Astro::Coord::ECI - Manipulate geocentric coordinates
use Astro::Coord::ECI; use Astro::Coord::ECI::Sun; use Astro::Coord::ECI::TLE; use Astro::Coord::ECI::Utils qw{rad2deg}; # 1600 Pennsylvania Avenue, in radians, radians, and KM my ($lat, $lon, $elev) = (0.678911227503559, -1.34456123391096, 0.01668); # Record the time my $time = time (); # Set up observer's location my $loc = Astro::Coord::ECI->geodetic ($lat, $lon, $elev); # Uncomment to turn off atmospheric refraction if desired. # $loc->set( refraction => 0 ); # Instantiate the Sun. my $sun = Astro::Coord::ECI::Sun->universal ($time); # Figure out if the Sun is up at the observer's location. my ($azimuth, $elevation, $range) = $loc->azel ($sun); print "The Sun is ", rad2deg ($elevation), " degrees above the horizon.\n";
See the Astro::Coord::ECI::TLE documentation for an example involving satellite pass prediction.
In version [% next_version %] I very belatedly took cognizance of the fact that in the Southern hemisphere the seasons are reversed. The result is an incompatible change in the text description of the equinox and solstice events generated by the Astro::Coord::ECI::Sun object if its station
attribute is set to a location south of the Equator. If this attribute is not set at all you get the Northern-hemisphere text.
This module was written to provide a base class for a system to predict satellite visibility. Its main task is to convert the Earth-Centered Inertial (ECI) coordinates generated by the NORAD models into other coordinate systems (e.g. latitude, longitude, and altitude above mean sea level), but a other functions have accreted to it. In addition, a few support routines have been exposed for testing, or whatever.
All distances are in kilometers, and all angles are in radians (including right ascension, which is usually measured in hours).
Times are normal Perl times, whether used as universal or dynamical time. Universal time is what is usually meant, unless otherwise stated.
Known subclasses include Astro::Coord::ECI::Moon to predict the position of the Moon, Astro::Coord::ECI::Star to predict the position of a star, or anything else that can be considered fixed on the celestial sphere, Astro::Coord::ECI::Sun to predict the position of the Sun, Astro::Coord::ECI::TLE to predict the position of a satellite given the NORAD orbital parameters, and Astro::Coord::ECI::TLE::Iridium (a subclass of Astro::Coord::ECI::TLE) to predict Iridium flares.
Note that in version 0.022_01 the velocity code got a substantial rework, which is still in progress. I am attempting give useful velocities where they are available, and no velocities at all where they are not. Unfortunately I have yet to locate any worked problems, so the velocity code is, in the most literal sense, untested.
In general, locations specified in Earth-fixed coordinates are considered to share the rotational velocity of the Earth, and locations specified in inertial coordinates are considered to have undefined velocities unless a velocity was specified explicitly or produced by the object's model. This involved a change in the behavior of the eci() method when velocity was not specified but location was. See the next paragraph for my excuse.
Caveat user: This class and its subclasses should probably be considered alpha code, meaning that the public interface may not be completely stable. I will try not to change it, but given sufficient reason I will do so. If I do so, I will draw attention to the change in the documentation.
The following methods should be considered public.
This method instantiates a coordinate object. Any arguments are passed to the set() method once the object has been instantiated.
This method calculates the angle between the $coord2 and $coord3 objects, as seen from $coord. The calculation uses the law of haversines, and does not take atmospheric refraction into account. The return is a number of radians between 0 and pi.
The algorithm comes from "Ask Dr. Math" on Drexel's Math Forum, http://mathforum.org/library/drmath/view/51879.html, which attributes it to the Geographic Information Systems FAQ, http://www.faqs.org/faqs/geography/infosystems-faq/, which in turn attributes it to R. W. Sinnott, "Virtues of the Haversine," Sky and Telescope, volume 68, number 2, 1984, page 159.
Prior to version 0.011_03 the law of cosines was used, but this produced errors on extremely small angles. The haversine law was selected based on Jean Meeus, "Astronomical Algorithms", 2nd edition, chapter 17 "Angular Separation".
This method ignores the station
attribute.
This method returns the name of the class that implements the named attribute, or undef if the attribute name is not valid.
This method takes another coordinate object, and computes its azimuth, elevation, and range in reference to the object doing the computing. The return is azimuth in radians measured clockwise from North (always positive), elevation above the horizon in radians (negative if below), and range in kilometers.
As a side effect, the time of the $coord object may be set from the $coord2 object.
Atmospheric refraction is taken into account using the correct_for_refraction()
method if the $coord
object's refraction
attribute is true. The invocant's correct_for_refraction()
method is the one used; that is, if $coord
and $coord2
have different correct_for_refraction()
methods, the $coord
object's method is used.
This method is implemented in terms of azel_offset(). See that method's documentation for further details.
This method computes the azimuth, elevation, and range if the $coord
object as seen from the position stored in the $coord
object's station
attribute. An exception will be thrown if the station
attribute is not set.
This method takes another coordinate object, and computes its azimuth, elevation, and range in reference to the object doing the computing. The return is azimuth in radians measured clockwise from North (always positive), elevation above the horizon in radians (negative if below), and range in kilometers.
If the optional $offset
argument is provided, the elevation is offset upward by the given fraction of the radius of the $coord2
object. Thus, an $offset
of 1
specifies the upper limb of the object, 0
specifies the center of the object, and -1
specifies the lower limb. This offset is applied before atmospheric refraction.
As a side effect, the time of the $coord object may be set from the $coord2 object.
By default, atmospheric refraction is taken into account in the calculation of elevation, using the correct_for_refraction()
method. This better represents the observed position in the sky when the object is above the horizon, but produces a gap in the data when the object passes below the horizon, since I have no refraction equations for rock. See the correct_for_refraction()
documentation for details.
The invocant's correct_for_refraction()
method is the one used; that is, if $coord
and $coord2
have different correct_for_refraction()
methods, the $coord
object's method is used.
If you want to ignore atmospheric refraction (and not have a gap in your data), set the refraction attribute of the $coord object to any value Perl considers false (e.g. 0).
The algorithm for position is the author's, but he confesses to having to refer to T. S. Kelso's "Computers and Satellites" column in "Satellite Times", November/December 1995, titled "Orbital Coordinate Systems, Part II" and available at http://celestrak.com/columns/v02n02/ to get the signs right.
