mirror of
https://gitlab.astro-wise.org/micado/atmosphericmodel.git
synced 2026-10-30 05:01:02 +00:00
582 lines
18 KiB
Python
582 lines
18 KiB
Python
#!/usr/bin/env Python3
|
||
import numpy as np
|
||
from scipy import integrate
|
||
import os
|
||
from .ciddorModel import Observatory
|
||
|
||
'''
|
||
@ Author: Joost van den Born
|
||
@ Contact: born@astron.nl
|
||
@ Description: This script offers a complete model of the atmospheric refraction
|
||
and dispersion. It uses the Ciddor1996 paper to calculate the
|
||
index of refraction for a wide range of conditions, with corresponding error propagation if needed.
|
||
The actual refraction is calculated by any of the following models:
|
||
- Plane parallel atmosphere
|
||
- Cassini spherical homogeneous atmosphere
|
||
- Mathar's Barometric exponential model
|
||
- Other methods described in Corbard et al., 2019.
|
||
'''
|
||
|
||
class refraction(Observatory):
|
||
'''
|
||
Class that sets the observer location.
|
||
These location attributes are then used to calculate the
|
||
appropriate atmospheric refraction.
|
||
|
||
See also:
|
||
Observatory()
|
||
dispersion()
|
||
...
|
||
|
||
Attributes
|
||
-----------
|
||
lat : float
|
||
Latitude of the observer in degrees
|
||
h : float
|
||
Altitude above mean sea level of the observer in meters
|
||
Other Observatory() attributes
|
||
|
||
Methods
|
||
-------
|
||
cassini(n1, n2, zenith, reduced_height=None)
|
||
Returns the atmospheric refraction assuming Cassini's homegeneous
|
||
atmosphere model.
|
||
cassiniError(z, l1, l2, T, p, RH, xc, dl1=0, dl2=0, dT=0.2, dP=20,
|
||
dRH=0.02, dCO2=20, dz=0, lat=None, h=None)
|
||
Returns the uncertainty in the atmospheric refraction, assuming
|
||
Cassini's homegeneous atmosphere model and the given parameters.
|
||
corbard(n1, n2, zenith, reduced_height=None)
|
||
Returns the atmospheric refraction following the method described
|
||
in Corbard et al., 2019.
|
||
gammaCoefficients(n1, reduced_height=None)
|
||
Helper function for matharExponential(), that calculates the gamma
|
||
coefficients.
|
||
H_isotherm(T)
|
||
Sets the reduced height for the conditions at the observer, assuming
|
||
an isothermal atmosphere.
|
||
matharExponential(n1, zenith, reduced_height=None)
|
||
Returns the atmospheric refraction assuming Mathar's barometric
|
||
exponential model.
|
||
oriani(n1, n2, zenith, reduced_height=None)
|
||
Returns the atmospheric refraction following the classic version
|
||
of Oriani's theorem.
|
||
planeParallel(n1, n2, zenith)
|
||
Returns the atmospheric refraction assuming a plane parallel
|
||
atmosphere.
|
||
psi(x)
|
||
Submethod for corbard().
|
||
refractionIntegral(n1, n2, zenith, R0=None, useStandardAtmosphere=True, heightData=None, rhoData=None)
|
||
Returns the atmospheric refraction following the refraction integral.
|
||
setReducedHeight(p, rho)
|
||
Sets the reduced atmospheric height.
|
||
set_H(p, rho)
|
||
Sets the reduced height for the conditions at the observer, assuming
|
||
an adiabatic atmosphere.
|
||
set_rc()
|
||
Sets the local radius of curvature of the Earth at the observer.
|
||
tan5(n1, n2, zenith, reduced_height=None)
|
||
Returns the atmospheric refraction following Oriani's theorem,
|
||
including a fifth order term in the expansion.
|
||
Other Observatory() methods
|
||
'''
|
||
def __init__(self, lat, h):
|
||
Observatory.__init__(self)
|
||
self.h = h # height above sea level in meters
|
||
self.lat = lat # Latitude of the observer in degrees
|
||
self.rc = self.set_rc() * 1000 # Radius of curvature of the earth at the observer
|
||
# self.H2 = self.H_isotherm(T) # Reduced height of isotherm atmosphere
|
||
|
||
|
||
def cassini(self, n1, zenith, reduced_height=None):
|
||
'''
|
||
Refraction of spherical atmosphere, derived from geometry using Sine law
|
||
|
||
Parameters
|
||
----------
|
||
n1 : float
|
||
Refractive index at the wavelength of interest
|
||
zenith : float
|
||
Angle of observation in degrees.
|
||
zenith = 0 means the observation is directly overhead.
|
||
reduced_height : float (optional)
|
||
Reduced height of the atmosphere. Default value is calculated
|
||
from the object attributes.
|
||
|
||
Returns
|
||
-------
|
||
R : float
|
||
Atmospheric refraction in degrees
|
||
'''
|
||
if reduced_height:
|
||
H = reduced_height
|
||
else:
|
||
H = self.H
|
||
|
||
_R1 = np.arcsin(n1 * self.rc * np.sin(np.deg2rad(zenith)) / (self.rc + H)) - np.arcsin(self.rc * np.sin(np.deg2rad(zenith)) / (self.rc + H) )
|
||
return np.degrees(_R1)
|
||
|
||
|
||
def cassiniError(self, n, dn, zenith, dz=0, reduced_height=None):
|
||
'''
|
||
Calculates the uncertainty in the refraction angle in degrees.
|
||
See Observatory().n_tph() and Observatory().dn_tph() on how to
|
||
calculate the refractive index and it's uncertainty.
|
||
|
||
Parameters
|
||
----------
|
||
n : float
|
||
Refractive index of air
|
||
dn : float
|
||
Uncertainty in the refractive index of air
|
||
zenith : float
|
||
Angle of observation in degrees.
|
||
zenith = 0 means the observation is directly overhead.
|
||
dz : float (optional)
|
||
Uncertainty in the zenith angle, in degrees.
|
||
reduced_height : float (optional)
|
||
Reduced height of the atmosphere. Default value is calculated
|
||
from the object attributes.
|
||
|
||
Returns
|
||
-------
|
||
dR : float
|
||
Uncertainty in the atmospheric refraction in degrees.
|
||
'''
|
||
if reduced_height:
|
||
_H = reduced_height
|
||
else:
|
||
_H = self.H
|
||
_zenith = np.deg2rad(zenith)
|
||
_dz = np.deg2rad(dz)
|
||
_R = self.rc
|
||
_dRdn = (_R * np.sin(_zenith)) / ((_R + _H) * np.sqrt(1 - ((n**2 * _R**2 * np.sin(_zenith)**2)/(_R + _H)**2)))
|
||
|
||
_dRdz = (_R * np.cos(_zenith)) / (_R + _H) * ( n / np.sqrt(1-(n**2 * _R**2 * np.sin(_zenith)**2)/(_R + _H)**2) - 1 / np.sqrt(1-(_R**2 * np.sin(_zenith)**2)/(_R + _H)**2) )
|
||
|
||
_dR = np.sqrt( (_dRdn * dn)**2 + (_dRdz * _dz)**2)
|
||
|
||
return np.degrees(_dR)
|
||
|
||
|
||
def corbard(self, n1, zenith, reduced_height=None):
|
||
'''
|
||
Corbard et al., 2019, mention an additional formula based on the error function.
|
||
Oriani's theorem can be derived from this equation by 'keeping only the three first
|
||
terms of its asymptotic expansion'.
|
||
|
||
Parameters
|
||
----------
|
||
n1 : float
|
||
Refractive index at the wavelength of interest
|
||
zenith : float
|
||
Angle of observation in degrees.
|
||
zenith = 0 means the observation is directly overhead.
|
||
reduced_height : float (optional)
|
||
Reduced height of the atmosphere. Default value is calculated
|
||
from the object attributes.
|
||
|
||
Returns
|
||
-------
|
||
R : float
|
||
Atmospheric refraction in degrees
|
||
'''
|
||
if reduced_height:
|
||
H = reduced_height
|
||
else:
|
||
H = self.H
|
||
|
||
_a = n1 - 1
|
||
_b = H / self.rc
|
||
_R = _a * ( (2 - _a) / (np.sqrt( 2*_b - _a )) ) * np.sin( np.deg2rad(zenith) ) * self.psi( (np.cos(np.deg2rad(zenith))) / np.sqrt(2*_b - _a) )
|
||
return np.degrees(_R)
|
||
|
||
|
||
def gammaCoefficients(self, n1, reduced_height=None):
|
||
'''
|
||
Submethod for matharExponential(). Calculates the gamma
|
||
coefficients.
|
||
|
||
Parameters
|
||
----------
|
||
n1 : float
|
||
Refractive index at the wavelength of interest at the observer
|
||
reduced_height : float (optional)
|
||
Reduced height of the atmosphere. Default value is calculated
|
||
from the object attributes.
