def airmassPP(zangle):
"""
Calculate airmass for plane parallel atmosphere.
Parameters
----------
zangle : float or array
The zenith angle in degrees.
Returns
-------
Airmass : float or array
The airmass assuming a plane parallel
atmosphere.
"""
return 1.0 / np.cos(PC.degtorad(zangle))
def airmassPP(zangle):
"""
Calculate airmass for plane parallel atmosphere.
Parameters
----------
zangle : float or array
The zenith angle in degrees.
Returns
-------
Airmass : float or array
The airmass assuming a plane parallel
atmosphere.
"""
return 1.0/np.cos(PC.degtorad(zangle))
def airmassSpherical(zangle, obsAltitude, rearth=6371.0, yatm=10.0):
"""
Calculate the airmass for a given zenith angle and observer altitude.
This routine uses a geometric formula for a homogeneous, spherical
atmosphere with an elevated observer.
.. note:: In this model, the airmass is *not* necessarily one
toward the zenith.
Parameters
----------
zangle : float
Zenith angle of an object in deg.
obsAltitude : float
Elevation of the observer in meter.
rearth : float, optional
Earth's radius in km.
yatm : float, optional
Height of the atmosphere in km.
Returns
-------
Airmass : float
The airmass.
"""
# Convert observer's altitude to km to have
# consistent units
obsAltitude = obsAltitude/1000.0
r = (rearth/yatm)
y = (obsAltitude/yatm)
# Find maximal zenith angle
zmax = 180.0 - PC.radtodeg(np.arcsin(rearth/(rearth + obsAltitude)))
if zangle > zmax:
raise(PE.PyAValError("Zenith angle is too large. The maximum allowed angle is " + str(zmax) + " deg.", \
solution="Use an angle within the limit. Check observer's altitude."))
# Convert into deg
zangle = PC.degtorad(zangle)
return np.sqrt( ( r + y )**2 * np.cos(zangle)**2 + 2.*r*(1.-y) - y**2 + 1.0 ) - \
(r+y)*np.cos(zangle)
def airmassSpherical(zangle, obsAltitude, rearth=6371.0, yatm=10.0):
"""
Calculate the airmass for a given zenith angle and observer altitude.
This routine uses a geometric formula for a homogeneous, spherical
atmosphere with an elevated observer.
.. note:: In this model, the airmass is *not* necessarily one
toward the zenith.
Parameters
----------
zangle : float
Zenith angle of an object in deg.
obsAltitude : float
Elevation of the observer in meter.
rearth : float, optional
Earth's radius in km.
yatm : float, optional
Height of the atmosphere in km.
Returns
-------
Airmass : float
The airmass.
"""
# Convert observer's altitude to km to have
# consistent units
obsAltitude = obsAltitude / 1000.0
r = (rearth / yatm)
y = (obsAltitude / yatm)
# Find maximal zenith angle
zmax = 180.0 - PC.radtodeg(np.arcsin(rearth / (rearth + obsAltitude)))
if zangle > zmax:
raise(PE.PyAValError("Zenith angle is too large. The maximum allowed angle is " + str(zmax) + " deg.", \
solution="Use an angle within the limit. Check observer's altitude."))
# Convert into deg
zangle = PC.degtorad(zangle)
return np.sqrt( ( r + y )**2 * np.cos(zangle)**2 + 2.*r*(1.-y) - y**2 + 1.0 ) - \
(r+y)*np.cos(zangle)
def helcorr(obs_long, obs_lat, obs_alt, ra2000, dec2000, jd, debug=False):
"""
Calculate barycentric velocity correction.
This function calculates the motion of an observer in
the direction of a star. In contract to :py:func:`baryvel`
and :py:func:`baryCorr`, the rotation of the Earth is
taken into account.
.. note:: This function was ported from the REDUCE IDL package.
See Piskunov & Valenti 2002, A&A 385, 1095 for a detailed
description of the package and/or visit
http://www.astro.uu.se/~piskunov/RESEARCH/REDUCE/
.. warning:: Contrary to the original implementation the longitude
increases toward the East and the right ascension is
given in degrees instead of hours. The JD is given as is,
in particular, nothing needs to be subtracted.
