ASLبشمهندس عسل

Mission geometry

حاسبة المدار وقوانين كبلر

Connect altitude, orbital shape, speed, period and Earth coverage.

المحتوى التفصيلي متاح بالإنجليزية، بينما الحسابات والوحدات والنتائج تعمل بنفس المحرك الهندسي.

Given / choose the units

Diagrams update as you edit. Run Calculate to refresh numerical results.

أدخل قيم التصميم ثم شغّل الحساب لمراجعة النتائج.

Swipe the diagram sideways to see the full view. With a keyboard, focus the diagram and use the arrow keys.

Interactive Kepler orbit instrumentEarth is at one focus. The satellite is 7,171 kilometres from Earth and moving at 7.456 kilometres per second.PerigeeApogeeKEPLER ORBIT / e = 0.0000Solid and dashed sectors each span 5% of one period.
Elapsed0 min
Orbit0%
Radius7,171 km
Altitude800 km
Speed7.456 km/s
Perigee / apogee7,171 / 7,171 km
Live position and radiusPerigee comparisonApogee comparison
Equal areas are swept in equal times, so the spacecraft moves faster near perigee and slower near apogee. Drag the timeline or play one orbit to compare position, speed and radius. The model is Keplerian and excludes drag and perturbations.

Understand the system

Radius and circular motion

Orbit equations use distance from Earth’s center, not height above the surface. For the final’s 800 km altitude, 6371 km Earth radius and μ = 398600 km³/s², radius is 7171 km, speed about 7.46 km/s and period 6043.39 s. The printed 6030 s is inconsistent with those constants.

r=RE+h,v=μ/r,T=2πr3/μr=R_E+h,\quad v=\sqrt{\mu/r},\quad T=2\pi\sqrt{r^3/\mu}

r, R_E, h: km; μ: km³/s²; v: km/s; T: s.

Increasing radius lowers circular speed and increases period.

Source: finalExam.pdf · PDF page 2

Common mistakes

  • Never substitute altitude for orbital radius.
  • Apogee radius must not be smaller than perigee radius.
  • GSO period alone does not make an orbit geostationary.
Formula reference and symbol definitions

Orbital radius

r=RE+hr=R_E+h

All lengths in km

Spherical Earth.

Example: At 800 km altitude, r = 6371 + 800 = 7171 km. Do not substitute 800 km for r.

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Circular speed

v=μ/rv=\sqrt{\mu/r}

μ: km³/s²; r: km; v: km/s

Circular two-body orbit.

Example: For r = 7000 km and μ = 398600 km³/s², v = √(398600/7000) ≈ 7.546 km/s.

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Orbital period

T=2πa3/μT=2\pi\sqrt{a^3/\mu}

T: s; a: km; μ: km³/s²

Unperturbed bound orbit; a=r for circular.

Example: For a = 7171 km and μ = 398600 km³/s², T = 2π√(7171³/398600) ≈ 6043.39 s.

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Kepler third law

T2=4π2a3/μT^2=4\pi^2a^3/\mu

T: s; a: km

Two-body model.

Example: Doubling a multiplies T² by 2³ = 8, so the period becomes √8 ≈ 2.828 times as long.

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Semi-major axis

a=(ra+rp)/2a=(r_a+r_p)/2

Apsis radii and a: km

Center distances, r_a ≥ r_p.

Example: For rₐ = 7600 km and rₚ = 7200 km, a = (7600 + 7200)/2 = 7400 km.

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Axis from period

a=(μT2/(4π2))1/3a=(\mu T^2/(4\pi^2))^{1/3}

T: s; μ: km³/s²; a: km

Two-body model.

Example: With T = 86164.09 s and μ = 398600.44 km³/s², a ≈ 42164.17 km.

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Eccentricity

e=(ra−rp)/(ra+rp)e=(r_a-r_p)/(r_a+r_p)

e: dimensionless; radii: same units

Bound ellipse, 0 ≤ e < 1.

Example: For apsis radii 7600 and 7200 km, e = 400/14800 ≈ 0.027027.

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Vis-viva velocity

v=μ(2/r−1/a)v=\sqrt{\mu(2/r-1/a)}

r,a: km; v: km/s

Unperturbed ellipse.

Example: At r = a = 7000 km, v = √[398600(2/7000 − 1/7000)] ≈ 7.546 km/s.

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Coverage cap

A=2πRE2(1−cos⁡θ)A=2\pi R_E^2(1-\cos\theta)

A: km²; θ: central rad

Spherical Earth; geometric horizon.

Example: For a central cap angle of 60°, A = 2πRₑ²(1 − 0.5) = πRₑ²: one quarter of the total spherical surface.

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Continue the signal path

GEO Look Angle Calculator →Doppler and Propagation Delay Tool · use this result →

Design guide

Use the result with engineering context

Technical content reviewed

When this tool is useful

  • Calculating circular and elliptical orbits with engineering models
  • Calculating kepler laws with engineering models
  • Calculating leo, meo, geo, gso and heo with engineering models

What the result includes

  • Connect altitude, orbital shape, speed, period and Earth coverage.
  • Explanations, assumptions and links to related satellite calculators

What the model does not guarantee

  • Analytical engineering models do not replace measurements or a mission-specific design review.
  • Spectrum permissions, launch operations and real satellite tracking are outside these calculators.

Worked approach

Circular and elliptical orbits

The ideal orbit is an ellipse with Earth at one focus. Equal swept areas in equal times imply faster motion near perigee and slower motion near apogee. Period squared is proportional to semi-major axis cubed under the two-body, unperturbed model.

Common decisions

Questions engineers ask

Which topics does Orbit & Kepler Calculator cover?

Circular and elliptical orbits; Kepler laws; LEO, MEO, GEO, GSO and HEO; Polar and sun-synchronous orbits; Coverage.

How should I check my work in Orbit & Kepler Calculator?

Read the stated assumptions and source references, check units and independently repeat a worked example before relying on the result.

Does Orbit & Kepler Calculator operate real satellite equipment?

No. This is a browser-based educational resource. It does not transmit RF signals, command spacecraft or certify a working link.

Related build evidence

Related embedded-system engineering

Aqua Sync is a separate connected-device project, not evidence of a satellite deployment.

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