Chapter 9. Spacecraft Classification
- Objectives:
- Upon completion of this chapter you will be able to state the characteristics of various types of robotic spacecraft and be able to identify any of JPL's past, current, or future spacecraft as belonging to one of eight basic categories.
Robotic spacecraft are specially designed and constructed systems that can function in specific hostile environments. Their complexity and capabilities vary greatly and their purposes are diverse. To make some sense of all these variables, this chapter arbitrarily designates eight broad classes of robotic spacecraft according to the missions the spacecraft are intended to perform:
- Flyby spacecraft
- Orbiter spacecraft
- Atmospheric spacecraft
- Lander spacecraft
- Penetrator spacecraft
- Rover spacecraft
- Observatory spacecraft
- Communications & Navigation spacecraft
We illustrate these eight classes by offering one prime example of each, pictured on this page and some additional linked examples. Be sure to select and read at least each prime example, plus an additional link or two. The JPL public website has an up-to-date listing of all past, current, future and proposed JPL robotic spacecraft missions. Spacecraft that carry human occupants are not considered here.
(1) Flyby Spacecraft
|
VOYAGER 1 & 2 |
Flyby spacecraft conducted the initial reconnaissance phase of solar system exploration. They follow a continuous solar orbit or escape trajectory, never to be captured into a planetary orbit. They must have the capability of using their instruments to observe targets they pass. Ideally, their optical instruments can pan to compensate for the target's apparent motion in the instruments' field of view. They must downlink data to Earth, storing data onboard during the periods when their antennas are off Earthpoint. They must be able to survive long periods of interplanetary cruise. Flyby spacecraft may be designed to be stabilized in 3 axes using thrusters or reaction wheels, or to spin continuously for stabilization.
Our prime example of the flyby spacecraft category is the pair of Voyager spacecraft, which conducted encounters in the Jupiter, Saturn, Uranus, and Neptune systems. Click the Voyager image for details of the twin Voyager 1 and 2 spacecraft. Other examples of flyby spacecraft include:
- Stardust Cometary Sample Return
- Mariner 2 to Venus
- Mariner 4 to Mars
- Mariner 5 to Venus
- Mariner 6 and 7 to Mars
- Mariner 10 to Mercury
- Pioneers 10 and 11 to Jupiter and Saturn
- New Horizons Pluto-Kuiper Belt Mission
(2) Orbiter Spacecraft
|
GALILEO |
A spacecraft designed to travel to a distant planet and enter into orbit about it, must carry a substantial propulsive capability to decelerate it at the right moment, to achieve orbit insertion. It has to be designed to live with the fact that solar occultations will occur, wherein the planet shadows the spacecraft, cutting off any solar panels' production of electrical power and subjecting the vehicle to extreme thermal variation. Earth occultations will also occur, cutting off uplink and downlink communications with Earth. Orbiter spacecraft are carrying out the second phase of solar system exploration, following up the initial reconnaissance with in-depth study of each of the planets. The extensive list includes Magellan, Galileo, Mars Global Surveyor, Mars Odyssey, Cassini, and Messenger.
Our prime example of the orbiter spacecraft category is Galileo which entered orbit about Jupiter in 1995 to carry out a highly successful study of the Jovian system. Click the Galileo image for details of the Galileo spacecraft. Other examples of orbiter spacecraft include:
- Messenger Mercury Orbiter
- Mariner 9 Mars Orbiter
- Cassini Saturn Orbiter
- Mars Global Surveyor
- Mars Odyssey
- TOPEX/Poseidon Earth Orbiter
- Ulysses Solar Polar Orbiter
- Jason Earth Orbiter
- Mars '01 Orbiter
- Magellan Venus Orbiter
- Jupiter Icy Moons Orbiter, JIMO (project cancelled)
- Mars Observer a spacecraft lost
(3) Atmospheric Spacecraft
|
HUYGENS |
Atmospheric spacecraft are designed for a relatively short mission to collect data about the atmosphere of a planet or satellite. One typically has a limited complement of spacecraft subsystems. For example, an atmospheric spacecraft may have no need for propulsion subsystems or attitude and articulation control system subsystems at all. It does require an electric power supply, which may simply be batteries, and telecommunications equipment for tracking and data relay. Its scientific instruments may take direct measurements of an atmosphere's composition, temperature, pressure, density, cloud content and lightning.
