[27] Orbits & Resolution

What does Kepler's first law say about the shape of an orbit?

Every orbit is an ellipse with the central body (the Sun for planets, Earth for satellites) at one focus. A circle is the special case with eccentricity (stretch) 0.

Kepler's first law puts Earth at one focus of a satellite's elliptical orbit, not at its center. What does that imply, and what are the closest and farthest points called?

The satellite's altitude changes around the orbit, from perigee (closest point) to apogee (farthest point).

What does Kepler's second law say?

A line from the central body (Sun or Earth) to the orbiting body sweeps out equal areas in equal times.

By Kepler's second law, where in an elliptical orbit does a satellite move fastest, and where slowest?

Fastest at perigee (closest to Earth), slowest at apogee (farthest).

Kepler's second law is equivalent to conservation of which quantity?

Angular momentum: r² × angular speed stays constant around the orbit.

By Kepler's second law, how does the time to sweep a small fixed angle depend on the distance r from the central body?

Time ∝ r², because the thin wedge swept out has area of about ½r²Δθ. Twice as far takes 4 times as long.

By Kepler's second law, in which part of an elliptical orbit does a satellite spend most of its time?

Near apogee, where it moves slowest. That is why highly elliptical orbits put apogee over the region they want to watch for hours.

What does Kepler's third law say?

The square of the orbital period is proportional to the cube of the semi-major axis: T² ∝ a³. The semi-major axis a is half the ellipse's longest diameter (the radius, for a circle).

According to Kepler's third law, what does a satellite's orbital period depend on?

Only the semi-major axis of the orbit (plus the mass of the central body), not the satellite's own mass.

Two satellites at the same altitude, one twice as heavy. Which has the longer period?

Neither. Orbital period and speed do not depend on the satellite's mass.

What is Newton's form of Kepler's third law for an Earth satellite?

T² = 4π²a³ / (GM), with GM for Earth ≈ 3.986 × 10¹⁴ m³/s².

How can Kepler's third law give the mass of a planet or star?

Measure an orbiter's period T and semi-major axis a, then M = 4π²a³ / (GT²). For two comparable bodies this gives their total mass m₁ + m₂.

When using Kepler's third law for a circular orbit at altitude h, what value goes in for a?

a = Earth's radius + h (about 6,371 km + h). Forgetting to add Earth's radius is the classic mistake.

Why isn't Earth's radius the same everywhere, and which radius is larger?

Earth's spin makes it an oblate spheroid that bulges at the equator: the equatorial radius (~6,378 km) is about 21 km larger than the polar radius (~6,357 km). The mean is 6,371 km.

When can Kepler's third law be written simply as T² = a³?

For bodies orbiting the Sun (or any 1 solar-mass star) with T in years and a in AU (1 AU = the mean Earth-Sun distance).

For two planets orbiting the same star that is NOT our Sun, how do you compare their periods with Kepler's third law?

Use the ratio T₂/T₁ = (a₂/a₁)^(3/2), which works for any shared central body.

By Kepler's third law, if an orbit's semi-major axis is 4 times larger, how much longer is the period?

8 times longer, since 4^(3/2) = 8.

How does a circular orbit's speed depend on its radius?

v = √(GM/r) with r measured from Earth's center, so v ∝ 1/√r. Lower orbits are faster. Also v = 2πr/T.

Does a satellite move faster or slower when it orbits closer to Earth, and why?

Faster. Gravity is stronger closer in, so a higher speed is needed for its free fall to keep matching Earth's curvature.

When a satellite passes closer to Earth, how is its time spent over a given region affected?

It moves faster, so it spends less time (shorter dwell) over the region.

Roughly how fast do satellites in low Earth orbit (LEO, a few hundred km up) and geostationary orbit (about 36,000 km up) travel?

LEO about 7.5 to 7.8 km/s, GEO only about 3.1 km/s.

What is a satellite's gravitational acceleration at distance r from Earth's center?

a = GM/r², independent of the satellite's mass. In LEO it is still about 90% of surface gravity.

