Satellite Look Angles (Azimuth & Elevation) Calculator

Spherical Trigonometric Derivation of Earth Station Look Angles

To establish a high-gain microwave link between an Earth station ground terminal and an orbiting spacecraft, the antenna reflector must be pointed with precision along the direct line of sight. The angular coordinates defining this orientation are called look angles:

• True Azimuth ($Az$): The horizontal compass bearing from True North ($000^\circ$ clockwise to $360^\circ$) to the satellite line of sight.
• Elevation Angle ($El$): The vertical angle between the local horizontal plane and the satellite bore-sight ($0^\circ$ to $90^\circ$).
• Polarization Tilt Angle (Skew $\psi$): The axial angle through which the antenna feed horn or orthomode transducer (OMT) must be rotated to match the satellite’s linear polarization vectors.

Spherical Earth Geometry and the Central Angle ($\gamma$)

Consider an Earth station at geodetic latitude $\phi_e$ and longitude $\lambda_e$, communicating with a satellite whose sub-satellite point on the equatorial surface is at latitude $\phi_s$ and longitude $\lambda_s$. On the spherical surface of the Earth, the North Pole ($N$), the Earth station ($E$), and the sub-satellite point ($S$) form a spherical triangle.

The angular arc connecting the Earth station to the sub-satellite point across Earth’s center is the central Earth angle ($\gamma$). By the spherical law of cosines:

For a Geostationary Earth Orbit (GEO), the orbit is circular and equatorial, fixing $\phi_s = 0^\circ$. The expression simplifies to:

Bore-Sight Elevation Angle Derivation

In the planar triangle formed by the center of the Earth ($O$), the Earth station ($E$), and the satellite ($S$):

• Distance from Earth center to station: $R_E$ (WGS-84 mean radius $6,378.137\text{ km} + h_e$).
• Distance from Earth center to satellite: $r_s = R_E + h_s$ ($42,164.137\text{ km}$ for GEO).
• Interior angle at Earth center: $\gamma$.

Applying plane trigonometry to the triangle and defining elevation above the local horizontal tangent plane yields:

The geometrical horizon limit occurs when $El = 0^\circ$, which requires $\cos\gamma = R_E / r_s \approx 6378 / 42164 \approx 0.15127$. The maximum theoretical central coverage angle is $\gamma_{\text{max}} = \arccos(0.15127) \approx 81.30^\circ$. If $\cos\gamma \le R_E / r_s$, the satellite lies below the mathematical horizon, and the link is occluded by Earth’s curvature.

Azimuth Quadrant Disambiguation Across Hemispheres

The horizontal offset angle α between the local meridian and the satellite direction is calculated from Napier’s analogies for right spherical triangles:

Because standard trigonometric functions do not uniquely identify the $360^\circ$ circle, the True Azimuth $Az$ is assigned according to hemisphere and relative longitude:

1. Northern Hemisphere (φ_e > 0): The equatorial satellite always lies to the south:
   • Satellite is East of Station (ΔL > 0): $Az = 180^\circ - \alpha$ (pointing South-East).
   • Satellite is West of Station (ΔL < 0): $Az = 180^\circ + \alpha$ (pointing South-West).
2. Southern Hemisphere (φ_e < 0): The equatorial satellite always lies to the north:
   • Satellite is East of Station (ΔL > 0): $Az = \alpha$ (pointing North-East).
   • Satellite is West of Station (ΔL < 0): $Az = 360^\circ - \alpha$ (pointing North-West).
3. Equator (φ_e = 0°): If the satellite is east (ΔL > 0), $Az = 090^\circ$; if west (ΔL < 0), $Az = 270^\circ$; if directly overhead (ΔL = 0), $El = 90^\circ$ (zenith).

Polarization Tilt Angle (Skew) Mechanics

Geostationary satellites transmit linearly polarized electromagnetic waves with electric field vectors parallel (Horizontal) or perpendicular (Vertical) to the equatorial plane. Because the Earth is a sphere, an observer located away from the sub-satellite meridian views the satellite from a tilted reference frame.

To align the terminal’s feed horn with the arriving wavefront, the feed must be rotated by the polarization skew angle (ψ):

Sign Convention: Looking directly into the dish aperture from the front toward the feed horn, a positive skew indicates clockwise rotation, while a negative skew indicates counter-clockwise rotation.

Cross-Polarization Isolation Degradation: If polarization skew is left unadjusted, the cross-polarization discrimination (XPD) collapses according to:

An angular skew alignment error of merely $5^\circ$ reduces XPD to $21.2\text{ dB}$. In modern dual-polarized Ku-band and Ka-band networks that reuse identical frequencies across orthogonal polarizations, a $5^\circ$ skew error causes severe co-channel interference and triggers carrier shutdown by the satellite network operations center (NOC).

Atmospheric Refraction & Low-Elevation Masks

At elevation angles below 10°, density gradients in the troposphere act as an optical lens, bending electromagnetic waves downward toward the denser surface layers. This refraction causes the satellite to appear slightly higher in the sky than its true geometric position.

The recommended ITU-R P.834 atmospheric refraction correction (ΔEl) for standard surface refractivity ($N_s = 312\text{ N-units}$) is:

At $El = 5^\circ$, refraction lifts the apparent elevation by approximately $0.16^\circ$ ($9.6\text{ arcminutes}$). Professional teleport operators enforce a strict minimum elevation angle mask (5° to 10°) to avoid tropospheric scintillation, excessive rain path attenuation, and thermal ground noise pickup from dish sidelobes.

Benchmark Look Angles From Global Teleports

The table below lists calculated pointing angles and slant ranges from key international satellite teleports to major commercial geostationary orbital positions: