Zenith Refraction Variation

How do temperature and pressure affect atmospheric refraction?

Atmospheric refraction distorts the sun’s apparent position when the sun is low in the sky. The Solar Position Algorithm 1 includes adjustments for pressure and temperature.

from pvlib.solarposition import get_solarposition
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd

lat, lon = 40, -80
times = pd.date_range('2019-06-01 04:50', '2019-06-01 05:00',
                      freq='1s', tz='Etc/GMT+5')
temperature_range = np.arange(-20, 21, 10).astype(int)  # degrees C
pressure_range = np.arange(0, 126e3, 25e3).astype(int)  # Pascals


def get_zenith(p, t):
    solar_position = get_solarposition(times, lat, lon,
                                       pressure=p, temperature=t)
    return solar_position.apparent_zenith


df_pressure = pd.DataFrame({
    p: get_zenith(p, 12) for p in pressure_range
})
df_pressure.plot()
plt.ylabel(r'Apparent SPA Zenith [$\degree$]')
plt.title('Dawn Variation by Pressure')

df_temperature = pd.DataFrame({
    t: get_zenith(101325, t) for t in temperature_range
})
df_temperature.plot()
plt.ylabel(r'Apparent SPA Zenith [$\degree$]')
plt.title('Dawn Variation by Temperature')
plt.show()

• 
• 

def signed_max(s):
    a = s.abs()
    return a.max() * np.sign(s[a.idxmax()])


baseline = get_zenith(0, 0)

temperature_range = np.arange(-65, 66, 5).astype(int)  # degrees C
pressure_range = np.arange(0, 126e3, 5e3).astype(int)  # Pascals

max_errors = pd.DataFrame([
    {'pressure': p,
     'temperature': t,
     'max_error': signed_max(get_zenith(p, t) - baseline)}
    for p in pressure_range
    for t in temperature_range
])
piv = max_errors.pivot('pressure', 'temperature', 'max_error')

plt.figure()
plt.pcolormesh(piv.columns, piv.index, piv, shading='auto')
plt.colorbar().set_label(r'Difference from baseline [$\degree$]')
plt.contour(piv.columns, piv.index, piv,
            levels=[-0.83, -0.67, -0.5, -0.33, -0.17],
            colors='k', linestyles='solid')

reference_points = {
    'Reykjavic, Iceland': (4.3, 101325),
    'Mount Everest Summit': (-27, 33.7e3),
    'Mogadishu, Somalia': (26.4, 101325),
    'Mars Surface': (-60, 600),
    'Denver, Colorado': (9, 83503),
    'La Rinconada, Peru': (3.1, 53302),
}
for name, (t, p) in reference_points.items():
    plt.scatter(t, p, label=name, edgecolor='k', s=50, zorder=3)

plt.legend(loc='upper center', ncol=3, bbox_to_anchor=(0.5, -0.15))
plt.xlabel(r'Annual Average Temperature [$\degree$C]')
plt.ylabel('Annual Average Pressure [Pa]')
plt.title('Zenith Prediction Variation')
plt.tight_layout()
plt.show()

References

1

I Reda and A Andreas, Solar Position Algorithm for Solar Radiation Applications (Revised), Technical Report NREL/TP-560-34302, National Renewable Energy Laboratory, 2008. doi: https://doi.org/10.2172/15003974

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