|
||
|
||
Returns
|
||
-------
|
||
g : array
|
||
List of values, corresponding to the gamma coefficients.
|
||
g[0] corresponds to g_1, g[1] to g_3, g[2] to g_5, and so on.
|
||
'''
|
||
|
||
if reduced_height:
|
||
H = reduced_height
|
||
else:
|
||
H = self.H
|
||
|
||
_chi0 = n1**2 - 1
|
||
_Khat = H / self.rc
|
||
|
||
_g1 = (1 - 5/24 * _Khat * _chi0**2 - 3/4 * _chi0 - 3/8 * _Khat**2 * _chi0 \
|
||
- _Khat - 6 * _Khat**3 + 5/8 * _chi0**2 + 63/128 * _chi0**4 + 3/8 \
|
||
* _Khat * _chi0 + 35/256 * _Khat * _chi0**3 + 2 * _Khat**2 + 5/36 \
|
||
* _Khat**2 * _chi0**2 - 35/64 * _chi0**3 + 9/16 * _Khat**3 * _chi0 \
|
||
- 120 * _Khat**5 + 24 * _Khat**4 - 231 / 512 * _chi0**5 - 63 / 640 \
|
||
* _Khat * _chi0**4 - 35 / 512 * _Khat**2 * _chi0**3 - 5 / 36 * _Khat**3\
|
||
* _chi0**2 - 9 / 8 * _Khat**4 * _chi0 )
|
||
|
||
_g3 = (35/96 * _chi0**3 - 5/12 * _chi0**2 + 333 * _Khat**4 * _chi0 - 2400 \
|
||
* _Khat**5 + 3605/4608 * _Khat**2 * _chi0**3 + 63/256 * _Khat * _chi0**4\
|
||
- 819/16 * _Khat**3 * _chi0 - 35/18 * _Khat**2 * _chi0**2 - 35/96 \
|
||
* _Khat * _chi0**3 + 69/8 * _Khat**2 * _chi0 + 5/8 * _Khat * _chi0**2 \
|
||
- 3/2 * _Khat * _chi0 + 1355/216 * _Khat**3 * _chi0**2 + 10 * _Khat**2 \
|
||
+ 336 * _Khat**4 - 2 * _Khat + _chi0/2 - 21/64 * _chi0**4 - 54 * _Khat**3 \
|
||
+ 77/256 * _chi0**5 ) * 1/2
|
||
|
||
_g5 = (-96 * _Khat**3 + 8 * _Khat**2 - 7/24 * _chi0**3 + 65/4 * _Khat**2 \
|
||
* _chi0**2 + 35/48 * _Khat * _chi0**3 + 30 * _Khat**2 * _chi0 - 5/3 \
|
||
* _Khat * _chi0**2 - 3 * _Khat * _chi0 + 20091/8 * _Khat**4 * _chi0 \
|
||
- 32605/216 * _Khat**3 * _chi0**2 - 35/9 * _Khat**2 * _chi0**3 + 984 \
|
||
* _Khat**4 + _chi0**2/3 - 10200 * _Khat**5 - 77/320 * _chi0**5 + 21/80 \
|
||
* _chi0**4 - 1089/4 * _Khat**3 * _chi0 - 7/16 * _Khat * _chi0**4 ) * 3/8
|
||
|
||
_g7 = (-17520 * _Khat**5 - 9/40 * _chi0**4 - 8755/12 * _Khat**3 * _chi0**2 \
|
||
+ 1056 * _Khat**4 + _chi0**3/4 + 23517/4 * _Khat**4 * _chi0 - 11/3 \
|
||
* _Khat * _chi0**2 + 21 * _Khat**2 * _chi0 - 7/4 * _Khat * _chi0**3 \
|
||
+ 505/9 * _Khat**2 * _chi0**2 - 801/2 * _Khat**3 * _chi0 + 33/160 \
|
||
* _chi0**5 - 48 * _Khat**3 + 63/80 * _Khat * _chi0**4 + 2345/96 \
|
||
* _Khat**2 * _chi0**3) * 5/16
|
||
|
||
_g9 = (_chi0**4 / 5 + 384 * _Khat**4 - 13440 * _Khat**5 + 5562 * _Khat**4 \
|
||
* _chi0 + 3125/36 * _Khat**2 * _chi0**3 - 11/60 * _chi0**5 - 25/6 \
|
||
* _Khat * _chi0**3 - 26950/27 * _Khat**3 * _chi0**2 + 340/9 * _Khat**2 \
|
||
* _chi0**2 - 180 * _Khat**3 * _chi0 - 9/5 * _Khat * _chi0**4) * 35/128
|
||
|
||
g = np.array([_g1, _g3, _g5, _g7, _g9])
|
||
g*= n1*_chi0/2
|
||
|
||
return g
|
||
|
||
|
||
def H_isotherm(self, T):
|
||
'''
|
||
Calculates the reduced height of the atmosphere,
|
||
assuming an isotherm distribution.
