a circle enclosing
the positions of the two or more galaxies.
- A summary table of
Redshift-Independent Distances showing the
mean distance modulus (in magnitudes) and metric distance (in
megaparsecs) with their standard deviations, and the minimum,
maximum, and median values. There is also a link to the Detailed Listing of
individual values in NED's redshift-independent database.
- A summary table of Classifications (Types and Attributes) with
one classification from each of the following categories, when
available: "Galaxy Morphology," "Radio Morphology", "Distance
Indicator", "Kinematics", "Activity Type", "Luminosity Class", and
"Hierarchy". A link to the Detailed Listing of the individual
classifications is provided, too.
- Three different estimates of the Foreground Galactic Extinction.
See this FAQ page for more
information and cautions on using the extinction values returned
by your search.
- The first extinction estimate gives the total absorption Aλ
in magnitudes at several optical and near-infrared wavelengths, based on the
Schlafly & Finkbeiner
(ApJ 737, 103, 2011) recalibration of the Schlegel, Finkbeiner & Davis
(ApJ 500, 525, 1998)
- The second extinction estimate is based on the the original prescription of
Schlegel, Finkbeiner & Davis
(ApJ 500, 525, 1998).
- Third is the total absorption in B magnitudes,
based on the prescription of
Burstein and Heiles
(ApJS 54, 33, 1984) and references therein), modified in the
area of M31 to show only the Galactic extinction, not the Galactic
extinction affected by the HI emission in M31. No extinction is
calculated when |b| < 10 degrees; a few other small gaps occur
in the Burstein-Heiles extinction maps (see
AJ 87, 1165, 1982) where no extinction can be calculated.
- NED's Basic Data for the object including:
- The heliocentric radial velocity or redshift, with its mean error
when known, and reference code (again linked to NED's bibliographic
- Additional parameters whose choice depends on the object type. The
self-explanatory parameter headings change depending on the
Preferred Object Type of the object. For example:
- major and minor axis diameters (in arcminutes),
- an indicative optical magnitude, often followed by a single
letter indicating the spectral band or filter (e.g. "B", "r")
- a morphological type.
- Small Associations of Galaxies (Pairs, Triples, Multiplets,
- optical magnitude of the brightest member, or the total
magnitude of the association,
- separation or diameter as appropriate,
- descriptive morphology,
- number of galaxies or population.
- Clusters of Galaxies:
- magnitude and rank of a cluster member (often the first-,
third-, or tenth-ranked galaxy),
- diameter in arcminutes,
- classification or type, and the classification system,
- indicative population or richness class.
- indicative optical magnitude,
- description (e.g. "blazar", "BSO", "Sy1").
- Infrared Sources:
- log(flux density S) in mJy; or magnitude, flagged with "m" or
with band pass or filter,
- wavelength in microns for the flux density,
- second log(flux density S) in mJy, or another magnitude,
- wavelength in microns for the second flux density,
- origin of the source (e.g. "IRAS FaintSrcCat(v2)").
- Radio Sources:
- log(flux density S, in mJy),
- frequency (in GHz) for the flux density,
- spectral index, for S=A*freq^Index^,
- radio morphology (e.g., head-tail, double-lobe),
- size of source (in arcmin).
|NED's Basic Data -- diameters, magnitudes, morphological
types, flux densities, spectral indices, cluster populations or
richnesses, and so forth -- are generally NOT on homogeneous systems.
While we have used data from large compilations such as the
Sloan Digital Sky Survey,
The Third Reference Catalogue of Bright Galaxies (RC3),
the Hewitt-Burbidge compendia of QSO's,
or the Abell catalog of rich galaxy clusters
whenever possible, there are many hundreds of thousands of extragalactic
sources represented in NED that are not listed in these more
specialized catalogs. Thus, we have taken basic
parameters for these other objects from available
literature sources. We give preference to large catalogs
(e.g. SDSS, 2MASS, APM, IRAS, CGCG, UGC, 87GB) over shorter lists, but the
data have not been reduced to homogeneous systems. The Basic Data values
are intended only to provide convenient, representative, first-look
parameters for the object. Most objects in NED have more extensive,
detailed, multi-wavelength measurements for redshifts, photometry, and
diameters available using links in the Index section of the results page.
