refraction.py

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# LSST Data Management System
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from astropy import units
from astropy.units import cds
import numpy as np
 
import lsst.geom
from lsst.afw.coord.weather import Weather
 
__all__ = ["refraction", "differentialRefraction"]
 
# The differential refractive index is the difference of the refractive index from 1.,
#    multiplied by 1E8 to simplify the notation and equations.
deltaRefractScale = 1.0E8
 
 
def refraction(wavelength, elevation, observatory, weather=None):
    """Calculate overall refraction under atmospheric and observing conditions.
 
    Parameters
    ----------
    wavelength : `float`
        wavelength is in nm (valid for 230.2 < wavelength < 2058.6)
    elevation : `lsst.geom.Angle`
        Elevation of the observation, as an Angle.
    observatory : `lsst.afw.coord.Observatory`
        Class containing the longitude, latitude,
        and altitude of the observatory.
    weather : `lsst.afw.coord.Weather`, optional
        Class containing the measured temperature, pressure, and humidity
        at the observatory during an observation
        If omitted, typical conditions for the observatory's elevation will be calculated.
 
    Returns
    -------
    refraction : `lsst.geom.Angle`
        The angular refraction for light of the given wavelength,
        under the given observing conditions.
 
    Notes
    -----
    The calculation is taken from [1]_.
 
    References
    ----------
    .. [1] R. C. Stone, "An Accurate Method for Computing Atmospheric
       Refraction," Publications of the Astronomical Society of the Pacific,
       vol. 108, p. 1051, 1996.
    """
    if wavelength < 230.2:
        raise ValueError("Refraction calculation is valid for wavelengths between 230.2 and 2058.6 nm.")
    if wavelength > 2058.6:
        raise ValueError("Refraction calculation is valid for wavelengths between 230.2 and 2058.6 nm.")
    latitude = observatory.getLatitude()
    altitude = observatory.getElevation()
    if weather is None:
        weather = defaultWeather(altitude*units.meter)
    reducedN = deltaN(wavelength, weather)/deltaRefractScale
    temperature = extractTemperature(weather, useKelvin=True)
    atmosScaleheightRatio = 4.5908E-6*temperature.to_value(units.Kelvin)
 
    # Account for oblate Earth
    # This replicates equation 10 of Stone 1996
    relativeGravity = (1. + 0.005302*np.sin(latitude.asRadians())**2.
                       - 0.00000583*np.sin(2.*latitude.asRadians())**2. - 0.000000315*altitude)
 
    # Calculate the tangent of the zenith angle.
    tanZ = np.tan(np.pi/2. - elevation.asRadians())
    atmosTerm1 = reducedN*relativeGravity*(1. - atmosScaleheightRatio)
    atmosTerm2 = reducedN*relativeGravity*(atmosScaleheightRatio - reducedN/2.)
    result = float(atmosTerm1*tanZ + atmosTerm2*tanZ**3.)*lsst.geom.radians
    return result
 
 
def differentialRefraction(wavelength, wavelengthRef, elevation, observatory, weather=None):
    """Calculate the differential refraction between two wavelengths.
 
    Parameters
    ----------
    wavelength : `float`
        wavelength is in nm (valid for 230.2 < wavelength < 2058.6)
    wavelengthRef : `float`
        Reference wavelength, typically the effective wavelength of a filter.
    elevation : `lsst.geom.Angle`
        Elevation of the observation, as an Angle.
    observatory : `lsst.afw.coord.Observatory`
        Class containing the longitude, latitude,
        and altitude of the observatory.
    weather : `lsst.afw.coord.Weather`, optional
        Class containing the measured temperature, pressure, and humidity
        at the observatory during an observation
        If omitted, typical conditions for the observatory's elevation will be calculated.
 
    Returns
    -------
    differentialRefraction : `lsst.geom.Angle`
        The refraction at `wavelength` minus the refraction at `wavelengthRef`.
    """
    refractionStart = refraction(wavelength, elevation, observatory, weather=weather)
    refractionEnd = refraction(wavelengthRef, elevation, observatory, weather=weather)
    return refractionStart - refractionEnd
 
 
def deltaN(wavelength, weather):
    """Calculate the differential refractive index of air.
 