Parameters
----------
obs_long : float
Longitude of observatory (degrees, **eastern** direction is positive)
obs_lat : float
Latitude of observatory [deg]
obs_alt : float
Altitude of observatory [m]
ra2000 : float
Right ascension of object for epoch 2000.0 [deg]
dec2000 : float
Declination of object for epoch 2000.0 [deg]
jd : float
Julian date for the middle of exposure.
Returns
-------
Barycentric correction : float
The barycentric correction accounting for the rotation
of the Earth, the rotation of the Earth's center around
the Earth-Moon barycenter, and the motion of the Earth-Moon
barycenter around the center of the Sun [km/s].
HJD : float
Heliocentric Julian date for middle of exposure.
Notes
-----
:IDL REDUCE - Documentation:
Calculates heliocentric Julian date, barycentric and heliocentric radial
velocity corrections from:
INPUT:
<OBSLON> Longitude of observatory (degrees, western direction is positive)
<OBSLAT> Latitude of observatory (degrees)
<OBSALT> Altitude of observatory (meters)
<RA2000> Right ascension of object for epoch 2000.0 (hours)
<DE2000> Declination of object for epoch 2000.0 (degrees)
<JD> Julian date for the middle of exposure
[DEBUG=] set keyword to get additional results for debugging
OUTPUT:
<CORRECTION> barycentric correction - correction for rotation of earth,
rotation of earth center about the earth-moon barycenter, earth-moon
barycenter about the center of the Sun.
<HJD> Heliocentric Julian date for middle of exposure
Algorithms used are taken from the IRAF task noao.astutils.rvcorrect
and some procedures of the IDL Astrolib are used as well.
Accuracy is about 0.5 seconds in time and about 1 m/s in velocity.
History:
written by Peter Mittermayer, Nov 8,2003
2005-January-13 Kudryavtsev Made more accurate calculation of the sidereal time.
Conformity with MIDAS compute/barycorr is checked.
2005-June-20 Kochukhov Included precession of RA2000 and DEC2000 to current epoch
"""
from PyAstronomy.pyaC import degtorad
# This reverts the original longitude convention. After this,
# East longitudes are positive
obs_long = -obs_long
if jd < 2.4e6:
PE.warn(
PE.PyAValError("The given Julian Date (" + str(jd) +
") is exceedingly small. Did you subtract 2.4e6?"))
# Covert JD to Gregorian calendar date
xjd = jd
year, month, day, ut = tuple(daycnv(xjd))
# Current epoch
epoch = year + month / 12. + day / 365.
# Precess ra2000 and dec2000 to current epoch, resulting ra is in degrees
ra = ra2000
dec = dec2000
ra, dec = precess(ra, dec, 2000.0, epoch)
# Calculate heliocentric julian date
rjd = jd - 2.4e6
hjd = helio_jd(rjd, ra, dec) + 2.4e6
# DIURNAL VELOCITY (see IRAF task noao.astutil.rvcorrect)
# convert geodetic latitude into geocentric latitude to correct
# for rotation of earth
dlat = -(11.*60.+32.743)*np.sin(2.0*degtorad(obs_lat)) \
+1.1633*np.sin(4.0*degtorad(obs_lat)) - 0.0026*np.sin(6.0*degtorad(obs_lat))
lat = obs_lat + dlat / 3600.0
# Calculate distance of observer from earth center
r = 6378160.0 * (0.998327073+0.001676438*np.cos(2.0*degtorad(lat)) \
-0.00000351 * np.cos(4.0*degtorad(lat)) + 0.000000008*np.cos(6.0*degtorad(lat))) \
+ obs_alt
# Calculate rotational velocity (perpendicular to the radius vector) in km/s
# 23.934469591229 is the sidereal day in hours for 1986
v = 2. * np.pi * (r / 1000.) / (23.934469591229 * 3600.)