- Quantities Derived from the object's
redshift (if known) and position on the sky:
- Calculated and Corrected Velocities, with errors:
The errors in the model parameters for each correction are added in
quadrature to the error in the galaxy's redshift as follows: 4% of the
GSR correction, 6% of the Local Group correction, 7% of the 3K CMB
correction, and 7% of the velocity field correction.
- V (Heliocentric) in km/s with its error (if known) and source,
calculated from V = cz. No relativisitc correction is applied
to these apparent redshifts (see John
Huchra's discussion of extragalactic redshifts for more
information), so velocities in excess of the speed of light can
be displayed here.
- V (Galactocentric GSR) in km/s calculated as in
- V (Local Group) in km/s based on the formulation by Karachentsev
and Makarov (AJ 111, 794, 1996).
- V (3K CMB) in km/s using the CMB dipole model presented by
Fixsen et al. (ApJ 473, 576, 1996).
- V (Virgo Infall only) based on the local velocity field model
given in Mould et al. (ApJ 529, 786, 2000)
using only the term for the influence of the Virgo Cluster.
- V (Virgo + Great Attractor Infall only) again based on the local
velocity field model given in Mould et al.
(ApJ 529, 786, 2000)
using the terms for the influence of Virgo Cluster and the Great
Attractor, but not the Shapley Supercluster (we thank Jim Condon
for his code for this model, on which we have based ours).
See Mould et al. Appendix A or click here
for parameters of their local velocity field model.
- V (Virgo + Great Attractor + Shapley Supercluster), once
again based on the local velocity field model given in Mould
et al. (ApJ 529, 786, 2000)
using the terms for the influence of the Virgo Cluster, the
Great Attractor, and the Shapley Supercluster.
- Hubble Flow Distances and Distance Moduli, with their errors,
calculated from the apparent corrected velocities using
the cosmological parameters that you chose on the search input page
(defaults are H0 = 73 km/s/Mpc, Ωmatter
= 0.27, and Ωvacuum = 0.73).
No derived distances are given if the corrected velocity is
- Scale at these Hubble Flow Distances in parsec/arcsec,
kiloparsec/arcsec, kiloparsec/arcmin, and megaparsec/degree.
- You may search for physical companions with known redshifts with
an "Environment Search". The default values are calculated to
correspond to 750 kpc at the distance of the galaxy, and to be
plus or minus 500 km/s. You may change these values by entering
you desired numbers in the boxes. Click "Submit Environment
Search" to initiate the search.
- Several quantities derived from the redshift corrected to the
frame defined by the 3K background, and further corrected using
the cosmological parameters that you chose on the search input page
(again, defaults are H0 = 73 km/s/Mpc,
Ωmatter = 0.27, and Ωvacuum =
We thank Chris Burns (OCIW) for the code behind these calculations.
Further explanation of the calculated quantities is available
through Ned Wright's
Cosmology Calculator web site, and through Alberto Cappi's
CosmoTools web site.
|Calculated distances and scales for galaxies with
heliocentric velocities less than 500-700 km/s -- particularly in
the Virgo Cluster area -- should be treated with considerable
caution. Many of these nearby galaxies have individually-determined
distances given above in the list of redshift-independent distances.
These are generally much more accurate than the calculated distances
presented here, and should certainly be used in preference to the
- If you wish to change the cosmological parameters used to calculate
the derived values, you may do so by entering them in the
appropriate boxes. WMAP Three-Year (Spergel et al.
ApJS 170, 377, 2007)
and Five-Year (Komatsu et al.
