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Collection of equations for refractive index, refraction and dispersion calculations of atmospheric air.

Project description


Note: It would be appreciated if a reference to the following work, for which this package was originally build, is included whenever this code is used for a publication: Quantification of the expected residual dispersion of the MICADO Near-IR imaging instrument, van den Born & Jellema, 2020, MNRAS, DOI: 10.1093/mnras/staa1870.

Author info: Joost van den Born,


A minimal working example to calculate the atmospheric dispersion at Cerro Armazones between 1.49 and 1.78 micron.

from AstroAtmosphere import *

# Parameters at Cerro Armazones
T   = 279.65    # k
P   = 71200     # Pa
H   = 0.22
xc  = 450       # ppm
lat = -24.5983  # degrees
h   = 3064      # m
l1  = 1.49      # micron
l2  = 1.78      # micron
z0  = 30

# Initializing dispersion model
at  = Observatory()

# Calculating indices of refraction for l1 and l2
n1  = at.n_tph(l=l1, T=T, p=P, RH=H, xc=xc)
n2  = at.n_tph(l=l2, T=T, p=P, RH=H, xc=xc)

# Density of the atmosphere (following CIPM-81/91 equations)
rho = at.rho(p=P, T=T, RH=H, xc=xc)

# Initializing refraction model and setting the reduced height
disp = dispersion(lat, h)
disp.setReducedHeight(P, rho)

# Calculation of the atmopheric dipsersion
atm_dispersion = disp.cassini(n1, n2, z0) * 3.6e6
print ('The dispersion is %.03f milli arc seconds' %(atm_dispersion))


The package is available through Git-Lab and through PyPI. The package may be installed using the file, but easier is to install using pip.

pip install AstroAtmosphere

Contents of the AstroAtmosphere package

The package contains the following files:

  • /data/1976USSA.txt : Data file of the 1976 US Standard Atmosphere.
  • : Contains the Observatory() class to calculate the refractive index of atmospheric air, using the Ciddor (1996) model.
  • : Contains various atmospheric models that can be used to calculate the atmospheric dispersion
  • : Contains several miscellaneous functions, e.g. to quickly calculate the refractive index.
  • : Contains a Python native port of the pyslalib.SLA_REFRO() routine, translated from the original FORTRAN code.
  • : Contains a dictionary with the average atmospheric conditions for various large observatories around the world.
  • : Contains various atmospheric models that can be used to calculate the atmospheric refraction
  • : Contains various functions from literature to calculate the refractive index. Some examples of works included are Ciddor (1996), Barrell & Sears (1939), SLALIB, Owens (1967) and Edlen (1966).

The Ciddor refractivity model

The refractivity model from Ciddor, 1996, calculates the index of refraction of air using the following parameters:

  • T - Temperature in Kelvin
  • p - Pressure in Pascal
  • RH - Relative humidity in fractional units
  • xc - CO2 density in parts per million

It is invoked by first initializing an Observatory() object. The atmospheric conditions can be parsed into the n_tph() function.

at  = Observatory()
n   = at.n_tph(l=0.55, T=273.15, p=101325, RH=0.2, xc=300)

Basic functionality of other refractivity models is included as well and is discussed below.

The atmospheric refraction and atmospheric dispersion models

If the atmospheric refraction is denoted as R(n), then atmospheric dispersion is defined as R(n1) - R(n2). Thus they are very similar in function and use.

The available models in this package:

  • refractionIntegral() - Full integration using the refractive integral (requires detailed information about the atmosphere, e.g. AstroAtmosphere/data/1976USSA.txt)
  • planeParallel() - Plane-parallel atmosphere refraction model
  • cassini() - Cassini's homogeneous atmosphere refraction model
  • oriani() - Oriani's theorem ( Atan(z) + Btan^3(z) )
  • tan5() - Oriani's theorem expanded ( Atan(z) + Btan^3(z) + Ctan^5(z) )
  • corbard() - Error function refraction model
  • matharExponential() - Barometric exponential model by R. Mathar.

The package also includes a python native port of the SLA_REFRO() FORTRAN routine, ported from pyslalib. See below for more information.

Input parameters for the spherical atmosphere models (all but the first):

  • T - Temperature [K]
  • p - Pressure [Pa]
  • RH - Relative humidity
  • xc - CO2 density [ppm]
  • lat - Latitude [deg]
  • h - Height above sea level [m]
  • l - Wavelength(s) [um]
  • z - Zenith angle [deg]

An example - To calculate the atmospheric refraction we set up a refraction() object. This requires the latitude and altitude of the observer. The atmospheric dispersion is calculated similarly, but by exchanging the refraction() object by a dispersion() object and parsing two refractive indices into the cassini() function.