# Calculating local mean sidereal time (see astronomical almanach)
tu = (rjd - 51545.0) / 36525.0
gmst = 6.697374558 + ut + \
(236.555367908*(rjd-51545.0) + 0.093104*tu**2 - 6.2e-6*tu**3)/3600.0
lmst = idlMod(gmst - obs_long / 15., 24)
# Projection of rotational velocity along the line of sight
vdiurnal = v * np.cos(degtorad(lat)) * np.cos(degtorad(dec)) * np.sin(
degtorad(ra - lmst * 15))
# BARICENTRIC and HELIOCENTRIC VELOCITIES
vh, vb = baryvel(xjd, 0)
# Project to line of sight
vbar = vb[0]*np.cos(degtorad(dec))*np.cos(degtorad(ra)) + vb[1]*np.cos(degtorad(dec))*np.sin(degtorad(ra)) + \
vb[2]*np.sin(degtorad(dec))
vhel = vh[0]*np.cos(degtorad(dec))*np.cos(degtorad(ra)) + vh[1]*np.cos(degtorad(dec))*np.sin(degtorad(ra)) + \
vh[2]*np.sin(degtorad(dec))
# Use barycentric velocity for correction
corr = (vdiurnal + vbar)
if debug:
print('')
print('----- HELCORR.PRO - DEBUG INFO - START ----')
print(
'(obs_long (East positive),obs_lat,obs_alt) Observatory coordinates [deg,m]: ',
-obs_long, obs_lat, obs_alt)
print('(ra,dec) Object coordinates (for epoch 2000.0) [deg]: ', ra,
dec)
print('(ut) Universal time (middle of exposure) [hrs]: ', ut)
print('(jd) Julian date (middle of exposure) (JD): ', jd)
print('(hjd) Heliocentric Julian date (middle of exposure) (HJD): ',
hjd)
print('(gmst) Greenwich mean sidereal time [hrs]: ', idlMod(gmst, 24))
print('(lmst) Local mean sidereal time [hrs]: ', lmst)
print('(dlat) Latitude correction [deg]: ', dlat)
print('(lat) Geocentric latitude of observer [deg]: ', lat)
print('(r) Distance of observer from center of earth [m]: ', r)
print(
'(v) Rotational velocity of earth at the position of the observer [km/s]: ',
v)
print(
'(vdiurnal) Projected earth rotation and earth-moon revolution [km/s]: ',
vdiurnal)
print('(vbar) Barycentric velocity [km/s]: ', vbar)
print('(vhel) Heliocentric velocity [km/s]: ', vhel)
print('(corr) Vdiurnal+vbar [km/s]: ', corr)
print('----- HELCORR.PRO - DEBUG INFO - END -----')
print('')
return corr, hjd
def helcorr(obs_long, obs_lat, obs_alt, ra2000, dec2000, jd, debug=False):
"""
Calculate barycentric velocity correction.
This function calculates the motion of an observer in
the direction of a star. In contract to :py:func:`baryvel`
and :py:func:`baryCorr`, the rotation of the Earth is
taken into account.
.. note:: This function was ported from the REDUCE IDL package.
See Piskunov & Valenti 2002, A&A 385, 1095 for a detailed
description of the package and/or visit
http://www.astro.uu.se/~piskunov/RESEARCH/REDUCE/
.. warning:: Contrary to the original implementation the longitude
increases toward the East and the right ascension is
given in degrees instead of hours. The JD is given as is,
in particular, nothing needs to be subtracted.
Parameters
----------
obs_long : float
Longitude of observatory (degrees, **eastern** direction is positive)
obs_lat : float
Latitude of observatory [deg]
obs_alt : float
Altitude of observatory [m]
ra2000 : float
Right ascension of object for epoch 2000.0 [deg]
dec2000 : float
Declination of object for epoch 2000.0 [deg]
jd : float
Julian date for the middle of exposure.
Returns
-------
Barycentric correction : float
The barycentric correction accounting for the rotation
of the Earth, the rotation of the Earth's center around
the Earth-Moon barycenter, and the motion of the Earth-Moon
barycenter around the center of the Sun [km/s].
HJD : float
Heliocentric Julian date for middle of exposure.
Notes
-----
:IDL REDUCE - Documentation:
Calculates heliocentric Julian date, barycentric and heliocentric radial
velocity corrections from:
INPUT:
<OBSLON> Longitude of observatory (degrees, western direction is positive)
<OBSLAT> Latitude of observatory (degrees)
<OBSALT> Altitude of observatory (meters)
<RA2000> Right ascension of object for epoch 2000.0 (hours)
<DE2000> Declination of object for epoch 2000.0 (degrees)
<JD> Julian date for the middle of exposure
[DEBUG=] set keyword to get additional results for debugging
OUTPUT:
<CORRECTION> barycentric correction - correction for rotation of earth,
rotation of earth center about the earth-moon barycenter, earth-moon
barycenter about the center of the Sun.