Why do astronauts and objects in LEO float if gravity there is still about 90% of surface gravity?

They are in continuous free fall around Earth, not outside its gravity.

Which three conic sections can a Keplerian orbit be?

Ellipse (circle included), parabola, hyperbola. Only the ellipse is a bound, repeating orbit.

How do you find the semi-major axis of an elliptical Earth orbit from its perigee and apogee altitudes?

a = (r_perigee + r_apogee) / 2, where each r = Earth's radius + that altitude.

How is an orbit's eccentricity found from its geometry?

e = c/a (center-to-focus distance over semi-major axis) = (r_a − r_p)/(r_a + r_p). e = 0 is a circle, 0 < e < 1 an ellipse.

Which equation gives a satellite's speed at any point in an elliptical orbit?

The vis-viva equation: v² = GM(2/r − 1/a).

Kepler's first and second laws say altitude and speed vary around an ellipse. Why do most Earth-imaging satellites use near-circular orbits?

A constant altitude keeps pixel size, the width of ground imaged and ground speed the same on every pass.

What does orbital inclination measure?

The tilt of the orbit plane relative to Earth's equator: 0° equatorial, 90° polar, above 90° retrograde (moving against Earth's rotation).

Does a circular orbit's inclination change its period if the radius stays the same?

No. The period depends only on the orbit's size (Kepler's third law), and tilting the orbit plane does not change its size.

Can a satellite coast in a free orbit that follows the 60° N latitude circle?

No. Gravity pulls toward Earth's center, so every orbit's plane passes through the center, and the equator is the only latitude circle an orbit can follow.

What is a sidereal day, and why does it matter for geostationary orbit?

Earth's rotation period relative to the stars, about 23 h 56 min. A geostationary period must equal it, not 24 h.

Why is a sidereal day about 4 minutes shorter than a 24-hour solar day?

Earth moves about 1° along its orbit each day, so it must turn that extra 1° to bring the Sun back overhead.

How fast does Earth rotate, in degrees per hour?

15° per hour (360° in about 24 h).

About how fast is Earth's surface moving at the equator because of rotation?

About 0.465 km/s (≈1,670 km/h) eastward, from 2πR ÷ one sidereal day.

For a satellite orbiting eastward (prograde) over the equator, is its speed relative to the ground more or less than its orbital speed?

Less, because the ground also moves east (relative ≈ orbital − 0.465 km/s). A westward (retrograde) satellite's ground-relative speed is greater.

What altitude range is Low Earth Orbit (LEO)?

About 160 to 2,000 km (about 100 to 1,200 miles).

A satellite orbits at 1,500 km altitude. Which orbit class is it in?

LEO, which extends up to about 2,000 km.

What is the typical orbital period of a LEO Earth-observing satellite?

About 90 to 100 minutes (LEO overall about 88 to 128 min).

How do you get the number of orbits per day from the orbital period?

Orbits per day = 1,440 min ÷ period in minutes. A ~99 min LEO makes about 14.5 orbits per day.

What altitude range is Medium Earth Orbit (MEO)?

About 2,000 km up to geostationary altitude (35,786 km).

Which orbit class do GPS satellites use?

MEO (about 20,200 km, 12-hour period). Not LEO.

Which orbit class, used by navigation satellites like GPS, sits in the Van Allen radiation belts, so its satellites need heavy radiation shielding?

MEO, which sits in or crosses the belts of trapped energetic particles.

What are MEO orbits mainly used for?

Navigation constellations such as GPS, Galileo, GLONASS and most of China's BeiDou.

What is the altitude of geostationary orbit (GEO)?

About 35,786 km (≈36,000 km) above the equator, about 42,164 km from Earth's center.

Roughly how many times farther from the ground is a geostationary satellite than a 700 km LEO satellite?

About 50 times.

What physical rule fixes the altitude of geostationary orbit?

Kepler's third law: it is the one circular orbit radius whose period equals one sidereal day.

If a planet rotated more slowly (a longer day), would its stationary orbit be higher or lower?

Higher, since the needed orbit radius grows as r ∝ T^(2/3).