If velocities are available for both bodies involved in the calculation, they will be returned after the position (i.e. you get a six-element array back instead of a three-element array). The velocity of a point specified in Earth-fixed coordinates (e.g. geodetic latitude, longitude, and altitude) is assumed to be the rotational velocity of the Earth at that point. The returned velocities are azimuthal velocity in radians per second (not radians of azimuth, which get smaller as you go farther away from the Equator), elevational velocity in radians per second, and velocity in recession in kilometers per second, in that order.
If velocities are available for both bodies and the $coord2
object has its frequency
attribute set, the returned array will contain seven elements, with the seventh being the Doppler shift.
The algorithm for recessional velocity comes from John A. Magliacane's Predict
program, available at http://www.qsl.net/kd2bd/predict.html. The transverse velocity computations are the author's, but use the same basic approach: vector dot product of the velocity vector with a unit vector in the appropriate direction, the latter generated by the appropriate (I hope!) vector cross products.
Velocity conversions to azel()
appear to me to be mostly sane. See A NOTE ON VELOCITIES, below, for details.
If velocities are available and you have provided a non-zero value for the frequency
attribute, you will get the Doppler shift as the seventh element of the returned array. The caveats about velocity in recession apply to the Doppler shift as well. The frequency can come from either the $coord
or $coord2
object, but the $coord2
is preferred, and getting frequency from the $coord
object is being put through a deprecation cycle and removed.
This method computes the azimuth, elevation, and range if the $coord
object as seen from the location stored in the $coord
object's station
attribute. The functionality is as above, except for the fact that in the above version the station is the invocant, whereas in this version the orbiting body is the invocant.
This version also returns velocities if they are available for both bodies, and Doppler shift if in addition the frequency
attribute of the invocant is set.
This method does a deep clone of an object, producing a different but identical object.
At the moment, it's really just a wrapper for Storable::dclone.
This method corrects the given angular elevation for atmospheric refraction. This is done only if the elevation has a reasonable chance of being visible; that is, if the elevation before correction is not more than two degrees below either the 'horizon' attribute or zero, whichever is lesser. Sorry for the discontinuity thus introduced, but I did not wish to carry the refraction calculation too low because I am uncertain about the behavior of the algorithm, and I do not have a corresponding algorithm for refraction through magma.
This method can also be called as a class method, in which case the correction is applied only if the uncorrected elevation is greater than minus two degrees. It is really only exposed for testing purposes (hence the cumbersome name). The azel() method calls it for you if the refraction attribute is true.
The algorithm for atmospheric refraction comes from Thorfinn Saemundsson's article in "Sky and Telescope", volume 72, page 70 (July 1986) as reported Jean Meeus' "Astronomical Algorithms", 2nd Edition, chapter 16, page 106, and includes the adjustment suggested by Meeus.
This method calculates the dip angle of the horizon due to the altitude of the body, in radians. It will be negative for a location above the surface of the reference ellipsoid, and positive for a location below the surface.
The algorithm is simple enough to be the author's.
This method sets the dynamical time represented by the object.
This method can also be called as a class method, in which case it instantiates the desired object.
The algorithm for converting this to universal time comes from Jean Meeus' "Astronomical Algorithms", 2nd Edition, Chapter 10, pages 78ff.
This method returns the dynamical time previously set, or the universal time previously set, converted to dynamical.
The algorithm comes from Jean Meeus' "Astronomical Algorithms", 2nd Edition, Chapter 10, pages 78ff.
This method sets the coordinates represented by the object in terms of "Earth-Centered, Earth-fixed (ECEF) coordinates" in kilometers, with the x axis being latitude 0 longitude 0, the y axis being latitude 0 longitude 90 degrees east, and the z axis being latitude 90 degrees north. The velocities in kilometers per second are optional, and will default to zero. ECEF velocities are considered to be relative to the surface of the Earth; when converting to ECI, the rotational velocity of the Earth will be added in.
Note that prior to version 0.022_01, the documentation said that the default velocity was the rotational velocity of the Earth. This was not correct in ECEF coordinates. The functionality of the code itself in this respect did not change.
The object itself is returned.
This method can also be called as a class method, in which case it instantiates the desired object.
This method returns the object's "Earth-Centered, Earth-fixed (ECEF) coordinates".
If the original coordinate setting was in an inertial system (e.g. eci, equatorial, or ecliptic) and the absolute difference between the current value of 'equinox_dynamical' and the current dynamical() setting is greater than the value of $Astro::Coord::ECI::EQUINOX_TOLERANCE, the coordinates will be precessed to the current dynamical time before conversion. Yes, this should be done any time the equinox is not the current time, but for satellite prediction precession by a year or so does not seem to make much difference in practice. The default value of $Astro::Coord:ECI::EQUINOX_TOLERANCE is 365 days. Note that if this behavior or the default value of $EQUINOX_TOLERANCE begins to look like a bug, it will be changed, and noted in the documentation.
Some velocity conversions involving ecef()
appear to me to be mostly sane. See A NOTE ON VELOCITIES, below, for details.
This method sets the coordinates represented by the object in terms of "Earth-Centered Inertial (ECI) coordinates" in kilometers, time being universal time, the x axis being 0 hours "Right Ascension" 0 degrees "Declination", y being 6 hours "Right Ascension" 0 degrees "Declination", and z being 90 degrees north "Declination". The velocities in kilometers per second are optional. If omitted, the object will be considered not to have a velocity. This is a change introduced with version 0.022_01. Before that, the velocity defaulted to 0.
The time argument is optional if the time represented by the object has already been set (e.g. by the universal() or dynamical() methods).
The object itself is returned.
This method can also be called as a class method, in which case it instantiates the desired object. In this case the time is not optional.
The algorithm for converting from ECI to geocentric coordinates and back is based on the description of ECI coordinates in T. S. Kelso's "Computers and Satellites" column in "Satellite Times", September/October 1995, titled "Orbital Coordinate Systems, Part I" and available at http://celestrak.com/columns/v02n01/.
This method returns the "Earth-Centered Inertial (ECI) coordinates" of the object at the given time. The time argument is actually optional if the time represented by the object has already been set.
If you specify a time, the time represented by the object will be set to that time. The net effect of specifying a time is equivalent to
($x, $y, $z, $xdot, $ydot, $zdot) = $coord->universal($time)->eci()
If the original coordinate setting was in a non-inertial system (e.g. ECEF or geodetic), the equinox_dynamical attribute will be set to the object's dynamical time.