<HJD> Heliocentric Julian date for middle of exposure
Algorithms used are taken from the IRAF task noao.astutils.rvcorrect
and some procedures of the IDL Astrolib are used as well.
Accuracy is about 0.5 seconds in time and about 1 m/s in velocity.
History:
written by Peter Mittermayer, Nov 8,2003
2005-January-13 Kudryavtsev Made more accurate calculation of the sidereal time.
Conformity with MIDAS compute/barycorr is checked.
2005-June-20 Kochukhov Included precession of RA2000 and DEC2000 to current epoch
"""
from PyAstronomy.pyaC import degtorad
# This reverts the original longitude convention. After this,
# East longitudes are positive
obs_long = -obs_long
if jd < 2.4e6:
PE.warn(PE.PyAValError("The given Julian Date (" + str(jd) + ") is exceedingly small. Did you subtract 2.4e6?"))
# Covert JD to Gregorian calendar date
xjd = jd
year, month, day, ut = tuple(daycnv(xjd))
# Current epoch
epoch = year + month/12. + day/365.
# Precess ra2000 and dec2000 to current epoch, resulting ra is in degrees
ra = ra2000
dec = dec2000
ra, dec = precess(ra, dec, 2000.0, epoch)
# Calculate heliocentric julian date
rjd = jd-2.4e6
hjd = helio_jd(rjd, ra, dec) + 2.4e6
# DIURNAL VELOCITY (see IRAF task noao.astutil.rvcorrect)
# convert geodetic latitude into geocentric latitude to correct
# for rotation of earth
dlat = -(11.*60.+32.743)*np.sin(2.0*degtorad(obs_lat)) \
+1.1633*np.sin(4.0*degtorad(obs_lat)) - 0.0026*np.sin(6.0*degtorad(obs_lat))
lat = obs_lat + dlat/3600.0
# Calculate distance of observer from earth center
r = 6378160.0 * (0.998327073+0.001676438*np.cos(2.0*degtorad(lat)) \
-0.00000351 * np.cos(4.0*degtorad(lat)) + 0.000000008*np.cos(6.0*degtorad(lat))) \
+ obs_alt
# Calculate rotational velocity (perpendicular to the radius vector) in km/s
# 23.934469591229 is the sidereal day in hours for 1986
v = 2.*np.pi * (r/1000.) / (23.934469591229*3600.)
# Calculating local mean sidereal time (see astronomical almanach)
tu = (rjd-51545.0)/36525.0
gmst = 6.697374558 + ut + \
(236.555367908*(rjd-51545.0) + 0.093104*tu**2 - 6.2e-6*tu**3)/3600.0
lmst = idlMod(gmst-obs_long/15, 24)
# Projection of rotational velocity along the line of sight
vdiurnal = v*np.cos(degtorad(lat))*np.cos(degtorad(dec))*np.sin(degtorad(ra-lmst*15))
# BARICENTRIC and HELIOCENTRIC VELOCITIES
vh, vb = baryvel(xjd,0)
# Project to line of sight
vbar = vb[0]*np.cos(degtorad(dec))*np.cos(degtorad(ra)) + vb[1]*np.cos(degtorad(dec))*np.sin(degtorad(ra)) + \
vb[2]*np.sin(degtorad(dec))
vhel = vh[0]*np.cos(degtorad(dec))*np.cos(degtorad(ra)) + vh[1]*np.cos(degtorad(dec))*np.sin(degtorad(ra)) + \
vh[2]*np.sin(degtorad(dec))
# Use barycentric velocity for correction
corr = (vdiurnal + vbar)
if debug:
print ''
print '----- HELCORR.PRO - DEBUG INFO - START ----'
print '(obs_long (East positive),obs_lat,obs_alt) Observatory coordinates [deg,m]: ', -obs_long, obs_lat, obs_alt
print '(ra,dec) Object coordinates (for epoch 2000.0) [deg]: ', ra,dec
print '(ut) Universal time (middle of exposure) [hrs]: ', ut
print '(jd) Julian date (middle of exposure) (JD): ', jd
print '(hjd) Heliocentric Julian date (middle of exposure) (HJD): ', hjd
print '(gmst) Greenwich mean sidereal time [hrs]: ', idlMod(gmst, 24)
print '(lmst) Local mean sidereal time [hrs]: ', lmst
print '(dlat) Latitude correction [deg]: ', dlat
print '(lat) Geocentric latitude of observer [deg]: ', lat
print '(r) Distance of observer from center of earth [m]: ', r
print '(v) Rotational velocity of earth at the position of the observer [km/s]: ', v
print '(vdiurnal) Projected earth rotation and earth-moon revolution [km/s]: ', vdiurnal
print '(vbar) Barycentric velocity [km/s]: ', vbar
print '(vhel) Heliocentric velocity [km/s]: ', vhel
print '(corr) Vdiurnal+vbar [km/s]: ', corr
print '----- HELCORR.PRO - DEBUG INFO - END -----'
print ''
return corr, hjd
def positionAngle(ra1, dec1, ra2, dec2, positive=True):
"""
Compute the position angle.