"HEO" can mean two different orbit types. What are they?

High Earth Orbit (at or above geostationary altitude, 35,786 km) or Highly Elliptical Orbit (such as Molniya).

What is a Molniya orbit?

A highly elliptical orbit with about a 12-hour period whose high apogee sits over high northern latitudes.

Why does a Molniya (highly elliptical) orbit give long views of high latitudes?

Apogee is over high latitudes, and by Kepler's second law the satellite moves slowly there, dwelling for hours each orbit.

Which orbit type gives persistent, GEO-like viewing of polar regions?

Highly elliptical (Molniya-type), because geostationary satellites see poorly beyond about 60 to 70° latitude.

What is the main drawback of a highly elliptical (Molniya) orbit for ground stations?

The satellite's distance and position in the sky keep changing, so ground antennas must steer to track it. A GEO satellite works with a fixed dish.

What is a Lagrange point?

A point where two large bodies' gravity (e.g. Sun and Earth) plus orbital motion balance, so a spacecraft stays fixed relative to both (e.g. DSCOVR at Sun-Earth L1).

What is a geosynchronous orbit?

Any orbit whose period equals one sidereal day (23 h 56 min), so the satellite returns to the same spot in the sky at the same time each day.

What makes a geostationary orbit a special case of geosynchronous?

It is also circular (eccentricity 0) and equatorial (inclination 0°), so the satellite appears fixed over one point on the equator.

Why must a geostationary satellite sit over the equator?

Every orbit's plane passes through Earth's center, so any tilted orbit carries the satellite north and south of the equator each day. Only a 0° orbit can stay over one spot.

Seen from the ground, what path does an inclined geosynchronous satellite trace each day?

A figure-8 (analemma) about a fixed longitude, swinging north and south by its inclination.

A satellite's period matches Earth's rotation but its orbit is inclined 30°. Is it geostationary or geosynchronous?

Geosynchronous. Geostationary requires 0° inclination (and a circular orbit).

In a geosynchronous orbit, what motion does eccentricity (rather than inclination) add?

An east-west back-and-forth drift (faster near perigee, slower near apogee), not north-south motion.

Which is true of geostationary orbit: 12-hour period, passes over both poles, or equatorial with a ~24-hour period?

Equatorial with a ~24-hour (one sidereal day) period. A 12-hour period is MEO/Molniya, and passing over the poles is a polar orbit.

How do GOES satellites (NOAA's main weather satellites) stay over the same spot on Earth?

They are geostationary: an equatorial orbit at 35,786 km whose period matches Earth's rotation.

GOES-East always views the same area. What does that tell you about its orbit?

It is geostationary.

Is GOES in low, medium or high Earth orbit?

High: geostationary at 35,786 km counts as High Earth Orbit.

What is the main coverage limitation of a geostationary satellite?

It always sees the same fixed disk of Earth (a bit less than half the globe) and views poorly beyond about 60 to 70° latitude, with no view of the poles.

What is the theoretical maximum latitude a geostationary satellite can see?

About 81°, where its line of sight just grazes Earth's limb (cos φ = R_Earth / r_GEO).

In a GOES image, why are pixels over Canada much larger and more distorted than near the equator below the satellite?

Canada is viewed far off nadir (far from the point directly below the satellite): a longer slant distance and oblique angle stretch each pixel.

What is a satellite's ground track?

The path traced on Earth's surface by the point directly beneath the satellite.

What is the point on Earth's surface directly beneath a satellite sensor called?

The nadir point.

What does off-nadir viewing do to pixels and to the signal?

Pixels get larger and more distorted, and light travels a longer path through the atmosphere (more attenuation).

For a sensor that sweeps its view side to side across its path, why is spatial resolution (the smallest detail it can see) best at nadir?

The ground is closest and viewed straight on there, so each pixel's footprint is smallest and least distorted.

Which viewing geometry gives the best vertical resolution for atmospheric profiles: nadir or limb?