|
||
|
||
Parameters
|
||
----------
|
||
T : float
|
||
Atmospheric temperature in Kelvin
|
||
|
||
Returns
|
||
-------
|
||
H : float
|
||
Reduced height of the atmosphere in meters
|
||
'''
|
||
_kb = 1.38064852e-23 # kg m2 s-2 K-1
|
||
_m = 4.809651698e-26 # kg (weight per molecule of air, assuming dry air)
|
||
|
||
_phi = np.deg2rad(self.lat)
|
||
_c1 = 5.2790414e-3
|
||
_c2 = 2.32718e-5
|
||
_c3 = 1.262e-7
|
||
_c4 = 7e-10
|
||
_g0 = 9.780327 # m/s^2
|
||
_g0_local = _g0 * (1 + _c1 * np.sin(_phi)**2 + _c2 * np.sin(_phi)**4 + _c3 * np.sin(_phi)**6 + _c4 * np.sin(_phi)**8)
|
||
|
||
return (_kb*T) / (_m * _g0_local)
|
||
|
||
|
||
def matharExponential(self, n1, zenith, reduced_height=None):
|
||
'''
|
||
Model on atmospheric refraction described by Richard Mathar in
|
||
"A Barometric Exponential Model of the Atmosphere’s Refractive Index: ZenithAngles and Second Order Aberration in the Entrance Pupil" (2020)
|
||
ArXiv id: 2004.11808 (https://arxiv.org/abs/2004.11808)
|
||
|
||
The code in this function is adapted from eqs 3, 16, 17 and 18 in this
|
||
work. It is an adaptation of the RefrExpo.cxx C++ file written by Mathar.
|
||
|
||
This model assumes the refractive index, n, is related to the susceptibility,
|
||
chi, following: n(r) = sqrt(1+chi(r)) (eq. 9).
|
||
|
||
R. Mathar then assumes that the susceptibility is proportional to the
|
||
air density and reduced height, following eq. 10:
|
||
chi(r) = chi0 * exp(-(r-rho)/K).
|
||
|
||
The refraction is described using a Taylor expansion; R in terms of
|
||
the odd powers of the tangent of the observed zenith angle (eq. 3), not
|
||
unlike Oriani's theorem. This requires coefficients g_i.
|
||
|
||
Parameters
|
||
----------
|
||
n1 : float
|
||
Refractive index at the wavelength of interest at the observer
|
||
zenith : float
|
||
Angle of observation in degrees.
|
||
zenith = 0 means the observation is directly overhead.
|
||
reduced_height : float (optional)
|
||
Reduced height of the atmosphere. Default value is calculated
|
||
from the object attributes.
|
||
|
||
Returns
|
||
-------
|
||
R : float
|
||
Atmospheric refraction in degrees
|
||
'''
|
||
|
||
if reduced_height:
|
||
_H = reduced_height
|
||
else:
|
||
_H = self.H
|
||
|
||
_zenith = np.deg2rad(zenith)
|
||
_rc = self.rc # rho
|
||
|
||
# Atmospheric refraction in radians
|
||
_gamma = self.gammaCoefficients(n1, reduced_height=_H)
|
||
_R = sum([_gamma[i]*np.tan(_zenith)**(2*i+1) for i in range(len(_gamma))])
|
||
|
||
return np.degrees(_R)
|
||
|
||
|
||
def oriani(self, n1, zenith, reduced_height=None):
|
||
'''
|
||
Classic refraction formula with two tan(z) terms. This does not
|
||
assume any information about the structure of the atmosphere.
|
||
Found with Oriani's theorem.
|
||
|
||
Parameters
|
||
----------
|
||
n1 : float
|
||
Refractive index at the wavelength of interest
|
||
zenith : float
|
||
Angle of observation in degrees.
|
||
zenith = 0 means the observation is directly overhead.
|
||
reduced_height : float (optional)
|
||
Reduced height of the atmosphere. Default value is calculated
|
||
from the object attributes.