Typically, atmospheric spacecraft are carried to their destination by another spacecraft. Galileo carried its atmospheric probe on an impact trajectory with Jupiter in 1995 and increased its spin rate to stabilize the probe's attitude for atmospheric entry. After probe release Galileo maneuvered to change from an impact trajectory to a Jupiter Orbit Insertion trajectory. An aeroshell protected the probe from the thousands of degrees of heat created by atmospheric friction during atmospheric entry, then parachutes deployed after the aeroshell was jettisoned. The probe completed its mission on battery power, and the orbiter relayed the data to Earth. The Pioneer 13 Venus Multiprobe Mission deployed four atmospheric probes that returned data directly to Earth during descent into the Venusian atmosphere in 1978.
Balloon packages are atmospheric probes designed for suspension from a buoyant gas bag to float and travel with the wind. The Soviet Vega 1 and Vega 2 missions to Comet Halley in 1986 deployed atmospheric balloons in Venus' atmosphere en route to the comet. DSN tracked the instrumented balloons to investigate winds in the Venusian atmosphere. (The Vega missions also deployed Venus landers.) While not currently funded, informal plans for other kinds of atmospheric spacecraft include battery powered instrumented airplanes and balloons for investigations in Mars' atmosphere.
Our prime example of the atmospheric spacecraft category is Huygens, which was carried to Saturn's moon Titan by the Cassini spacecraft. Click the Huygens image for details of the Huygens spacecraft. Other examples of atmospheric spacecraft include:
- Galileo Atmospheric Probe
- Mars Balloon
- Titan "Aerover" Blimp
- Vega Venus Balloon
- JPL Planetary Aerovehicles Development
- Pioneer 13 Venus Multiprobe Mission
(4) Lander Spacecraft
|
PATHFINDER |
Lander spacecraft are designed to reach the surface of a planet and survive long enough to telemeter data back to Earth. Examples have been the highly successful Soviet Venera landers which survived the harsh conditions on Venus while carrying out chemical composition analyses of the rocks and relaying color images, JPL's Viking landers at Mars, and the Surveyor series of landers at Earth's moon, which carried out similar experiments. The Mars Pathfinder project, which landed on Mars on July 4, 1997, was intended to be the first in a series of landers on the surface of Mars at widely distributed locations to study the planet's atmosphere, interior, and soil. The lander, carrying its own instruments, was later named the Carl Sagan Memorial Mars Station. Pathfinder also deployed a rover, Sojourner. A system of actively-cooled, long-lived Venus landers designed for seismology investigations, is being envisioned for a possible future mission.
Our prime example of the lander spacecraft category is Mars Pathfinder. Click the Pathfinder image for details of the Pathfinder spacecraft. Other examples of lander spacecraft include:
- Viking Mars Landers
- Venera 13 Venus Lander
- Surveyor Moon Landers
(5) Penetrator Spacecraft
|
DEEP SPACE 2 |
Surface penetrators have been designed for entering the surface of a body, such as a comet, surviving an impact of hundreds of Gs, measuring, and telemetering the properties of the penetrated surface. As of April 2006, no Penetrator spacecraft have been successfully operated. Penetrator data would typically be telemetered to an orbiter craft for re-transmission to Earth. The Comet Rendezvous / Asteroid Flyby (CRAF) mission included a cometary penetrator, but the mission was cancelled in 1992 due to budget constraints.
Our prime example of a penetrator spacecraft is the twin Deep Space 2 penetrators which piggybacked to Mars aboard the Mars Polar Lander and were to slam into Martian soil December 3, 1999. They were never heard from. Click the Deep Space 2 image for details of the penetrator spacecraft. Other examples of penetrator spacecraft include:
- Deep Impact Mission to a comet
- Ice Pick Mission to Europa
- Lunar-A Mission to Earth's Moon
(6) Rover Spacecraft
|
SOJOURNER |
Electrically-powered rover spacecraft are being designed and tested by JPL as part of the Mars exploration effort. The Mars Pathfinder project included a small, highly successful mobile system referred to as a micro-rover by the name of Sojourner. Mars rovers are also being developed by Russia with a measure of support from The Planetary Society. Rover craft need to be be semi-autonomous. While they are steerable from Earth, the delay inherent in radio communications between Earth and Mars means they must be able to make at least some decisions on their own as they move. Their purposes range from taking images and soil analyses to collecting samples for return to Earth.