Velocities will be returned if they were originally provided. This is a change introduced in version 0.022_01. Prior to that version, velocities were always returned.
Some velocity conversions involving eci()
appear to me to be mostly sane. See A NOTE ON VELOCITIES, below, for details.
This method sets the Cartesian "Ecliptic" coordinates represented by the object in terms of X
(kilometers toward the vernal equinox), Y
(90 degrees along the ecliptic from X
), and Z
(toward ecliptic North). The time is universal time. The object itself is returned.
The time argument is optional if the time represented by the object has already been set (e.g. by the universal() or dynamical() methods).
This method can also be called as a class method, in which case it instantiates the desired object. In this case the time is not optional.
Caveat: you can pass in velocities (before the $time
) but they are unsupported, meaning that I can not at this point say whether they will be transformed sanely, much less correctly. Caveat programmer.
This method returns the Cartesian ecliptic coordinates of the object at the given time. The time is optional if the time represented by the object has already been set (e.g. by the universal() or dynamical() methods).
Caveat: velocities are unsupported by this method. That means you may (or may not, depending on the coordinates originally set) get them back, but I have not looked into whether they are actually correct.
This method sets the "Ecliptic" coordinates represented by the object in terms of "Ecliptic latitude" and "Ecliptic longitude" in radians, and the range to the object in kilometers, time being universal time. The object itself is returned.
The time argument is optional if the time represented by the object has already been set (e.g. by the universal() or dynamical() methods).
The latitude should be a number between -PI/2 and PI/2 radians inclusive. The longitude should be a number between -2*PI and 2*PI radians inclusive. The increased range (one would expect -PI to PI) is because in some astronomical usages latitudes outside the range + or - 180 degrees are employed. A warning will be generated if either of these range checks fails.
This method can also be called as a class method, in which case it instantiates the desired object. In this case the time is not optional.
The algorithm for converting from ecliptic latitude and longitude to right ascension and declination comes from Jean Meeus' "Astronomical Algorithms", 2nd Edition, Chapter 13, page 93.
This method returns the ecliptic latitude and longitude of the object at the given time. The time is optional if the time represented by the object has already been set (e.g. by the universal() or dynamical() methods).
Caveat: velocities are not returned by this method.
This method returns the ecliptic longitude of the body at its current time setting, in radians. It is really just a convenience method, since it is equivalent to ( $coord->ecliptic() )[1]
, and in fact that is how it is implemented.
This method sets the "Equatorial" coordinates represented by the object (relative to the center of the Earth) in terms of "Right Ascension" and "Declination" in radians, and the range to the object in kilometers, time being universal time. The object itself is returned.
If equatorial()
is called in this way, the station
attribute will be ignored, for historical reasons.
The right ascension should be a number between 0 and 2*PI radians inclusive. The declination should be a number between -PI/2 and PI/2 radians inclusive. A warning will be generated if either of these range checks fails.
The time argument is optional if the time represented by the object has already been set (e.g. by the universal() or dynamical() methods).
This method can also be called as a class method, in which case it instantiates the desired object. In this case the time is not optional.
You may optionally pass velocity information in addition to position information. If you do this, the velocity in right ascension (in radians per second), declination (ditto), and range (in kilometers per second in recession) are passed after the position arguments, and before the $time argument if any.
This method returns the "Equatorial" coordinates of the object at the relative to the center of the Earth. The station
attribute is ignored. The time argument is optional if the time represented by the object has already been set (e.g. by the universal() or dynamical() methods).
If velocities are available from the object (i.e. if it is an instance of Astro::Coord::ECI::TLE) the return will contain velocity in right ascension, declination, and range, the first two being in radians per second and the third in kilometers per second in recession.
Caveat: these velocities are believed by me to be sane if they are derived from ECI coordinates. This method does not support setting velocities. See A NOTE ON VELOCITIES, below, for details.
This method is retained (for the moment) for historical reasons, but equatorial_apparent()
is preferred.
This method returns the apparent equatorial coordinates of the object represented by $coord2, as seen from the location represented by $coord.
As a side effect, the time of the $coord object may be set from the $coord2 object.
If the refraction attribute of the $coord object is true, the coordinates will be corrected for atmospheric refraction using the correct_for_refraction() method.
If velocities are available from both objects (i.e. if both objects are Astro::Coord::ECI::TLE objects) the return will contain velocity in right ascension, declination, and range, the first two being in radians per second and the third in kilometers per second in recession.
Caveat: these velocities are believed by me not to be correct.
This method returns the apparent equatorial coordinates of the $coord
object as seen from the object in the station
attribute of the $coord
object.
This method will return velocities if available, but I have no idea whether they are correct.
This method computes the unreduced equatorial position of the second ECI object as seen from the first. If the second argument is undefined, computes the equatorial position of the first object as seen from the center of the Earth. Unlike the equatorial() method itself, this method is an accessor only. This method would probably not be exposed except for the anticipation of the usefulness of $range and $v_r in satellite conjunction computations, and the desirability of not taking the time to make the two atan2() calls that are unneeded in this application.
The 'unreduced' in the name of the method is intended to refer to the fact that the $rasc and $decl are not the right ascension and declination themselves, but the arguments to atan2() needed to compute them, and $v_rasc and $v_decl are in km/sec, rather than being divided by the range to get radians per second.
This method ignores the 'station'
attribute.
The returned data are:
$rasc is an array reference, which can be converted to the right ascension in radians by mod2pi(atan2($rasc->[0], $rasc->[1])).
$decl is an array reference, which can be converted to the declination in radians by atan2($decl->[0], $decl->[1]).
$range is the range in kilometers.
$v_rasc is the velocity in the right ascensional direction in kilometers per second. It can be converted to radians per second by dividing by $range.
$v_decl is the velocity in the declinational direction in kilometers per second. It can be converted to radians per second by dividing by $range.
$v_r is the velocity in recession in kilometers per second. Negative values indicate that the objects are approaching.
The velocities are only returned if they are available from the input objects.
Caveat: these velocities are believed by me not to be correct.
This method sets the value of the equinox_dynamical attribute, and returns the modified object. If called without an argument, it returns the current value of the equinox_dynamical attribute.
Yes, it is equivalent to $coord->set (equinox_dynamical => $value) and $coord->get ('equinox_dynamical'). But there seems to be a significant performance penalty in the $self->SUPER::set () needed to get this attribute set from a subclass. It is possible that more methods like this will be added, but I do not plan to eliminate the set() interface.