    Parameters
    ----------
    wavelength : `float`
        wavelength is in nanometers
    weather : `lsst.afw.coord.Weather`
        Class containing the measured temperature, pressure, and humidity
        at the observatory during an observation
 
    Returns
    -------
    deltaN : `float`
        The difference of the refractive index of air from 1.,
        calculated as (n_air - 1)*10^8
 
    Notes
    -----
    The differential refractive index is the difference of
    the refractive index from 1., multiplied by 1E8 to simplify
    the notation and equations. Calculated as (n_air - 1)*10^8
 
    This replicates equation 14 of [1]_
 
    References
    ----------
    .. [1] R. C. Stone, "An Accurate Method for Computing Atmospheric
       Refraction," Publications of the Astronomical Society of the Pacific,
       vol. 108, p. 1051, 1996.
    """
    waveNum = 1E3/wavelength  # want wave number in units 1/micron
    dryAirTerm = 2371.34 + (683939.7/(130. - waveNum**2.)) + (4547.3/(38.9 - waveNum**2.))
    wetAirTerm = 6487.31 + 58.058*waveNum**2. - 0.71150*waveNum**4. + 0.08851*waveNum**6.
    return (dryAirTerm*densityFactorDry(weather) + wetAirTerm*densityFactorWater(weather))
 
 
def densityFactorDry(weather):
    """Calculate dry air pressure term to refractive index calculation.
 
    Parameters
    ----------
    weather : `lsst.afw.coord.Weather`
        Class containing the measured temperature, pressure, and humidity
        at the observatory during an observation
 
    Returns
    -------
    densityFactor : `float`
        Returns the relative density of dry air
        at the given pressure and temperature.
 
    Notes
    -----
    This replicates equation 15 of [1]_
 
    References
    ----------
    .. [1] R. C. Stone, "An Accurate Method for Computing Atmospheric
       Refraction," Publications of the Astronomical Society of the Pacific,
       vol. 108, p. 1051, 1996.
    """
    temperature = extractTemperature(weather, useKelvin=True)
    waterVaporPressure = humidityToPressure(weather)
    airPressure = weather.getAirPressure()*units.pascal
    dryPressure = airPressure - waterVaporPressure
    eqn = dryPressure.to_value(cds.mbar)*(57.90E-8 - 9.3250E-4/temperature.to_value(units.Kelvin)
                                          + 0.25844/temperature.to_value(units.Kelvin)**2.)
    densityFactor = (1. + eqn)*dryPressure.to_value(cds.mbar)/temperature.to_value(units.Kelvin)
    return densityFactor
 
 
def densityFactorWater(weather):
    """Calculate water vapor pressure term to refractive index calculation.
 
    Parameters
    ----------
    weather : `lsst.afw.coord.Weather`
        Class containing the measured temperature, pressure, and humidity
        at the observatory during an observation
 
    Returns
    -------
    densityFactor : `float`
        Returns the relative density of water vapor
        at the given pressure and temperature.
 
    Notes
    -----
    This replicates equation 16 of [1]_
 
    References
    ----------
    .. [1] R. C. Stone, "An Accurate Method for Computing Atmospheric
       Refraction," Publications of the Astronomical Society of the Pacific,
       vol. 108, p. 1051, 1996.
    """
    temperature = extractTemperature(weather, useKelvin=True)
    waterVaporPressure = humidityToPressure(weather)
    densityEqn1 = (-2.37321E-3 + 2.23366/temperature.to_value(units.Kelvin)
                   - 710.792/temperature.to_value(units.Kelvin)**2.
                   + 7.75141E-4/temperature.to_value(units.Kelvin)**3.)
    densityEqn2 = waterVaporPressure.to_value(cds.mbar)*(1. + 3.7E-4*waterVaporPressure.to_value(cds.mbar))
    relativeDensity = waterVaporPressure.to_value(cds.mbar)/temperature.to_value(units.Kelvin)
    densityFactor = (1 + densityEqn2*densityEqn1)*relativeDensity
 
    return densityFactor
 
 
def humidityToPressure(weather):
    """Convert humidity and temperature to water vapor pressure.
 