Our prime example of a rover spacecraft is of course the famous Sojourner Rover, shown here in an image from the surface of Mars. Click the Sojourner image for details of the rover spacecraft. Other examples of rover spacecraft include:
- Mars Exploration Rovers
- Lunokhod and Marsokhod Russian Rovers
- JPL Inflatable Rovers
- Red Rover Student activity
(7) Observatory Spacecraft
|
Spitzer (SIRTF) |
An observatory spacecraft does not travel to a destination to explore it. Instead, it occupies an Earth orbit, or a solar orbit, from where it can observe distant targets free of the obscuring and blurring effects of Earth's atmosphere.
NASA's Great Observatories program studies the universe at wavelengths from infra-red to gamma-rays. The program includes four Observatory Spacecraft: the familiar Hubble Space Telescope (HST), the Chandra X-Ray Observatory (CXO, previously known as AXAF), the Compton Gamma Ray Observatory (GRO), and the Space Infrared Telescope Facility (SIRTF) renamed Spitzer in flight.
The HST is still operating as of July 2004. GRO has completed its mission and was de-orbited in June 2000. CXO was launched in July 1999 and continues to operate. SIRTF launched in January 2003 and is currently operating. In the coming decades many new kinds of observatory spacecraft will be deployed to take advantage of the tremendous gains available from operating in space.
Our prime example of an observatory spacecraft is the Spitzer Space Infrared Telescope Facility. Click the SIRTF image for details of the observatory spacecraft. Other examples of observatory spacecraft include:
- HST Hubble Space Telescope
- Chandra X-ray Observatory
- Compton Gamma-ray Observatory
- IRAS Infrared Astronomical Satellite
- TPF Terrestrial Planet Finder
- NGST Next-Generation Space Telescope
- SIM Space Interferometry Mission
- Planck Cosmic Background Radiation Field survey
(8) Communications & Navigation Spacecraft
|
TDRSS |
Communications and navigation spacecraft are abundant in Earth orbit, but they are largely incidental to JPL's missions. The Deep Space Network's Ground Communications Facility does make use of Earth-orbiting communications spacecraft to transfer data among its sites in Spain, Australia, California, and JPL. The Deep Space Network uses Earth-orbiting Global Positioning System navigation spacecraft to maintain an accurate time reference throughout the network.
In the future, communications and navigation spacecraft may be deployed at Mars, Venus, or other planets, dedicated to communications with orbiters, rovers, penetrators, and atmospheric spacecraft operating in their vicinity. This task is currently carried out to some extent by various orbiter spacecraft that are also equipped for limited communications relay. The purpose of dedicated Mars communications orbiters would be to augment the Deep Space Network's capabilities to communicate with the resident spacecraft. None are in place as of July, 2004. This concept is revisited in Chapter 18.
The communications spacecraft example offered here is NASA's Tracking and Data Relay Satellite System, TDRSS. NASA missions supported by the system include the Hubble Telescope, the Space Shuttle, GRO, Landsat, TOPEX, and EUVE and the International Space Station. Click the TDRSS image for details of this communications spacecraft. Other examples of communications and navigation spacecraft include:
- Milstar
- Global Positioning System (GPS)
- DirecTV
- Globalstar
For Further Reference
Following is a list of links to brief illustrated descriptions of the spacecraft missions mentioned above:
- Voyager
- Ulysses
- Sojourner
- Pathfinder
- Magellan
- Huygens
- Deep Space 2
- Pioneer
- Topex
- Spitzer
- Mars Observer
- Cassini
- Galileo Atmospheric Probe
- Galileo
- Mars Baloon
Here is a list of virtually every lunar and planetary mission ever flown or attempted by any nation, and those on schedule for future launch. The list is arranged by launch date, and each entry is linked to a page of facts about the mission.