The position angle is measured from the first position
from North through East. If the `positive` flag is set
True (default) the result will be given as an angle from
0 to 360 degrees. If the flag is set False, the scale is
-180 to 180 degrees with negative number increasing from
North through West; note that this is the behavior of
the posAng IDL routine.
Parameters
----------
ra1 : float
Right ascension of first object [deg].
dec1 : float
Declination of first object [deg].
ra2 : float
Right ascension of second object [deg].
dec2 : float
Declination of second object [deg].
positive : boolean, optional
If True (default), the output will be
given as an angle between 0 and 360
degrees. Otherwise, the angle ranges
from -180 to +180 degrees.
Returns
-------
Position angle : float
The position angle in degrees.
Notes
-----
.. note:: This function was ported from the IDL Astronomy User's Library.
:IDL - Documentation:
NAME:
POSANG
PURPOSE:
Computes rigorous position angle of source 2 relative to source 1
EXPLANATION:
Computes the rigorous position angle of source 2 (with given RA, Dec)
using source 1 (with given RA, Dec) as the center.
CALLING SEQUENCE:
POSANG, U, RA1, DC1, RA2, DC2, ANGLE
INPUTS:
U -- Describes units of inputs and output:
0: everything radians
1: RAx in decimal hours, DCx in decimal
degrees, ANGLE in degrees
RA1 -- Right ascension of point 1
DC1 -- Declination of point 1
RA2 -- Right ascension of point 2
DC2 -- Declination of point 2
OUTPUTS:
ANGLE-- Angle of the great circle containing [ra2, dc2] from
the meridian containing [ra1, dc1], in the sense north
through east rotating about [ra1, dc1]. See U above
for units.
PROCEDURE:
The "four-parts formula" from spherical trig (p. 12 of Smart's
Spherical Astronomy or p. 12 of Green' Spherical Astronomy).
EXAMPLE:
For the star 56 Per, the Hipparcos catalog gives a position of
RA = 66.15593384, Dec = 33.94988843 for component A, and
RA = 66.15646079, Dec = 33.96100069 for component B. What is the
position angle of B relative to A?
IDL> RA1 = 66.15593384/15.d & DC1 = 33.95988843
IDL> RA2 = 66.15646079/15.d & DC2 = 33.96100069
IDL> posang,1,ra1,dc1,ra2,dc2, ang
will give the answer of ang = 21.4 degrees
NOTES:
(1) If RA1,DC1 are scalars, and RA2,DC2 are vectors, then ANGLE is a
vector giving the position angle between each element of RA2,DC2 and
RA1,DC1. Similarly, if RA1,DC1 are vectors, and RA2, DC2 are scalars,
then DIS is a vector giving the position angle of each element of RA1,
DC1 and RA2, DC2. If both RA1,DC1 and RA2,DC2 are vectors then ANGLE
is a vector giving the position angle between each element of RA1,DC1
and the corresponding element of RA2,DC2. If then vectors are not the
same length, then excess elements of the longer one will be ignored.