ApJS 180, 330, 2009)
values may be entered by clicking the appropriate buttons. You
may also choose which redshift/velocity correction is used as input
for the calculations of luminosity distance and cosmology-corrected
quantities (the default is the redshift corrected to the 3K cosmic
microwave background). After you have made your selections, click
the "Submit Changed Hubble Parameters" button; the page will
refresh with the new derived values displayed.
- Quick-Look Photometry and Luminosities.
This section contains photometric data -- with errors, if available in the
original publication -- selected from NED's photometry database. The
brightest data point in each of ten major band passes is shown when
available. The bands are
| Bandpass || Frequency || Wavelength (microns) |
| Gamma-ray || ν > 2e19 || λ < 1.5e-5 |
| X-ray || 2e19 > ν > 2e16 || 1.5e-5 < λ < 0.015 |
| UV || 2e16 > ν > 9e14 || 0.015 < λ < 0.333 |
| Optical || 9e14 > ν > 3e14 || 0.333 < λ < 1.0 |
| Near-IR || 3e14 > ν > 6e13 || 1.0 < λ < 5.0 |
| Mid-IR || 6e13 > ν > 7.5e12 || 5.0 < λ < 40 |
| Far-IR || 7.5e12 > ν > 1e12 || 40 < λ < 300 |
| Sub-mm || 1e12 > ν > 3e11 || 300 < λ < 1000 |
| Millimeter || 3e11 > ν > 3e10 || 1000 < λ < 10000 |
| Radio || 3e10 > ν || 10000 < λ |
Also included are absolute magnitudes or luminosities
calculated using the average value of the
redshift-independent distance, or the distance calculated from the
redshift (if no redshift-independent distance
is available). No k-correction has been applied, nor have the data been
corrected for foreground extinction.
Finally, solar luminosities (νLν☉) have
assuming standard values for the solar bolometric luminosity and flux
(see e.g. Allen's Astrophysical Quantities, ed. A.N. Cox, 2000, page 340).
The calculation is
νLν☉ = νLν / 3.845 x 1026
νLν = ν * fν * 4πD2
where νLν☉ is in solar units,
is in watts, fν is the object's flux density in
watts/m2/Hz, and D is its distance in meters.
Should you need a luminosity referred to the sun in a specific bandpass
for which you have the object's absolute magnitude, you may calculate it
L/L☉ = 10-0.4*(M - M☉)
where M☉ is the sun's absolute magnitude in the bandpass.
The sun's absolute magnitude in several popular bands is in the following
table (see Allen's Astrophysical Quantities, ed. A.N. Cox, 2000, page 341,
Bohlin, and Castelli)
| Bandpass || M☉ |
| U || +5.66 |
| B || +5.47 |
| V || +4.82 |
| R || +4.28 |
| I || +3.94 |
| J || +3.71 |
| H || +3.37 |
| K || +3.33 |
and for the SDSS bands
| Bandpass || M☉ |
| u || +6.55 |
| g || +5.12 |
| r || +4.68 |
| i || +4.57 |
| z || +4.54 |
- Quick-Look Angular and Physical Diameters.
The largest visual and near-infrared diameters are listed, as available.
- Links to External Archives and
This section of your NED results contains two subsections distinguished by the nature of their hyperlinks.
Links listed under "Data Directly Related to Object Names" are made as
The list of NED names for an object returned by your search is compared
with a list of related on-line astronomical services that are
specific to the survey or catalog associated with the NED names.
For example, if the
object has an NVSS name, a 2MASS name, and a UGC name, only links to
data services referring to NVSS, 2MASS, and UGC are listed.
Click on the links to access these external data services.
Because these links are keyed on object names based on multi-wavelength
cross-identifications, the data returned are
relevant to your object with a high degree of certainty and reliability
(but see Notes and Caveats below).
Services currently accessed include catalog queries through VizieR (CDS)
and LEDA (Lyon Extragalactic Database);
and image and catalog data for large surveys such as 2MASS (IRSA, IPAC),
FIRST (NRAO), and NVSS (also at NRAO).