This method sets the "Geocentric" coordinates represented by the object in terms of "Geocentric latitude" psiprime and "Longitude" lambda in radians, and distance from the center of the Earth rho in kilometers.
The latitude should be a number between -PI/2 and PI/2 radians inclusive. The longitude should be a number between -2*PI and 2*PI radians inclusive. The increased range (one would expect -PI to PI) is because in some astronomical usages latitudes outside the range + or - 180 degrees are employed. A warning will be generated if either of these range checks fails.
This method can also be called as a class method, in which case it instantiates the desired object.
This method should not be used with map coordinates because map latitude is "Geodetic latitude", measured in terms of the tangent of the reference ellipsoid, whereas geocentric coordinates are, essentially, spherical coordinates.
The algorithm for conversion between geocentric and ECEF is the author's.
This method returns the "Geocentric latitude", "Longitude", and distance to the center of the Earth.
This method sets the "Geodetic" coordinates represented by the object in terms of its "Geodetic latitude" psi and "Longitude" lambda in radians, and its height h above mean sea level in kilometers.
The latitude should be a number between -PI/2 and PI/2 radians inclusive. The longitude should be a number between -2*PI and 2*PI radians inclusive. The increased range (one would expect -PI to PI) is because in some astronomical usages latitudes outside the range + or - 180 degrees are employed. A warning will be generated if either of these range checks fails.
The ellipsoid argument is the name of a "Reference Ellipsoid" known to the class, and is optional. If passed, it will set the ellipsoid to be used for calculations with this object. If not passed, the default ellipsoid is used.
This method can also be called as a class method, in which case it instantiates the desired object.
The conversion from geodetic to geocentric comes from Jean Meeus' "Astronomical Algorithms", 2nd Edition, Chapter 11, page 82.
This is the method that should be used with map coordinates.
This method returns the geodetic latitude, longitude, and height above mean sea level.
The ellipsoid argument is the name of a "Reference ellipsoid" known to the class, and is optional. If not specified, the most-recently-set ellipsoid will be used.
The conversion from geocentric to geodetic comes from Kazimierz Borkowski's "Accurate Algorithms to Transform Geocentric to Geodetic Coordinates", at http://www.astro.uni.torun.pl/~kb/Papers/geod/Geod-BG.htm. This is best viewed with Internet Explorer because of its use of Microsoft's Symbol font.
This method returns the named attributes of the object. If called in list context, you can give more than one attribute name, and it will return all their values.
If called as a class method, it returns the current default values.
See "Attributes" for a list of the attributes you can get.
This method reports the coordinates of $coord
in the East-North-Up coordinate system, as seen from the position stored in the station
attribute of $coord
.
Velocity conversions to enu()
appear to me to be mostly sane. See A NOTE ON VELOCITIES, below, for details.
This method reports the coordinates of $coord2
in the North-East-Up coordinate system, as seen from the position stored in the station
attribute of $coord
. This is a left-handed coordinate system.
Velocity conversions to neu()
appear to me to be mostly sane. See A NOTE ON VELOCITIES, below, for details.
This method returns the Heliocentric ecliptic Cartesian position of the object. You can optionally pass time as an argument; this is equivalent to
@coords = $coord->universal( $time ) ->heliocentric_ecliptic_cartesian();
At this time this object is not able to derive Heliocentric ecliptic Cartesian coordinates from other coordinates (say, ECI).
This method sets the object's position in Heliocentric ecliptic Cartesian coordinates. Velocities may not be set at the moment, though this is planned. You may also pass an optional time as the last argument.
The invocant is returned.
This method returns the Heliocentric ecliptic coordinates of the object.
This method sets the local mean time of the object. This is not local standard time, but the universal time plus the longitude of the object expressed in seconds. Another definition is mean solar time plus 12 hours (since the solar day begins at noon). You will get an exception of some sort if the position of the object has not been set, or if the object represents inertial coordinates, or on any subclass whose position depends on the time.
This method returns the local mean time of the object. It will raise an exception if the time has not been set.
Note that this returns the actual local time, not the GMT equivalent. This means that in formatting for output, you call
strftime $format, gmtime $coord->local_mean_time ();
This method returns the local time (defined as solar time plus 12 hours) of the given object. There is no way to set the local time.
This is really just a convenience routine - the calculation is simply the local mean time plus the equation of time at the given time.
Note that this returns the actual local time, not the GMT equivalent. This means that in formatting for output, you call
strftime $format, gmtime $coord->local_time ();
This method returns the location of the object in the Maidenhead Locator System. Height above the reference ellipsoid is not part of the system, but is returned anyway, in kilometers. The $precision is optional, and is an integer greater than 0.
The default precision is 4, but this can be modified by setting $Astro::Coord::ECI::DEFAULT_MAIDENHEAD_PRECISION
to the desired value. For example, if you want the default precision to be 3 (which it probably should have been in the first place), you can use
no warnings qw{ once }; local $Astro::Coord::ECI::DEFAULT_MAIDENHEAD_PRECISION = 3;
Note that precisions greater than 4 are not defined by the standard. This method extends the system by alternating letters (base 24) with digits (base 10), but this is unsupported since the results will change, possibly without notice, if the standard is extended in a manner incompatible with this implementation.
Conversion of latitudes and longitudes to Maidenhead Grid is subject to truncation error, perhaps more so since latitude and longitude are specified in radians. An attempt has been made to minimize this by using Perl's stringification of numbers to ensure that something that looks like 42
is not handled as 41.999999999385
. This probably amounts to shifting some grid squares very slightly to the north-west, but in practice it seems to give better results for points on the boundaries of the grid squares.
This method sets the geodetic location in the Maidenhead Locator System. Height above the reference ellipsoid is not part of the system, but is accepted anyway, in kilometers, defaulting to 0.
The actual location set is the center of the specified grid square.
Locations longer than 8 characters are not defined by the standard. This method extends precision by assuming alternate letters (base 24) and digits (base 10), but this will change, possibly without notice, if the standard is extended in a manner incompatible with this implementation.
The object itself is returned, to allow call chaining.
This method returns the mean angular velocity of the body in radians per second. If the $coord object has a period() method, this method just returns two pi divided by the period. Otherwise it returns the contents of the angularvelocity attribute.
This method returns the next time the given $body
passes the given $azimuth
as seen from the given $coord
, calculated to the nearest second. The start time is the current time setting of the $body
object.
