Calculator Inputs
Use elevation mode for normal slant range work. Use coordinate mode when the ground station and sub-satellite point are known.
Formula Used
Let R be Earth radius, h be satellite altitude, and r = R + h be orbital radius.
For elevation angle E, the slant range is:
d = √(r² - R² cos²E) - R sinE
The central angle is found from:
ψ = cos⁻¹((R + d sinE) / r)
The surface arc distance is:
s = Rψ
For coordinate mode, the central angle comes from spherical geometry:
cosψ = sinφ₁ sinφ₂ + cosφ₁ cosφ₂ cos(λ₂ - λ₁)
The coordinate slant range is:
d = √(R² + r² - 2Rr cosψ)
Signal delay is:
Delay = d / signal speed
How to Use This Calculator
- Select the calculation method.
- Enter Earth radius. The default mean value is 6,371 km.
- Enter satellite altitude above Earth surface.
- Enter elevation angle for elevation mode.
- Enter ground and sub-satellite coordinates for coordinate mode.
- Enter signal speed. Light speed is used by default.
- Press the calculate button.
- Review slant range, delay, visibility, and horizon values.
- Use CSV or PDF buttons to save the result.
Example Data Table
| Example | Altitude | Elevation | Approximate slant range | One-way delay |
|---|---|---|---|---|
| Low orbit satellite near zenith | 408 km | 80° | 413.907 km | 1.381 ms |
| Navigation satellite | 20,200 km | 45° | 21,681.340 km | 72.321 ms |
| Geostationary satellite, low elevation | 35,786 km | 10° | 40,581.179 km | 135.364 ms |
| Geostationary satellite overhead | 35,786 km | 90° | 35,786.000 km | 119.369 ms |
Understanding Distance to Satellite Calculations
Why Slant Range Matters
Satellite distance is not always equal to altitude. Altitude is measured from Earth surface to the satellite. Slant range is different. It is the straight line from a ground receiver to the satellite. This value changes with elevation angle. A satellite overhead has the shortest range for its altitude. A satellite near the horizon has a longer path.
Role of Elevation Angle
Elevation angle is measured above the local horizon. A value of zero degrees means the satellite is on the horizon. A value of ninety degrees means it is overhead. Low elevation signals travel farther. They also pass through more atmosphere. This can increase delay, refraction, fading, and blockage. Higher elevation angles usually give stronger and cleaner links.
Using Earth Geometry
The calculator treats Earth as a sphere. It uses Earth radius, satellite altitude, and spherical geometry. This model is useful for physics classes, antenna planning, orbital studies, and communication checks. Real mission planning can require ellipsoid Earth models, atmospheric bending, precise ephemeris data, and local terrain. Still, the spherical model gives a strong first estimate.
Signal Delay and Communication
Distance also affects signal delay. Radio waves travel close to the speed of light. The one-way delay is range divided by signal speed. The round-trip delay is twice that value. This is important for satellite internet, radar, telemetry, navigation, and command links. Geostationary links have larger delay than low orbit links because the distance is much greater.
Coordinate Method
Coordinate mode uses the ground station position and the sub-satellite point. The sub-satellite point is where a line from Earth center through the satellite meets Earth surface. The calculator finds the central angle between both surface points. Then it computes the direct range. It also estimates elevation and visibility. If elevation is negative, the satellite is below the horizon.
Frequently Asked Questions
1. What is satellite slant range?
Slant range is the direct straight-line distance between a ground receiver and a satellite. It changes with altitude and elevation angle.
2. Is slant range the same as altitude?
No. Altitude is measured vertically above Earth surface. Slant range is measured from the observer to the satellite.
3. Why does low elevation increase distance?
At low elevation, the satellite is closer to the horizon. The line of sight crosses a longer diagonal path.
4. What Earth radius should I use?
The default 6,371 km is a common mean Earth radius. You may change it for special models or local studies.
5. What is the horizon slant range?
It is the maximum direct line distance when the satellite is exactly at zero degrees elevation.
6. How is signal delay calculated?
The calculator divides slant range by signal speed. It also doubles that value to estimate round-trip delay.
7. Can this calculator handle GPS satellites?
Yes. Enter a GPS altitude near 20,200 km and your chosen elevation angle. The result estimates range and delay.
8. Is this accurate for professional orbit control?
No. It is an educational spherical model. Professional systems need precise orbit data, timing corrections, and atmospheric models.