(2) Note that POSANG is not commutative -- the position angle between
A and B is theta, then the position angle between B and A is 180+theta
PROCEDURE CALLS:
ISARRAY()
HISTORY:
Modified from GCIRC, R. S. Hill, RSTX, 1 Apr. 1998
Use V6.0 notation W.L. Mar 2011
"""
# Convert into rad
rarad1 = pyaC.degtorad(ra1)
rarad2 = pyaC.degtorad(ra2)
dcrad1 = pyaC.degtorad(dec1)
dcrad2 = pyaC.degtorad(dec2)
radif = rarad2 - rarad1
angle = arctan2(sin(radif),
cos(dcrad1) * tan(dcrad2) - sin(dcrad1) * cos(radif))
result = pyaC.radtodeg(angle)
if positive and (result < 0.0):
result += 360.0
return result
def positionAngle(ra1, dec1, ra2, dec2, positive=True):
"""
Compute the position angle.
The position angle is measured from the first position
from North through East. If the `positive` flag is set
True (default) the result will be given as an angle from
0 to 360 degrees. If the flag is set False, the scale is
-180 to 180 degrees with negative number increasing from
North through West; note that this is the behavior of
the posAng IDL routine.
Parameters
----------
ra1 : float
Right ascension of first object [deg].
dec1 : float
Declination of first object [deg].
ra2 : float
Right ascension of second object [deg].
dec2 : float
Declination of second object [deg].
positive : boolean, optional
If True (default), the output will be
given as an angle between 0 and 360
degrees. Otherwise, the angle ranges
from -180 to +180 degrees.
Returns
-------
Position angle : float
The position angle in degrees.
Notes
-----
.. note:: This function was ported from the IDL Astronomy User's Library.
:IDL - Documentation:
NAME:
POSANG
PURPOSE:
Computes rigorous position angle of source 2 relative to source 1
EXPLANATION:
Computes the rigorous position angle of source 2 (with given RA, Dec)
using source 1 (with given RA, Dec) as the center.
CALLING SEQUENCE:
POSANG, U, RA1, DC1, RA2, DC2, ANGLE
INPUTS:
U -- Describes units of inputs and output:
0: everything radians
1: RAx in decimal hours, DCx in decimal
degrees, ANGLE in degrees
RA1 -- Right ascension of point 1
DC1 -- Declination of point 1
RA2 -- Right ascension of point 2
DC2 -- Declination of point 2
OUTPUTS:
ANGLE-- Angle of the great circle containing [ra2, dc2] from
the meridian containing [ra1, dc1], in the sense north
through east rotating about [ra1, dc1]. See U above
for units.
PROCEDURE:
The "four-parts formula" from spherical trig (p. 12 of Smart's
Spherical Astronomy or p. 12 of Green' Spherical Astronomy).
EXAMPLE:
For the star 56 Per, the Hipparcos catalog gives a position of
RA = 66.15593384, Dec = 33.94988843 for component A, and
RA = 66.15646079, Dec = 33.96100069 for component B. What is the
position angle of B relative to A?
IDL> RA1 = 66.15593384/15.d & DC1 = 33.95988843
IDL> RA2 = 66.15646079/15.d & DC2 = 33.96100069
IDL> posang,1,ra1,dc1,ra2,dc2, ang
will give the answer of ang = 21.4 degrees
NOTES:
(1) If RA1,DC1 are scalars, and RA2,DC2 are vectors, then ANGLE is a
vector giving the position angle between each element of RA2,DC2 and
RA1,DC1. Similarly, if RA1,DC1 are vectors, and RA2, DC2 are scalars,
then DIS is a vector giving the position angle of each element of RA1,
DC1 and RA2, DC2. If both RA1,DC1 and RA2,DC2 are vectors then ANGLE
is a vector giving the position angle between each element of RA1,DC1
and the corresponding element of RA2,DC2. If then vectors are not the
same length, then excess elements of the longer one will be ignored.
(2) Note that POSANG is not commutative -- the position angle between
A and B is theta, then the position angle between B and A is 180+theta
PROCEDURE CALLS:
ISARRAY()
HISTORY:
Modified from GCIRC, R. S. Hill, RSTX, 1 Apr. 1998
Use V6.0 notation W.L. Mar 2011
"""
# Convert into rad
rarad1 = pyaC.degtorad(ra1)
rarad2 = pyaC.degtorad(ra2)
dcrad1 = pyaC.degtorad(dec1)
dcrad2 = pyaC.degtorad(dec2)
radif = rarad2 - rarad1
angle = arctan2(sin(radif), cos(dcrad1)*tan(dcrad2)-sin(dcrad1)*cos(radif))
result = pyaC.radtodeg(angle)
if positive and (result < 0.0):
result+= 360.0
return result