The second subsection lists links under "General Archive Resources".
These are not necessarily tied to a specific name prefix.
When the service supports a name-based query, the primary NED name
(the first name listed in the Object Names columns above) is used;
query is based on the position of the object. These external
or may not contain observations relevant to the NED objects in your list;
you will have to run the query and explore the results.
(Again, see Notes and Caveats below).
On-line resources currently queried here include infrared
mission archives at IRSA (2MASS, MSX, IRAS, WISE), optical and UV mission archives
at MAST (HST, IUE, EUVE, GALEX, etc.), high energy mission archives at
HEASARC (HEAO, GINGA, EXOSAT, EINSTEIN, XTE, ROSAT, etc.),
the IMPReSS tool used to visualize observation coverage maps (GSFC),
the Astrobrowse tool (HEASARC) used to explore astronomical resources,
the SDSS Sky Server,
the UZC Spectral Archive at SAO,
the NVSS image archive at NRAO,
the ATNF Observation Log,
the VLA Observation Log,
and a name-based object query to SIMBAD.
Notes and Caveats
NED does not automatically poll all of the above services and
include a link only if data are available. This would take too much time --
we want to display your NED query results as quickly as possible.
While many queries to external services will return relevant data,
some queries may return nothing because observations for the given
object (or near its position) may not be available in all archives.
Results from an external archive may be unexpected. For example, a query
to MAST on "3C 279", returns not only IUE, EUVE, and HST observations of
this QSO, but also an observation of the planet Mars with the GHRS
instrument on HST. This is not an error, but is due to the fact that MAST
-- and many other archive services -- do position searches rather than
name searches. The NED name is converted to a position and a default
radius, and all observations within that search radius are returned. In
this example, it may be of interest to know that Mars has appeared close
to 3C 279. This could spur, for example, an archival investigation into
possible occultation measurements with resultant information about the
apparent size of the QSO, Mars's atmosphere, etc.
Objects with name prefixes or suffixes added by NED,
such as "NED01", square brackets, colons, etc., are stripped off before
building the link to the external archive. Typically, this means that
the search will return data relevant to the parent object and not to an
object with name prefixes or suffixes. For example, the link to a catalog
search for "NGC 3690 NED01" becomes "NGC 3690", so the data returned by
the external service will refer to NGC 3690.
Also note that the availability of these external resources is
under control of the corresponding external servers. Thus, the response
time to your query will depend on the state of the external server -- its
Internet connectivity and its usage load, and even whether it is currently
on-line or not.
Finally, a few of the SDSS spectroscopically-observed objects will not link
correctly with the SDSS server through the "External Archives and Services"
link that appears for them in this section (an example is
SDSS J132958.87+471557.4, a knot near the nucleus of NGC 5195 = Messier
These are usually found in sections of the SDSS that have been masked
because of the presence of nearby bright objects (in the case of this
example, Messier 051 itself).
Notes on Coordinates in NED
The positional uncertainty is given as the semi-major and semi-minor
axes of the 95% confidence ellipse, and the position angle (East of
North) of the major axis of the ellipse. The two-dimensional 95%
confidence ellipse corresponds to 2.5 sigma (where sigma is the
standard deviation). Note that this is not the same as the
one-dimensional 95% confidence interval; that corresponds to 1.96
sigma, assuming a normal (Gaussian) distribution of residuals about
the mean value.