# Density of the atmosphere (following CIPM-81/91 equations)
rho = at.rho(p=101325, T=273.15, RH=0.2, xc=300)

# Initializing refraction model and setting the reduced height
ref = refraction(lat=30, h=2200)
ref.setReducedHeight(p=101325, rho=rho)

# Calculating the atmospheric refraction
atm_ref = ref.cassini(n, zenith=30)

Analytical error propagation

Full analytical error propagation is available for the Ciddor dispersion model (Observatory.dn_tph()), for the Cassini refraction model (refraction.cassiniError()) and for the Cassini dispersion model (dispersion.cassiniError()). The results agree well with a Monte Carlo simulation, suggesting that if additional errors have not been considered, they are of neglible importance.

Uncertainties that are considered:

Ciddor dispersion model:

Param. Description Unit
dl Wavelength um
dT Temperature K
dP Pressure Pa
dRH Relative Humidity
dCO2 CO2 density ppm

Cassini refraction model:

Param. Description Unit
dn Refractive index
dz Zenith angle deg

Cassini dispersion model:

Param. Description Unit
dl1, dl2 Wavelength(s) um
dT Temperature K
dP Pressure Pa
dRH Relative Humidity
dCO2 CO2 density ppm
dz Zenith angle deg

An example of its use is given in the code snippet here:

 # Initializing dispersion object
disp = dispersion(lat, h)
disp.setReducedHeight(P, rho)

# Atmospheric dispersion
dispersion_obs = disp.cassini(n1, n2, zenith_obs)

# Uncertainty in the atmospheric dispersion
dispersion_unc = disp.cassiniError(zenith_obs, l1, l2, T, P, H, xc, 
    dl1=0.001, dl2=0.001, dT=0.2, dP=30, dRH=0.03, dCO2=25, dz=1/3600) * 3.6e6

Other refractivity models

The package also includes limited functionality of other refractivity models. These are not as extensive, work only under standard atmospheric conditions, or over a limited wavelength range, compared to the Ciddor model.

Included are

Model name Function Remarks
Barrell & Sears (1939) BarrellAndSears() Outdated model, especially poor for short wavelengths.
Ciddor (1996) Ciddor() Only at standard atmosphere. CO2 density can be changed.
Edlen (1953) Edlen1953() The equation by Edlèn, 1953, combined with modification from Barrel (1951), to include temperature, pressure and humidity dependence.
Edlen (1966) Edlen1966() Only at standard atmosphere. Function requires wavelength only.
Owens (1967) Owens() -
Bonsch & Potulski (1998) BonschPotulski() Function does not include humidity
Birch & Downs (1994) BirchDowns() Only at standard atmosphere. Function requires wavelength only.
Peck & Reeder (1972) PeckReeder() Only at standard atmosphere. Function requires wavelength only.
Hohenkerk & Sinclair (1985) HohenkerkAndSinclair() Based on Barrell & Sears equation. Generally, gives slightly larger values.
Hohenkerk & Sinclair (1985) slalib() Uses pySLALIB to retrieve the refractive index at standard atmospheric conditions.

The last function extracts the refractive index from the pyslalib.sla_refro() function and thus requires pyslalib. This is an optional dependency for the setup of AstroAtmosphere.

Quick functions

Some functions can make use of the observatories dictionary as input. These include default average conditions of the atmosphere at these locations. When these are used, only the wavelength and zenith angle are required to calculate the refraction and dispersion.

 # Quick refractive index
nq = quick_refractive_index(l=1.49, conditions='STANDARD')

# Quick refraction
rq = quick_refraction(l1, zenith_obs, conditions='STANDARD')

# Quick dispersion
dq = quick_dispersion(l1, l2, zenith_obs, conditions='STANDARD')

Presently, several observatories are known. These can all be found in the observatories dictionary. Contact the author to add additional standard conditions.

Observatory Flag Notes
Standard conditions STANDARD
Cerro Armazones CERRO_ARMAZONES Location of the future ELT
Cerro Paranal CERRO_PARANAL Location of the VLT, VISTA, VST, NGTS and SSO
Roque de los Muchachos Observatory (La Palma) LA_PALMA GTC, INT, WHT, NOT, and others.
La Silla Observatory LA_SILLA ESO 3.6m telescope, NTG, BlackGEM and others.
Las Campanas Observatory LAS_CAMPANAS Site of the future GMT.


This package also contains a python native port of the SLA_REFRO() routine of the pyslalib package, which was originally written in FORTRAN by P.T. Wallace. It can be called directly using sla_refro(). When using the same units as the other refraction functions, you might prefer using slalib_refraction() or slalib_dispersion().

Finally, the dispersion model used by Alexei Filippenko in 1982 is also included. This model is a combination of the (modified) Edlèn equation and a plane-parallel atmospheric geometry. This calculation can be accessed by calling the Filippenko1982() function.

Project details

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