This method returns the next time the $coord
object passes the given $azimuth
as seen from the location in the $coord
object's station
attribute, calculated to the nearest second. The start time is the current time setting of the $coord
object.
This method calculates the next time the given body passes above or below the given elevation (in radians) as seen from $coord
. The $elev
argument may be omitted (or passed as undef), and will default to 0. If the $upper
argument is true, the calculation will be based on the upper limb of the body (as determined from its angulardiameter
attribute); if false, the calculation will be based on the center of the body. The $upper
argument defaults to true if the $elev
argument is zero or positive, and false if the $elev
argument is negative.
The algorithm is successive approximation, and assumes that the body will be at its highest at meridian passage. It also assumes that if the body hasn't passed the given elevation in 183 days it never will. In this case it returns undef in scalar context, or an empty list in list context.
This method calculates the next time the $coord
object passes above or below the given elevation (in radians) as seen from the position found in the $coord
object's station
attribute. The $elev
argument may be omitted (or passed as undef), and will default to 0. If the $upper
argument is true, the calculation will be based on the upper limb of the body (as determined from its angulardiameter
attribute); if false, the calculation will be based on the center of the body. The $upper
argument defaults to true if the $elev
argument is zero or positive, and false if the $elev
argument is negative.
The algorithm is successive approximation, and assumes that the body will be at its highest at meridian passage. It also assumes that if the body hasn't passed the given elevation in 183 days it never will. In this case it returns undef in scalar context, or an empty list in list context.
This method calculates the next meridian passage of the given $body
over (or under) the location specified by the $coord
object. The $body
object must be a subclass of Astro::Coord::ECI.
The optional $want
argument should be specified as true (e.g. 1) if you want the next passage above the observer, or as false (e.g. 0) if you want the next passage below the observer. If this argument is omitted or undefined, you get whichever passage is next.
The start time of the search is the current time setting of the $coord
object.
The returns are the time of the meridian passage, and an indicator which is true if the passage is above the observer (i.e. local noon if the $body
represents the Sun), or false if below (i.e. local midnight if the $body
represents the Sun). If called in scalar context, you get the time only.
The current time of both $coord
and $body
objects are left at the returned time.
The algorithm is by successive approximation. It will croak if the period of the $body
is close to synchronous, and will probably not work well for bodies in highly eccentric orbits. The calculation is to the nearest second, and the time returned is the first even second after the body crosses the meridian.
This method calculates the next meridian passage of the $coord
object over (or under) the location specified by the $coord
object's station
attribute.
The optional $want
argument should be specified as true (e.g. 1) if you want the next passage above the observer, or as false (e.g. 0) if you want the next passage below the observer. If this argument is omitted or undefined, you get whichever passage is next.
The start time of the search is the current time setting of the $coord
object.
The returns are the time of the meridian passage, and an indicator which is true if the passage is above the observer (i.e. local noon if the $coord
object represents the Sun), or false if below (i.e. local midnight if the $coord
object represents the Sun). If called in scalar context, you get the time only.
The current time of both $coord
and its station
are left at the returned time.
The algorithm is by successive approximation. It will croak if the period of the $body
is close to synchronous, and will probably not work well for bodies in highly eccentric orbits. The calculation is to the nearest second, and the time returned is the first even second after the body crosses the meridian.
This method is a convenience wrapper for precess_dynamical(). The functionality is the same except that the time is specified in universal time.
This method precesses the coordinates of the object to the given equinox, specified in dynamical time. The starting equinox is the value of the 'equinox_dynamical' attribute, or the current time setting if the 'equinox_dynamical' attribute is any false value (i.e. undef, 0, or ''). A warning will be issued if the value of 'equinox_dynamical' is undef (which is the default setting) and the object represents inertial coordinates. As of version 0.013_02, the time setting of the object is unaffected by this operation.
As a side effect, the value of the 'equinox_dynamical' attribute will be set to the dynamical time corresponding to the argument.
As of version 0.061_01, this does nothing to non-inertial objects -- that is, those whose position was set in Earth-fixed coordinates.
If the object's 'station' attribute is set, the station is also precessed.
The object itself is returned.
The algorithm comes from Jean Meeus, "Astronomical Algorithms", 2nd Edition, Chapter 21, pages 134ff (a.k.a. "the rigorous method").
This class method can be used to define or redefine reference ellipsoids.
Nothing bad will happen if you call this as an object method, but it still just creates a reference ellipsoid definition -- the object is unaffected.
It is not an error to redefine an existing ellipsoid.
This class method returns the definition of the named reference ellipsoid. It croaks if there is no such ellipsoid.
You can also call this as an object method, but the functionality is the same.
The following reference ellipsoids are known to the class initially:
CLARKE-1866 - 1866. semimajor => 6378.2064 km, flattening => 1/294.98. Source: http://www.colorado.edu/geography/gcraft/notes/datum/elist.html GRS67 - Geodetic Reference System 1967. semimajor => 6378.160 km, flattening => 1/298.247. Source: http://www.colorado.edu/geography/gcraft/notes/datum/elist.html GRS80 - Geodetic Reference System 1980. semimajor => 6378.137 km, flattening => 1/298.25722210088 (flattening per U.S. Department of Commerce 1989). Source: http://biology.usgs.gov/fgdc.metadata/version2/spref/horiz/geodet/faq.htm IAU68 - International Astronomical Union, 1968. semimajor => 6378.160 km, flattening => 1/298.25. Source: http://maic.jmu.edu/sic/standards/ellipsoid.htm IAU76 - International Astronomical Union, 1976. semimajor => 6378.14 km, flattening => 1 / 298.257. Source: Jean Meeus, "Astronomical Algorithms", 2nd Edition NAD83 - North American Datum, 1983. semimajor => 6378.137 km, flattening => 1/298.257. Source: http://biology.usgs.gov/fgdc.metadata/version2/spref/horiz/geodet/faq.htm (NAD83 uses the GRS80 ellipsoid) sphere - Just in case you were wondering how much difference it makes (a max of 11 minutes 32.73 seconds of arc, per Jean Meeus). semimajor => 6378.137 km (from GRS80), flattening => 0. WGS72 - World Geodetic System 1972. semimajor => 6378.135 km, flattening=> 1/298.26. Source: http://biology.usgs.gov/fgdc.metadata/version2/spref/horiz/geodet/faq.htm WGS84 - World Geodetic System 1984. semimajor => 6378.137 km, flattening => 1/1/298.257223563. Source: http://www.colorado.edu/geography/gcraft/notes/datum/elist.html
Reference ellipsoid names are case-sensitive.