|
||
|
||
Returns
|
||
-------
|
||
R : float
|
||
Atmospheric refraction in degrees
|
||
'''
|
||
if reduced_height:
|
||
H = reduced_height
|
||
else:
|
||
H = self.H
|
||
|
||
_a = n1 - 1
|
||
_b = H / self.rc
|
||
_R = _a * (1 - _b) * np.tan( np.deg2rad(zenith) ) - _a * (_b - _a/2) * np.tan( np.deg2rad(zenith) )**3
|
||
return np.degrees(_R)
|
||
|
||
|
||
def planeParallel(self, n1, zenith):
|
||
'''
|
||
Refraction based on a flat atmosphere, easily derived from Snell's law.
|
||
|
||
Parameters
|
||
----------
|
||
n1 : float
|
||
Refractive index at the wavelength of interest
|
||
zenith : float
|
||
Angle of observation in degrees.
|
||
zenith = 0 means the observation is directly overhead.
|
||
|
||
Returns
|
||
-------
|
||
R : float
|
||
Atmospheric refraction in degrees
|
||
'''
|
||
_R = np.arcsin(n * np.sin( np.deg2rad(zenith) )) - np.deg2rad(zenith)
|
||
return np.degrees(_R)
|
||
|
||
|
||
def psi(self, x):
|
||
'''
|
||
Submethod for corbard(). Described explicitly in Corbard et al.,2019.
|
||
|
||
Parameters
|
||
----------
|
||
x : float
|
||
Input of the function
|
||
|
||
Returns
|
||
-------
|
||
y : float
|
||
Output of the function
|
||
'''
|
||
_f = lambda a: np.exp(-a**2)
|
||
_int = integrate.quad(_f, x, np.inf)
|
||
return np.exp(x**2) * _int[0]
|
||
|
||
|
||
def refractionIntegral(self, n1, zenith, R0=None, useStandardAtmosphere=True,
|
||
heightData=None, rhoData=None):
|
||
'''
|
||
Using data that gives temperature, pressure and density as a
|
||
function of height, the gladstone-dale relation is invoked to
|
||
calculate the path of a light ray along the atmosphere.
|
||
This is allows us to also include atmospheric activity in
|
||
different atmosphere layers.
|
||
Recommended use with the US Standard Atmosphere from 1976.
|
||
USSA1976 data generated from code supplied by http://www.pdas.com/atmos.html
|
||
The method described in eq. 3 in Auer & Standish (2000) is used for the
|
||
calculation.
|
||
|
||
Parameters
|
||
----------
|
||
n1 : float
|
||
Refractive index at the wavelength of interest
|
||
zenith : float
|
||
Angle of observation in degrees.
|
||
zenith = 0 means the observation is directly overhead.
|
||
R0 : float (optional)
|
||
Local radius of curvature of the Earth in km.
|
||
useStandardAtmosphere : bool
|
||
Flag to use the included USSA1976 data
|
||
heightData : 1D Array
|
||
If "useStandardAtmosphere" is False, the function will ask
|
||
for alternative atmospheric data to integrate over. This
|
||
gives the altitudinal points
|
||
rhoData : 1D Array
|
||
If useStandardAtmosphere is False, the function will ask
|
||
for alternative atmospheric data to integrate over. This
|
||
gives the atmospheric density at the points defined by
|
||
"heightData".
|
||
|
||
Returns
|
||
-------
|
||
R : float
|
||
Atmospheric refraction in degrees
|
||
|
||
'''
|
||
if not R0:
|
||
R0 = self.rc
|
||
if useStandardAtmosphere:
|
||
_datafileLocation = os.path.join(os.path.dirname(__file__), 'data/1976USSA.txt')
|
||
_h, _, _, _, _Temp, _Press, _Dens,_,_,_ = np.genfromtxt(_datafileLocation, unpack=True, skip_header=6)
|
||
else:
|
||
_Dens = rhoData
|
||
_h = heightData
|
||
|
||
zenith = np.deg2rad(zenith)
|
||
_Dens = np.asarray(_Dens)
|
||
_h = np.asarray(_h)
|
||
_R = R0 + _h * 1000
|
||
|
||
# Gladstone-Dale relation between refractive index and density
|
||
_n_h = 1 + (n0 - 1) / _Dens[0] * _Dens
|
||
|
||
# Calculation of the zenith angle for given height and refractive index, from the refractive invariant
|
||
_z = np.arcsin( (n0 * R0) / (_n_h * _R) * np.sin(zenith) )
|
||
|
||
# Calculation of log space derivative of n and r
|
||
_lnn = np.log(_n_h)
|
||
_lnr = np.log(_R)
|
||
_dlnn = np.asarray([_lnn[i+1] - _lnn[i] for i in range(len(_lnn)-1)])
|
||
_dlnr = np.asarray([_lnr[i+1] - _lnr[i] for i in range(len(_lnr)-1)])
|
||
_dz = np.asarray([(_z[i] - _z[i+1]) for i in range(len(_z)-1)])
|
||
|
||
# Calculation of the integrand in eq. 3 in Auer & Standish
|
||
_Integrand = -1*(_dlnn/_dlnr) / (1 + _dlnn/_dlnr) * _dz
|
||
|
||
|
||
return np.degrees(np.sum(_Integrand))
|
||
|
||
|
||
def setReducedHeight(self, p, rho):
|
||
'''
|
||
Sets the reduced atmospheric height.