A few large lists of positions which do NOT appear in print are given
bibliographic reference codes which point to papers explaining the
measurement and reduction techniques. For example, positions taken
from the Guide Star Catalog carry a reference pointing to "The Guide
Star Catalog. II. Photometric and Astrometric Models and Solutions"
in AJ 99, 2059, 1990; and positions for reference galaxies in the Lick
Northern Proper Motion survey carry a reference to "Lick Northern
Proper Motion Program. I. Goals, Organization, and Methods" in AJ 94,
Some unpublished positions measured by members of the NED team are
referenced as "Internal NED Report", followed by the team member's
On most NED pages, the displayed precision of a position is dependent
upon the 95% confidence ellipse (2.5 times the standard deviation)
assigned to the semi-minor axis of the position uncertainty ellipse. Here
is a table, with examples, showing the displayed coordinate precision for
different ranges of uncertainty.
| 95% CI || CI/2.5 || RA || Dec |
|(arcsec) || (arcsec) || || |
|>= 750 || 300 || 04h30m33s || +05d15m |
|>= 75 || 30 || 04h30m31s || +05d15.0m |
|>= 12.5 || 5 || 04h30m31.6s || +05d15m00s |
|>= 1.25 || 0.5 || 04h30m31.60s || +05d14m59.6s |
|>= 0.125 || 0.05 || 04h30m31.605s || +05d14m59.59s |
|>= 0.0125 || 0.005 || 04h30m31.6048s || +05d14m59.594s |
|>= 0.00125 || 0.0005 || 04h30m31.60485s || +05d14m59.5943s |
|>= 0.000125|| 0.00005 || 04h30m31.604856s || +05d14m59.59428s |
| < 0.000125|| 0.00005 || 04h30m31.6048560s|| +05d14m59.594280s |
For positions displayed as decimal degrees, we use the following rules:
| CI/2.5 || Digits displayed |
|(degrees) || after decimal point |
| 0.5 || 1 |
| 0.05 || 2 |
| 0.005 || 3 |
| 0.0005 || 4 |
| . || |
| . || |
| . || |
| 5.0E(-n) || n+1 |
|where n <= 8.|| |
Only the calculator and the initial object list page displayed after an
object search do not follow these rules. The
calculator always displays 7 significant digits in RA seconds and 6 in Dec
(i.e. a 95% confidence ellipse of >= 0.00125 arcsec), while the object
list always gives 3 significant digits in RA seconds and 2 in Dec (95% CI
of >= 12.5 arcsec).
If you request data via a batch job, the positions will
be in the fixed format
regardless of the accuracy stored in NED.
Notes on the Local Velocity Field Model
V (Local Infall) is calculated from Equation A2 of Mould et al.
(ApJ 529, 786, 2000), with signs corrected:
V(Local Infall) = VSun + ΔVLG + ΔVVirgo + ΔVGA + ΔVShapley
Each of the Δ terms represents one of the velocity corrections for the
attraction of a nearby mass concentration. As explained above, we adopt the
formulation for the Local Group correction from Karachentsev and Makarov
(AJ 111, 794, 1996).
Otherwise, we follow Mould et al. explicitly by adopting their three
attractors as detailed in their Table A1 reproduced, with corrections, here:
| Cluster || RA || Dec
|| VH ||
VLG || VField || Radius
|| VRange |
| Virgo || 12h28m19s || +12d40.0m || 1035 ||
957 || 200 || 10 || 600 - 2300 |
| Great Attractor || 13h20.0m || -44d00m || 4600 ||
4380 || 400 || 10 || 2600 - 6600 |
| Shapley || 13h30.0m || -31d00m || 13800 ||
13600 || 85 || 12 || 10000 - 16000 |
All velocities in the table are in km/s. VH is the observed mean
heliocentric velocity of the cluster, VLG is the mean cluster
velocity corrected to the center of the Local Group using the standard
300 sin(l)cos(b) calculation, VField is the adopted model infall
velocity at the position of the Local Group, "Radius" is the adopted radius of
the cluster in degrees, and VRange is the heliocentric velocity
range within which cluster galaxies are found.
This model assumes that all
galaxies within "Radius" of the center of the cluster (given by RA and Dec),
and within the given velocity range, are cluster members. Any galaxy
meeting these criteria is assigned the mean heliocentric velocity of the
cluster for calculation of its distance.
Note that the declinations of the Great Attractor and the Shapley
Supercluster given in Table A1 of Mould et al. should be negative, and
that the minus signs in their Equation A2 should all be positive.
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