    Parameters
    ----------
    weather : `lsst.afw.coord.Weather`
        Class containing the measured temperature, pressure, and humidity
        at the observatory during an observation
 
    Returns
    -------
    pressure : `astropy.units.Quantity`
        The water vapor pressure in Pascals
        calculated from the given humidity and temperature.
 
    Notes
    -----
    This replicates equations 18 & 20 of [1]_
 
    References
    ----------
    .. [1] R. C. Stone, "An Accurate Method for Computing Atmospheric
       Refraction," Publications of the Astronomical Society of the Pacific,
       vol. 108, p. 1051, 1996.
    """
    humidity = weather.getHumidity()
    x = np.log(humidity/100.0)
    temperature = extractTemperature(weather)
    temperatureEqn1 = (temperature + 238.3*units.Celsius)*x + 17.2694*temperature
    temperatureEqn2 = (temperature + 238.3*units.Celsius)*(17.2694 - x) - 17.2694*temperature
    dewPoint = 238.3*temperatureEqn1/temperatureEqn2
    waterVaporPressure = (4.50874 + 0.341724*dewPoint + 0.0106778*dewPoint**2 + 0.184889E-3*dewPoint**3
                          + 0.238294E-5*dewPoint**4 + 0.203447E-7*dewPoint**5)*133.32239*units.pascal
 
    return waterVaporPressure
 
 
def extractTemperature(weather, useKelvin=False):
    """Thin wrapper to return the measured temperature from an observation.
 
    Parameters
    ----------
    weather : `lsst.afw.coord.Weather`
        Class containing the measured temperature, pressure, and humidity
        at the observatory during an observation
    useKelvin : bool, optional
        Set to True to return the temperature in Kelvin instead of Celsius
        This is needed because Astropy can't easily convert
        between Kelvin and Celsius.
 
    Returns
    -------
    temperature : `astropy.units.Quantity`
        The temperature in Celsius, unless `useKelvin` is set.
    """
    temperature = weather.getAirTemperature()*units.Celsius
    if useKelvin:
        temperature = temperature.to(units.Kelvin, equivalencies=units.temperature())
    return temperature
 
 
def defaultWeather(altitude):
    """Set default local weather conditions if they are missing.
 
    Parameters
    ----------
    weather : `lsst.afw.coord.Weather`
        Class containing the measured temperature, pressure, and humidity
        at the observatory during an observation
    altitude : `astropy.units.Quantity`
        The altitude of the observatory, in meters.
 
    Returns
    -------
    default : `lsst.afw.coord.Weather`
        Updated Weather class with any `nan` values replaced by defaults.
    """
    if isinstance(altitude, units.quantity.Quantity):
        altitude2 = altitude
    else:
        altitude2 = altitude*units.meter
    p0 = 101325.*units.pascal  # sea level air pressure
    g = 9.80665*units.meter/units.second**2  # typical gravitational acceleration at sea level
    R0 = 8.31447*units.Joule/(units.mol*units.Kelvin)  # gas constant
    T0 = 19.*units.Celsius  # Typical sea-level temperature
    lapseRate = -6.5*units.Celsius/units.km  # Typical rate of change of temperature with altitude
    M = 0.0289644*units.kg/units.mol  # molar mass of dry air
 
    temperature = T0 + lapseRate*altitude2
    temperatureK = temperature.to(units.Kelvin, equivalencies=units.temperature())
    pressure = p0*np.exp(-(g*M*altitude2)/(R0*temperatureK))
    humidity = 40.  # Typical humidity at many observatory sites.
    weather = Weather((temperature/units.Celsius).value, (pressure/units.pascal).value, humidity)
    return weather