The default model is WGS84.
This method returns true if the $coord object represents the given class. It is pretty much like isa (), but if called on a container class (i.e. Astro::Coord::ECI::TLE::Set), it returns true based on the class of the members of the set, and dies if the set has no members.
The $class argument is optional. If not specified (or undef), it is pretty much like ref $coord || $coord (i.e. it returns the class name), but with the delegation behavior described in the previous paragraph if the $coord object is a container.
There. This took many more words to explain than it did to implement.
This method sets various attributes of the object. If called as a class method, it changes the defaults.
For reasons that seemed good at the time, this method returns the object it was called on (i.e. $coord in the above example).
See "Attributes" for a list of the attributes you can set.
This method sets the time represented by the object, in universal time (a.k.a. CUT, a.k.a. Zulu, a.k.a. Greenwich).
This method can also be called as a class method, in which case it instantiates the desired object.
This method returns the universal time previously set.
This class has the following attributes:
This attribute specifies the horizon used for almanac calculations, as radians above or below the plans of the observer. It can also be set to the following string:
* height
adjusts the horizon in proportion to the observer's height above sea level. This assumes a spherical Earth and an unobstructed horizon.
When doing almanac calculations, it is the almanac_horizon
of the observer that is used, not the almanac_horizon
of the observed body.
The default is 0
.
This attribute represents the angular velocity of the Earth's surface in radians per second. The initial value is 7.292114992e-5, which according to Jean Meeus is the value for 1996.5. He cites the International Earth Rotation Service's Annual Report for 1996 (Published at the Observatoire de Paris, 1997).
Subclasses may place appropriate values here, or provide a period() method.
This attribute turns on debugging output. The only supported value of this attribute is 0. That is to say, the author makes no guarantees of what will happen if you set it to some other value, nor does he guarantee that this behavior will not change from release to release.
The default is 0.
This attribute exists to support classes/instances which represent astronomical bodies. It represents the diameter of the body, and is used by the azel() method when computing the upper limb of the body. It has nothing to do with the semimajor attribute, which always refers to the Earth, and is used to calculate the latitude and longitude of the body.
The default is 0.
This attribute represents the location of the edge of the Earth's shadow cast by the illuminating body, in terms of the apparent radius of that body. That is, the edge of the umbra is specified by 1, the middle of the penumbra by 0, and the edge of the penumbra by -1. It was added for the benefit of the Astro::Coord::ECI::TLE class, on the same dubious logic that the twilight attribute was added.
The default is 1.
This attribute represents the name of the reference ellipsoid to use. It must be set to one of the known ellipsoids. If you set this, flattening and semimajor will be set also. See the documentation to the known_ellipsoid() method for the initially-valid names, and how to add more.
The default is 'WGS84'.
This attribute represents the time of the "Equinox" to which the coordinate data are referred. Models implemented by subclasses should set this to the "Equinox" to which the model is referred. When setting positions directly the user should also set the desired equinox_dynamical if conversion between inertial and Earth-fixed coordinates is of interest. If this is not set, these conversions will use the current time setting of the object as the "Equinox".
In addition, if you have a position specified in earth-fixed coordinates and convert it to inertial coordinates, this attribute will be set to the dynamical time of the object.
Unlike most times in this package, this attribute is specified in dynamical time. If your equinox is universal time $uni, set this attribute to $uni + dynamical_delta ($uni). The dynamical_delta subroutine is found in Astro::Coord::ECI::Utils.
The default is undef.
This attribute represents the flattening factor of the reference ellipsoid. If you set the ellipsoid attribute, this attribute will be set to the flattening factor for the named ellipsoid. If you set this attribute, the ellipsoid attribute will become undefined.
The default is appropriate to the default ellipsoid.
This attribute represents the frequency on which the body transmits, in Hertz. If specified, the azel()
method will return the Doppler shift if velocities are available.
The default is undef
.
This attribute represents the distance the effective horizon is above the geometric horizon. It was added for the Astro::Coord::ECI::TLE::Iridium class, on the same dubious logic that the twilight attribute was added.
The default is the equivalent of 20 degrees.
This is an informational attribute, and its setting (or lack thereof) does not affect the functioning of the class. Certain subclasses will set this when they are instantiated. See the subclass documentation for details.
This attribute returns true (in the Perl sense) if the object was most-recently set to inertial coordinates (i.e. eci, ecliptic, or equatorial) and false otherwise. If coordinates have not been set, it is undefined (and therefore false).
This is an informational attribute, and its setting (or lack thereof) does not affect the functioning of the class. Certain subclasses will set this when they are instantiated. See the subclass documentation for details.
Setting this attribute to a true value includes refraction in the calculation of the azel() method. If set to a false value, atmospheric refraction is ignored.
The default is true (well, 1 actually).
This attribute records the coordinate system in which the coordinates of the object were set.
This attribute represents the semimajor axis of the reference ellipsoid. If you set the ellipsoid attribute, this attribute will be set to the semimajor axis for the named ellipsoid. If you set this attribute, the ellipsoid attribute will become undefined.
For subclasses representing bodies other than the Earth, this attribute will be set appropriately.
The default is appropriate to the default ellipsoid.
This attribute represents the default observing location for relative coordinate systems. It must be set to an Astro::Coord::ECI
object, or undef
.
The intent is that all methods which compute positions or events as seen from a user-specified location should make use of this. In other words, if you find one that does not, you have found a bug.
Cascading station objects are not supported. That is, if you have Astro::Coord::ECI
objects $a
, $b
, $c
and so on, if you
$a->set( station => $b ),
then $b->set( station => $c )
is not supported, nor is $c->set( station => $a )
. Because of the work involved in preventing this in general, I have decided to rely on just politely recommending that It Is Better Not. But specific pernicious cases will throw exceptions, and I reserve the right to do the work and throw exceptions in all cases if it becomes a problem.
This attribute represents the elevation of the center of the Sun's disk at the beginning and end of twilight.
Some of the usual values are:
civil twilight: -6 degrees nautical twilight: -12 degrees astronomical twilight: -18 degrees
Note that the documentation at this point used to say (or at least imply) that the strings 'civil'
, 'nautical'
, and 'astronomical'
were acceptable. This has never been the case to my knowledge, so since I have received no bug reports, I have considered the bug to be in the documentation, and removed the relevant text as of version 0.051_01.
