Satellites
around Earth

Pick an altitude. A circular orbit around Earth then has one period, one speed, and one ground track. The period is P = 2π √(a³ / μ). The speed is v = √(μ / r). μ is the Earth Fact Sheet GM, 0.39860 × 10⁶ km³/s². The altitude is height above that sheet’s equatorial radius, 6,378.137 km. Geostationary is the circle whose period equals the same sheet’s sidereal rotation, 23.9345 hours, not the 24.0000-hour mean solar day. That circle sits at 35,786.05 km. The slider runs from 160 km to 410,000 km, far enough to reach the Moon Fact Sheet distances. An ellipse is optional: perigee and apogee, and vis-viva speeds at each end. The year of a solar circle is the Kepler desk. Leaving the circle is the escape desk. The half-ellipse up to the geosynchronous radius is the Hohmann desk. The sidereal day itself is the sidereal desk.

Educational calculator. Simplified two-body model. Circular unless an ellipse is set. No drag, no oblateness, and no third body in the arithmetic. Not for mission planning. The drawing is top-down from the north, in the equatorial plane. Cited inclinations are not drawn. The horizon ignores refraction. A radio horizon sits farther out than this geometric one. The 160 km end of the slider is a classroom floor, not a cited boundary.

Interactive · One altitude

True scale · Earth and the orbit in one ruler · north up

Earth
Orbit
Geostationary
Satellite

True scale. The frame fits the current orbit, so a low orbit is a thin ring on a large Earth.

Altitude above the equatorial radius. The slider is logarithmic.

Altitude km

What this means

A lower orbit is faster, and it finishes sooner. Gravity is stronger closer in, so the sideways speed that keeps missing the Earth is higher: v = √(μ / r). The path is also shorter. The period grows as a to the power 3/2, so the time drops faster than the radius. At the cited ISS altitude of about 420 km the circle takes 92.97 minutes and moves at 7.657 km/s. At geostationary altitude the same formulas give 23 h 56.07 min and 3.075 km/s. Nothing in the arithmetic pushes the low satellite. It is falling the whole way, and the ground curves away as fast as it falls.

Geostationary sits where one orbit lasts one sidereal day. The ground’s turn against the stars is the fact sheet’s 23.9345 hours, not the 24.0000-hour mean solar day. Set P equal to that sidereal period and solve r³ = μ P² / (4π²). Subtract the equatorial radius and the altitude is 35,786.05 km. A circle timed to 24.0000 hours would sit 76.89 km higher, and it would drift. Geostationary also means a circle in the equatorial plane. This page matches the period. It does not draw the inclination. The Hohmann desk uses this same GM and the same 23.9345 hours, then subtracts the volumetric mean radius, 6,371.000 km. The circle in space is the same. The printed altitude differs by those two radii, 7.137 km.

Low orbits decay. This page does not time the decay. NASA’s Earth-observing notes say the station’s altitude drops from atmospheric drag, and that reboosts raise it. The same notes say each orbit takes 90–93 minutes depending on the exact altitude, and that the ground track steps west by about 22.9°. Hubble’s public pages say the telescope still meets enough air to descend, which is why shuttle crews boosted it. Those are qualitative facts. A lifetime in days is not printed here, because it depends on solar activity and on the satellite, and this model has no atmosphere in the formula.

Literacy · one radius, several readings

Circular orbit: One radius, one speed, √(μ / r). The period is 2π √(r³ / μ). Around the Sun, in years and AU, that period is the P² = a³ of the Kepler desk. Here μ is Earth’s, and the radius is kilometers.
Geostationary: The circular period equals Earth’s sidereal rotation, so the satellite and the ground turn together. On these constants that altitude is 35,786.05 km. A real geostationary satellite is also equatorial. Geosynchronous is the wider word: the period matches, and the inclination may not be zero. This button is the period.
Ground track: During one orbit the Earth turns east by 360° × P / 23.9345 h. The next equatorial crossing is that far west of the previous one. At geostationary altitude the turn is exactly one rotation, so the crossing does not step. No J2, so no extra nodal drift. A cited inclination changes which latitudes the track reaches. It is not in this angle.
Horizon: From a sphere, the limb is the tangent. The line-of-sight distance is √(r² − R²). The distance along the ground is R × arccos(R / r). The visible cap is (1 − R / r) / 2 of the surface. It never reaches a half. Refraction is left out.
Light time: Straight down and straight back: 2 × altitude / c, with c = 299,792.458 km/s. A station off to the side is a longer slant. The readout is the nadir round trip. The one-way time is half of it.
Ellipse: Perigee and apogee set a = (r_p + r_a) / 2 and e = (r_a − r_p) / (r_a + r_p). The period still comes from a alone. The speeds are vis-viva, v = √[μ (2/r − 1/a)], faster at perigee. The vis-viva desk is that formula for the planets. The Moon button uses the fact-sheet perigee and apogee. Earth’s GM alone does not reproduce the sheet’s 27.3217-day month or its 1.082 and 0.970 km/s. The sheet’s Moon GM is not added.

