Solar zenith angle

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The solar zenith angle is the angle measured from directly overhead to the geometric centre of the sun's disc, as described using a horizontal coordinate system. The solar elevation angle is the altitude of the sun, the angle between the horizon and the centre of the sun's disc. If we write θs for the solar zenith angle, then the solar elevation angle αs = 90° – θs.[1]


Solar zenith angle[edit]

The solar zenith angle, \theta_s is estimated using results from spherical trigonometry by[2][3]

 \cos \theta_s = \sin \varphi \sin \delta + \cos \varphi \cos \delta \cos h


Solar elevation angle[edit]

The solar elevation angle is the altitude of the sun, the angle between the horizon and the centre of the sun's disc. The approximate value can be calculated with the following formula:

\sin \alpha_\mathrm{s} = \cos h \cos \delta \cos \varphi + \sin \delta \sin \varphi



The calculated values are approximations due to the distinction between common/geodetic latitude and geocentric latitude. However, the two values differ by less than 12 minutes of arc.

The formulas neglect the effect of atmospheric refraction.

The formula for solar elevation angle was derived using the trigonometric addition formulas from the solar zenith angle formula.[4]



Sunset and sunrise occur (approximately) when the zenith angle is 90°, where the hour angle h0 satisfies[3]

\cos h_0 = -\tan \phi \tan \delta.

Precise times of sunset and sunrise occur when the upper limb of the Sun appears, as refracted by the atmosphere, to be on the horizon.


A weighted daily average zenith angle, used in computing the local albedo of the Earth, is given by

\overline{\cos \theta_s} = \frac{\int_{-h_0}^{h_0} Q \cos \theta_s \text{d}h}{\int_{-h_0}^{h_0} Q \text{d}h}

where Q is the instantaneous insolation.[3]

Summary of special angles[edit]

For example, the solar elevation angle is :

  • 90° if you are on the equator, a day of equinox, at a solar hour of twelve
  • near 0° at the sunset or at the sunrise
  • between -90° and 0° during the night

An exact calculation is given in position of the Sun. Other approximations exist elsewhere.[5]

See also[edit]


  1. ^ Schowengerdt, R. A. (2007). "Optical radiation models". Remote Sensing. pp. 45–88. doi:10.1016/B978-012369407-2/50005-X. ISBN 9780123694072. 
  2. ^ Jacobson, Mark Z. (2005). Fundamentals of Atmospheric Modeling (2nd ed.). Cambridge University Press. p. 317. ISBN 0521548659. 
  3. ^ a b c Hartmann, Dennis L. (1994). Global Physical Climatology. Academic Press. p. 30. ISBN 0080571638. 
  4. ^ Woolf, Harold M. (1968). "On the computation of solar elevation angles and the determination of sunrise and sunset times". NASA technical memorandu, X-1646 (Washington, D.C.): 3. 
  5. ^ livioflores-ga. "Equation to know where the Sun is at a given place at a given date-time". Google. Retrieved 9 March 2013.