Limb: the sensor looks sideways through the edge of the atmosphere, so each view samples a thin altitude layer (e.g. Aura's MLS). A nadir view sees the whole column at once.

What is a swath?

The width of the strip of ground imaged in one pass.

What is IFOV (instantaneous field of view)?

The angle seen by a single detector element (in mrad or µrad). Its projection on the ground is one pixel (called the footprint for radar and lidar).

What is FOV (field of view)?

The total angle seen by the whole sensor. Its projection on the ground is the swath width.

IFOV vs FOV: which one sets pixel size and which sets swath width?

IFOV (one detector) sets pixel size. FOV (whole sensor) sets swath width.

What is ground sample distance (GSD)?

The ground distance between adjacent pixel centers, i.e. how much ground one pixel represents.

What effect does a lower (smaller) GSD have on an image?

Each pixel covers less ground, so finer detail (higher spatial resolution) is visible.

In remote sensing, what is a spectral band (channel)?

A range of wavelengths that a sensor records as its own separate image, such as a red band near 0.65 µm or a thermal IR band near 11 µm.

What are the four types of resolution in remote sensing?

Spatial, spectral, radiometric and temporal.

What does spatial resolution describe?

The size of the smallest ground feature a sensor can distinguish (roughly the pixel size, or GSD).

Which type of resolution is the ability to tell apart small differences in the wavelength (frequency) of light?

Spectral resolution: the number and narrowness of the bands.

What does radiometric resolution describe?

The ability to distinguish small differences in brightness (energy) at a given wavelength, set by bit depth and noise.

What does temporal resolution describe?

How often the same place is observed (revisit time), set by the orbit's repeat pattern, swath width and pointing.

Which resolution type is improved by adding more bits per pixel?

Radiometric.

Which resolution type is improved by using more, narrower bands?

Spectral.

Which resolution type is improved by lowering the orbit or shrinking the IFOV?

Spatial.

Which resolution type is improved by adding satellites or widening the swath?

Temporal.

What is a polar orbit?

An orbit with inclination near 90° that passes over or near both poles.

Can a near-polar LEO satellite also be geosynchronous?

No. Geosynchronous needs a 23 h 56 min period (35,786 km up), while LEO orbits take about 90 to 100 min.

Why does a polar orbit give near-global coverage?

Earth rotates beneath the orbit plane, so each pass covers a new north-south strip until all longitudes have been seen.

What is local solar time?

Time set by the Sun's position at that spot, with 12:00 when the Sun is highest. Unlike clock time, it does not depend on time zones.

What is a sun-synchronous orbit (SSO)?

A near-polar orbit whose plane turns to keep pace with the Sun, so the satellite crosses the equator at the same local solar time every day (for example, always about 1:30 pm).

Why is a sun-synchronous orbit ideal for climate monitoring and change detection?

The sun angle and shadows are the same on every pass, so differences between images are real changes, not lighting changes.

Why would an imager prefer a sun-synchronous orbit over an ordinary polar orbit?

An ordinary polar orbit drifts through local times, so lighting changes between images. Sun-synchronous keeps illumination consistent.

Images of the same place taken on the same date in different years show identical sun angle and shadows. What kind of orbit does that suggest?

Sun-synchronous: it passes over each spot at the same local solar time, so the lighting repeats.

What inclination does a typical sun-synchronous orbit have?

About 98°: near-polar and slightly retrograde.

At what altitude are typical sun-synchronous orbits: 705 km, 2,200 km, 35,780 km, or 384,000 km?

705 km. Sun-synchronous orbits are LEO, usually about 600 to 800 km. (35,786 km is GEO, 384,000 km is the Moon.)

Sun-synchronous chain, step 1: what feature of Earth makes an inclined satellite orbit's plane slowly rotate (precess)?

Earth's equatorial bulge (its oblateness, called the J2 effect).

Sun-synchronous chain, step 2: how does Earth's equatorial bulge make an inclined orbit's plane rotate?

The extra mass at the equator keeps tugging the satellite toward the equator's plane, a steady twist that swings the orbit plane slowly around Earth's axis, like a tilted spinning top.