Sources · public pages

Cited, not invented

NASA NSSDCA — Earth Fact Sheet — GM 0.39860 × 10⁶ km³/s²; equatorial radius 6,378.137 km; volumetric mean radius 6,371.000 km; sidereal rotation period 23.9345 hours; length of day 24.0000 hours. This desk uses the GM, the equatorial radius, and the sidereal rotation. The volumetric radius is named only so the Hohmann altitude can be compared. Last updated 15 November 2024.
NIST CODATA — speed of light in vacuum — 299,792,458 m/s exactly, so 299,792.458 km/s. The light-time line divides by that c.
NASA — International Space Station lithograph, 2023 — average altitude of about 227 nautical miles (420 kilometers). The ISS button is that 420 km. The period and the speed are computed. The lithograph’s 17,500 mph is a round public figure and is not this 7.657 km/s.
NASA — Space Station Facts and Figures — orbiting Earth about every 90 minutes, and 16 orbits in 24 hours. Those are the round public sentences. This circle at 420 km is 92.97 minutes and 15.489 orbits per mean solar day.
NASA Earth Science and Remote Sensing Unit — orbit tutorial — inclination 51.6°; each orbit 90–93 minutes depending on altitude; generally 15.5 to 15.9 orbits per day; the track shifts west by approximately 22.9°; altitude drops from atmospheric drag and reboosts raise it. The 51.6° is not used in the formulas. The 22.9° is the tutorial’s approximate shift, not this page’s 23.31° at 420 km.
ESA — International Space Station fact sheet — orbital altitude 370–460 km, inclination 51.6°, orbital velocity 7.7–7.6 km/s. The band is cited. It is not a second preset. On this GM and equatorial radius, 370 km computes to 91.95 minutes and 460 km computes to 93.79 minutes.
NASA Science — About Hubble — low-Earth orbit at an altitude of about 300 miles (483 km), inclined 28.5° to the equator; about 95 minutes; about 17,000 mph. The button is that 483 km. The period here is 94.27 minutes. The 28.5° is not in the formulas. The 17,000 mph is NASA’s round figure.
NASA Science — Orbiting Hubble — the orbit has been as high as 360 miles (579 km) and descends because of atmospheric drag; shuttle crews boosted it. No lifetime is taken from this page.
FCC — FCC-21-48A1 — SpaceX’s third modification: 1,584 satellites at 550 km, inclination 53°, 72 planes. The Starlink button is that authorized shell, not a live catalog and not the only shell.
GPS.gov — Space segment — medium Earth orbit at an altitude of approximately 20,200 km (12,550 miles). Each satellite circles the Earth twice a day.
IS-GPS-200N — AREF = 26,559,710 meters, the nominal semi-major axis when the ephemeris correction ΔA is zero. The GPS button uses that radius. The altitude 20,181.57 km is AREF minus the equatorial radius. The reference inclination in that specification is 55°. It is not in the formulas. With this desk’s fact-sheet GM the sidereal count is 2.000 orbits per sidereal day.
NASA NSSDCA — Moon Fact Sheet — semimajor axis 0.3844 × 10⁶ km; perigee 0.3633 × 10⁶ km; apogee 0.4055 × 10⁶ km; eccentricity 0.0549; revolution period 27.3217 days; max orbital velocity 1.082 km/s; min 0.970 km/s. The button uses the perigee and apogee. The period and the speeds on the card are this desk’s Earth-only GM. They are not those sheet lines. The sheet notes the distance roughly ranges from 357,000 km to 407,000 km over the year. This ellipse does not.

Classroom picture of one Earth orbit. GM, equatorial radius, and the sidereal day are the Earth Fact Sheet. Preset sizes are the cited altitudes or radii. Periods, speeds, the ground track, the horizon, and the light time are computed. Simplified two-body model. Circular unless an ellipse is set. Not for mission planning. Earthrise preview: NASA / Bill Anders (public domain).

Keep going

The ellipse

Presets	ISS is about 420 km. Hubble is about 483 km. Starlink is the authorized 550 km shell. GPS is the interface specification’s nominal semi-major axis. Geostationary is computed. The Moon is the fact-sheet perigee and apogee.
Altitude	Height above the equatorial radius, 6,378.137 km. The slider is logarithmic from 160 km to 410,000 km. The number accepts the same range.
Snap	Sets a circular orbit at the geostationary altitude, 35,786.05 km. The period is 23 h 56.07 min.
Ellipse	Opens perigee and apogee. The period uses the average radius. Speeds are vis-viva. Equal ends stay a circle.
Log radius	Draws radius on a log scale out to 410,000 km, so a low orbit and the Moon’s distance share the frame. True scale fits the current orbit and keeps one ruler.

Period and speed	P = 2π √(a³ / μ). On a circle, v = √(μ / r). On an ellipse, v = √[μ (2/r − 1/a)] at each end.
Orbits per day	Mean solar day, 24.0000 hours, and sidereal day, 23.9345 hours.
Ground track	How far west the next equatorial crossing falls. Geostationary prints 0°, because Earth has turned exactly once.
Surface and horizon	The spherical cap, the tangent line of sight, and the arc along the ground. No refraction.
Light	Round trip straight down and back, 2 × altitude / c. One way is half.

Diagram	Left and Right nudge the altitude, or the perigee when the ellipse is open. Shift plus an arrow nudges the apogee.
Buttons	Enter or Space selects. The pressed preset is aria-pressed. Targets are at least 44 pixels on a side.
Phone	Below 900 pixels the long section row becomes a single Sections control. Open it for the same links. The site menu is the Menu button in the top bar.
This dialog	The question mark, or ?, opens it. Escape closes it. Tab stays inside until you close it. Click the dim page behind the card to close it.

Satellites
around Earth

Cited, not invented

Vis-viva

Hohmann

Escape

Kepler

Tides

Sidereal

Eccentricity

Solar map

Satellite desk

Controls

Readouts

Keyboard and touch

Satellitesaround Earth

Cited, not invented

Vis-viva

Hohmann

Escape

Kepler

Tides

Sidereal

Eccentricity

Solar map

Satellites
around Earth