Slant Range

The Slant Range function computes the slant range from a ground station to an object (satellite or aircraft) based on the station and object altitudes and the elevation angle.

INSTRUCTIONS: Choose units and enter the following:

• (sA) Station Altitude above Mean Earth Radius
• (α) Elevation Angle from the Station above the horizon to object.
• (oA) Object Altitude above Mean Earth Radius

Slant Range (R):  The range is returned in kilometers.  However this can be automatically converted to compatible units via the pull-down menu.

The Math / Science

The formula for the slant range uses both the law of sines and the law of cosines.   

In the obtuse triangle shown (right):

• Re = Mean Earth Radius
• b = Re + sA
• a = Re + oA
• c is the slant range
• CA is the Earth Central Angle
• BA is the subtended angle (β)
• AA = arcsine( a*sin(BA) / b )  
• α = 90° -  AA

Law of Sines:

      `a/(sin("AA")) = b / (sin("BA")) = c / (sin("CA"))`

Law of Cosines:

       `c = sqrt( a^2 + b^2 - 2a*b*cos(CA) )`

Earth Model Calculators

• Slant Range using Position Vectors
• Slant Range from Elevation Angle (α)
• Slant Range using Subtended Angle (β)
• Distance to Horizon
• Angle of Satellite Visibility
• Grazing Angle
• XYZ to Latitude, Longitude, Altitude
• Earth WGS-84 Equatorial radius in meters: 6378137.0 m
• Earth WGS-84 Polar radius in meters: 6356752.31424518 m 
• Earth Flattening Factor (WGS-84): 0.00335281066474748071984552861852
• Ellipse flattening factor: F = (Er - Pr)/Er  
• Geocentric to Geodetic Latitude
• Geodetic to Geocentric Latitude

Oblate Spheroid Calculators and Data

• Earth WGS-84 Equatorial radius in meters: 6378137.0 m
• Earth WGS-84 Polar radius in meters: 6356752.31424518 m 
• Earth Flattening Factor (WGS-84): 0.00335281066474748071984552861852
• Ellipse flattening factor: F = (Er - Pr)/Er  
• Geocentric to Geodetic Latitude
• Geodetic to Geocentric Latitude
• Slant Range