The default is -6 degrees (or, actually, the equivalent in radians).
The velocities returned by the methods in this class are a work in progress. Velocities set are always returned. But conversions are more problematic. For one thing, I have not come across decent worked examples, so I am limited to sanity checking.
I consider a velocity to be sane if the difference between the position at time t
and the position at time t + 1
lies between the velocity at time t
and the velocity at time t + 1
.
In order to understand what velocity conversions are sane, you need to understand how coordinate conversions work internally. I will try to make this plain with some lame ASCII art, which depicts the relationships among those coordinate systems where velocity conversion appears to be at least mostly sane in the above sense of the word:
ecliptic() equatorial() ^ | eci() ^ | v ecef() geocentric() geodetic() | v neu() -> enu() | v azel()
If the object's position and velocity were set in one set of units, other units are obtained by following the arrows. The arrows below ecef() are one-way because it is not currently possible to set a position in these units. Similarly, the arrow from eci()
to equatorial()
is one-way because there is currently no way to set an equatorial velocity. There are no arrows to geocentric()
, geodetic()
and ecliptic()
because these conversions do not support velocities.
Partly because this module straddles the divide between geography and astronomy, the establishment of terminology and conventions was a thorny and in the end somewhat arbitrary process. Because of this, documentation of salient terms and conventions seemed to be in order.
This term refers to the distance of a location above mean sea level.
Altitude input to and output from this module is in kilometers.
Maps use "elevation" for this quantity, and measure it in meters. But we're using "Elevation" for something different, and I needed consistent units.
This term refers to distance around the horizon measured clockwise from North.
Azimuth output from this module is in radians.
Astronomical sources tend to measure from the South, but I chose the geodetic standard, which seems to be usual in satellite tracking software.
Declination is the angle a point makes with the plane of the equator projected onto the sky. North declination is positive, south declination is negative.
Declination input to and output from this module is in radians.
A dynamical time is defined theoretically by the motion of astronomical bodies. In practice, it is seen to be related to Atomic Time (a.k.a. TAI) by a constant.
There are actually two dynamical times of interest: TT (Terrestrial Time, a.k.a. TDT for Terrestrial Dynamical Time), which is defined in terms of the geocentric ephemerides of solar system bodies, and TDB (Barycentric Dynamical Time), which is defined in terms of the barycentre (a.k.a "center of mass") of the solar system. The two differ by the relativistic effects of the motions of the bodies in the Solar system, and are generally less than 2 milliseconds different. So unless you are doing high-precision work they can be considered identical, as Jean Meeus does in "Astronomical Algorithms".
For practical purposes, TT = TAI + 32.184 seconds. If I ever get the gumption to do a re-implementation (or alternate implementation) of time in terms of the DateTime object, this will be the definition of dynamical time. Until then, though, formula 10.2 on page 78 of Jean Meeus' "Astronomical Algorithms" second edition, Chapter 10 (Dynamical Time and Universal Time) is used.
Compare and contrast this to "Universal time". This explanation leans heavily on http://star-www.rl.ac.uk/star/docs/sun67.htx/node226.html, which contains a more fulsome but eminently readable explanation.
This is a Cartesian coodinate system whose origin is the center of the reference ellipsoid. The X axis passes through 0 degrees "Latitude" and 0 degrees "Longitude". The Y axis passes through 90 degrees east "Latitude" and 0 degrees "Longitude", and the Z axis passes through 90 degrees north "Latitude" (a.k.a the North Pole).
All three axes are input to and output from this module in kilometers.
Also known as "XYZ coordinates", e.g. at http://www.ngs.noaa.gov/cgi-bin/xyz_getxyz.prl
This is the Cartesian coordinate system in which NORAD's models predict the position of orbiting bodies. The X axis passes through 0 hours "Right Ascension" and 0 degrees "Declination". The Y axis passes through 6 hours "Right Ascension" and 0 degrees "Declination". The Z axis passes through +90 degrees "Declination" (a.k.a. the North Pole). By implication, these coordinates are referred to a given "Equinox".
All three axes are input to and output from this module in kilometers.
The Ecliptic is the plane of the Earth's orbit, projected onto the sky. Ecliptic coordinates are a spherical coordinate system referred to the ecliptic and expressed in terms of "Ecliptic latitude" and "Ecliptic longitude". By implication, Ecliptic coordinates are also referred to a specific "Equinox".
Ecliptic longitude is the angular distance of a point above the plane of the Earth's orbit.
Ecliptic latitude is input to and output from this module in radians.
Ecliptic longitude is the angular distance of a point east of the point where the plane of the Earth's orbit intersects the plane of the equator. This point is also known as the vernal "Equinox" and the first point of Ares.
Ecliptic longitude is input to and output from this module in radians.
This term refers to an angular distance above the horizon.
Elevation output from this module is in radians.
This is the prevailing meaning of the term in satellite tracking. Astronomers use "altitude" for this quantity, and call the corresponding coordinate system "altazimuth." But we're using "Altitude" for something different.
Equatorial coordinates are a spherical coordinate system referred to the plane of the equator projected onto the sky. Equatorial coordinates are specified in "Right Ascension" and "Declination", and implicitly referred to a given "Equinox"
The "Ecliptic", "Equatorial", and "Earth-Centered Inertial (ECI) coordinates" are defined in terms of the location of the intersection of the celestial equator with the "Ecliptic". The actual location of this point changes in time due to precession of the Earth's axis, so each of these coordinate systems is implicitly qualified by ("referred to" appears to be the usual terminology) the relevant time. By a process of association of ideas, this time is referred to as the equinox of the data.
When referring to a coordinate system, this term means that the coordinate system assumes the Earth is spherical.
Geocentric latitude is the angle that the ray from the center of the Earth to the location makes with the plane of the equator. North latitude is positive, south latitude is negative.
Geocentric latitude is input to and output from this module in radians.
When referring to a coordinate system, this term means that the coordinate system assumes the Earth is an ellipsoid of revolution (or an oblate spheroid if you prefer). A number of standard "Reference Ellipsoids" have been adopted over the years.
Geodetic latitude is the latitude found on maps. North latitude is positive, south latitude is negative.
Geodetic latitude is input to and output from this module in radians.
Technically speaking, Geodetic latitude is the complement of the angle the plane of the horizon makes with the plane of the equator. In this software, the plane of the horizon is determined from a "Reference Ellipsoid".
See either "Ecliptic latitude", "Geocentric latitude" or "Geodetic latitude". When used without qualification, "Geodetic latitude" is meant.
When unqualified, this term refers to the angular distance East or West of the standard meridian. East longitude is positive, West longitude is negative.
Longitude is input to or output from this module in radians.
For "Ecliptic longitude", see that entry.
Jean Meeus reports in "Astronomical Algorithms" that the International Astronomical Union has waffled on the sign convention. I have taken the geographic convention.
This term refers to the angle the plane of the equator makes with the plane of the Earth's orbit.
Obliquity is output from this module in radians.
This term refers to a specific ellipsoid adopted as a model of the shape of the Earth. It is defined in terms of the equatorial radius in kilometers, and a dimensionless flattening factor which is used to derive the polar radius, and defined such that the flattening factor of a sphere is zero.
See the documentation on the reference_ellipsoid() method for a list of reference ellipsoids known to this class.
This term refers to the angular distance of a point east of the intersection of the plane of the Earth's orbit with the plane of the equator.
Right Ascension is input to and output from this module in radians.
In astronomical literature it is usual to report right ascension in hours, minutes, and seconds, with 60 seconds in a minute, 60 minutes in an hour, and 24 hours in a circle.
This term can refer to a number of scales, but the two of interest are UTC (Coordinated Universal Time) and UT1 (Universal Time 1, I presume). The latter is in effect mean solar time at Greenwich, though its technical definition differs in detail from GMT (Greenwich Mean Time). The former is a clock-based time, whose second is the SI second (defined in terms of atomic clocks), but which is kept within 0.9 seconds of UT1 by the introduction of leap seconds. These are introduced (typically at midyear or year end) by prior agreement among the various timekeeping bodies based on observation; there is no formula for computing when a leap second will be needed, because of irregularities in the Earth's rotation.
Jean Meeus' "Astronomical Algorithms", second edition, deals with the relationship between Universal time and "Dynamical time" in Chapter 10 (pages 77ff). His definition of "Universal time" seems to refer to UT1, though he does not use the term.
This software considers Universal time to be equivalent to Perl time. Since we are interested in durations (time since a given epoch, to be specific), this is technically wrong in most cases, since leap seconds are not taken into account. But in the case of the bodies modeled by the Astro::Coord::ECI::TLE object, the epoch is very recent (within a week or so), so the error introduced is small. It is larger for astronomical calculations, where the epoch is typically J2000.0, but the angular motions involved are smaller, so it all evens out. I hope.
Compare and contrast "Dynamical time". This explanation leans heavily on http://star-www.rl.ac.uk/star/docs/sun67.htx/node224.html.
See "Earth-Centered, Earth-fixed (ECEF) coordinates".
The author wishes to acknowledge and thank the following individuals and organizations.
Kazimierz Borkowski, whose "Accurate Algorithms to Transform Geocentric to Geodetic Coordinates", at http://www.astro.uni.torun.pl/~kb/Papers/geod/Geod-BG.htm, was used for transforming geocentric to geodetic coordinates.
Dominik Brodowski (http://www.brodo.de/), whose SGP C-lib (available at http://www.brodo.de/space/sgp/) provided a reference implementation that I could easily run, and pick apart to help get Astro::Coord::ECI::TLE working. Dominik based his work on Dr. Kelso's Pascal implementation.
Felix R. Hoots and Ronald L. Roehric, the authors of "SPACETRACK REPORT NO. 3 - Models for Propagation of NORAD Element Sets," which provided the basis for the Astro::Coord::ECI::TLE module.
T. S. Kelso, who compiled this report and made it available at http://celestrak.com/, whose "Computers and Satellites" columns in "Satellite Times" magazine were invaluable to an understanding and implementation of satellite tracking software, whose support, encouragement, patience, and willingness to provide answers on arcane topics were a Godsend, who kindly granted permission to use his azimuth/elevation algorithm, and whose Pascal implementation of the NORAD tracking algorithms indirectly provided a reference implementation for me to use during development.
John A. Magliacane, whose open-source Predict
program, available at http://www.qsl.net/kd2bd/predict.html, gave me a much-needed leg up on the velocity calculations in the azel()
method.
Jean Meeus, whose book "Astronomical Algorithms" (second edition) formed the basis for this module and the Astro::Coord::ECI::Sun, Astro::Coord::ECI::Moon, and Astro::Coord::ECI::Star modules, and without whom it is impossible to get far in computational astronomy. Any algorithm not explicitly credited above is probably due to him.
Dr. Meeus' publisher, Willmann-Bell Inc (http://www.willbell.com/), which kindly and patiently answered my intellectual-property questions.
Goran Gasparovic of MIT, who asked for (and supplied information for) the ability to display results in apparent equatorial coordinates, rather than azimuth and elevation.
Dr. Kevin Cornelius of Ouachita Baptist University, whose handout "Relationships Among Unit Vectors" for his Mathematical Physics class (PHYS 4053) provided a much surer and more expeditious way to handling velocity conversion from spherical to Cartesian (and vice versa) than my own ancient and rickety matrix math.
Functionality involving velocities is untested, and is quite likely to be wrong.
Bugs can be reported to the author by mail, or through http://rt.cpan.org/.
The Astro package by Chris Phillips. This contains three function-based modules: Astro::Coord, which provides various astronomical coordinate conversions, plus the calculation of various ephemeris variables; Astro::Time contains time and unit conversions, and Astro::Misc contains various calculations unrelated to position and time.
The Astro-Coords package by Tim Jenness. This contains various modules to do astronomical calculations, and includes coordinate conversion and calculation of planetary orbits based on orbital elements. Requires SLALIB from http://www.starlink.rl.ac.uk/Software/software_store.htm.
The Astro::Nova module by Steffen Mueller, which wraps (and bundles) the libnova celestial mechanics, astrometry and astrodynamics library found at http://libnova.sourceforge.net/.
The http://www.heavens-above.com/ web site, which does not use this code, but does provide similar functionality.
Thomas R. Wyant, III (wyant at cpan dot org)
Copyright (C) 2005-2015 by Thomas R. Wyant, III
This program is free software; you can redistribute it and/or modify it under the same terms as Perl 5.10.0. For more details, see the full text of the licenses in the directory LICENSES.
This program is distributed in the hope that it will be useful, but without any warranty; without even the implied warranty of merchantability or fitness for a particular purpose.