|
||
|
||
Parameters
|
||
----------
|
||
p : float
|
||
Atmospheric pressure in Pascal
|
||
rho : float
|
||
Density of the atmospheric air at the observer
|
||
'''
|
||
self.H = self.set_H(p, rho) # Reduced height of the atmosphere assuming ideal gas law
|
||
|
||
|
||
def set_H(self, p, rho):
|
||
'''
|
||
Calculates the reduced height of the atmosphere, assuming ideal gas law.
|
||
|
||
Parameters
|
||
----------
|
||
p : float
|
||
Atmospheric pressure in Pascal
|
||
rho : float
|
||
Density of the atmospheric air at the observer
|
||
|
||
Returns
|
||
-------
|
||
H : float
|
||
Reduced height of the atmosphere in meters
|
||
'''
|
||
_phi = np.deg2rad(self.lat)
|
||
_c1 = 5.2790414e-3
|
||
_c2 = 2.32718e-5
|
||
_c3 = 1.262e-7
|
||
_c4 = 7e-10
|
||
_g0 = 9.780327 # m/s^2
|
||
_g0_local = _g0 * (1 + _c1 * np.sin(_phi)**2 + _c2 * np.sin(_phi)**4 + _c3 * np.sin(_phi)**6 + _c4 * np.sin(_phi)**8)
|
||
_g = _g0_local - (3.0877e-6 - 4.3e-9 * np.sin(_phi)**2) * self.h + 7.2e-13 * self.h**2
|
||
return p / (rho * _g)
|
||
|
||
|
||
def set_rc(self):
|
||
'''
|
||
Returns the radius of the curvature at the observer in km,
|
||
taking into account the latitude of the observer and the corresponding
|
||
ellipsoidality of the earth.
|
||
|
||
Returns
|
||
-------
|
||
rc : float
|
||
Local radius of curvature of the Earth in kilometers.
|
||
'''
|
||
_A = 6378.137 # km
|
||
_B = 6356.752 # km
|
||
_phi = np.deg2rad(self.lat)
|
||
|
||
_rc0 = (_A*_B)**2 / (_A**2 * np.cos(_phi)**2 + _B**2 * np.sin(_phi)**2)**(3/2)
|
||
return _rc0 + self.h/1000
|
||
|
||
|
||
def tan5(self, n1, zenith, reduced_height=None):
|
||
'''
|
||
Same as oriani's theorem, but including a higher order term.
|
||
Found in Corbard et al., 2019.
|
||
|
||
Parameters
|
||
----------
|
||
n1 : float
|
||
Refractive index at the wavelength of interest
|
||
zenith : float
|
||
Angle of observation in degrees.
|
||
zenith = 0 means the observation is directly overhead.
|
||
reduced_height : float (optional)
|
||
Reduced height of the atmosphere. Default value is calculated
|
||
from the object attributes.
|
||
|
||
Returns
|
||
-------
|
||
R : float
|
||
Atmospheric refraction in degrees
|
||
'''
|
||
if reduced_height:
|
||
H = reduced_height
|
||
else:
|
||
H = self.H
|
||
|
||
_a = n1 - 1
|
||
_b = H / self.rc
|
||
_R = _a * (1 - _b) * np.tan( np.deg2rad(zenith) ) - _a * (_b - _a/2) * np.tan( np.deg2rad(zenith) )**3 + 3 * _a * (_b - _a/2)**2 * np.tan( np.deg2rad(zenith) )**5
|
||
return np.degrees(_R)
|
||
|