The JPL website also describes all JPL's current, future, proposed, and past missions, as well as a complete alphabetical listing of them.
Chapter 4. Interplanetary Trajectories
- Objectives:
- Upon completion of this chapter you will be able to describe the use of Hohmann transfer orbits in general terms and how spacecraft use them for interplanetary travel. You will be able to describe the general concept of exchanging angular momentum between planets and spacecraft to achieve gravity assist trajectories.
When travelling among the planets, it's a good idea to minimize the propellant mass needed by your spacecraft and its launch vehicle. That way, such a flight is possible with current launch capabilities, and costs will not be prohibitive. The amount of propellant needed depends largely on what route you choose. Trajectories that by their nature need a minimum of propellant are therefore of great interest.
Hohmann Transfer Orbits
To launch a spacecraft from Earth to an outer planet such as Mars using the least propellant possible, first consider that the spacecraft is already in solar orbit as it sits on the launch pad. This existing solar orbit must be adjusted to cause it to take the spacecraft to Mars: The desired orbit's perihelion (closest approach to the sun) will be at the distance of Earth's orbit, and the aphelion (farthest distance from the sun) will be at the distance of Mars' orbit. This is called a Hohmann Transfer orbit. The portion of the solar orbit that takes the spacecraft from Earth to Mars is called its trajectory.
From the above, we know that the task is to increase the apoapsis (aphelion) of the spacecraft's present solar orbit. Recall from Chapter 3...
A spacecraft's apoapsis altitude can be raised by increasing the spacecraft's energy at periapsis. |
Well, the spacecraft is already at periapsis. So the spacecraft lifts off the launch pad, rises above Earth's atmosphere, and uses its rocket to accelerate in the direction of Earth's revolution around the sun to the extent that the energy added here at periapsis (perihelion) will cause its new orbit to have an aphelion equal to Mars' orbit. The acceleration is tangential to the existing orbit. How much energy to add? See the Rocket & Space Technology website.
After this brief acceleration away from Earth, the spacecraft has achieved its new orbit, and it simply coasts the rest of the way. The launch phase is covered further in Chapter 14.
Earth to Mars via Least Energy Orbit
Getting to the planet Mars, rather than just to its orbit, requires that the spacecraft be inserted into its interplanetary trajectory at the correct time so it will arrive at the Martian orbit when Mars will be there. This task might be compared to throwing a dart at a moving target. You have to lead the aim point by just the right amount to hit the target. The opportunity to launch a spacecraft on a transfer orbit to Mars occurs about every 25 months.
To be captured into a Martian orbit, the spacecraft must then decelerate relative to Mars using a retrograde rocket burn or some other means. To land on Mars, the spacecraft must decelerate even further using a retrograde burn to the extent that the lowest point of its Martian orbit will intercept the surface of Mars. Since Mars has an atmosphere, final deceleration may also be performed by aerodynamic braking direct from the interplanetary trajectory, and/or a parachute, and/or further retrograde burns.
Inward Bound
To launch a spacecraft from Earth to an inner planet such as Venus using least propellant, its existing solar orbit (as it sits on the launch pad) must be adjusted so that it will take it to Venus. In other words, the spacecraft's aphelion is already the distance of Earth's orbit, and the perihelion will be on the orbit of Venus.
This time, the task is to decrease the periapsis (perihelion) of the spacecraft's present solar orbit. Recall from Chapter 3...
A spacecraft's periapsis altitude can be lowered by decreasing the spacecraft's energy at apoapsis. |
To achieve this, the spacecraft lifts off of the launch pad, rises above Earth's atmosphere, and uses its rocket to accelerate opposite the direction of Earth's revolution around the sun, thereby decreasing its orbital energy while here at apoapsis (aphelion) to the extent that its new orbit will have a perihelion equal to the distance of Venus's orbit. Once again, the acceleration is tangential to the existing orbit. How much energy to add? See the
Rocket & Space Technology website.
Of course the spacecraft will continue going in the same direction as Earth orbits the sun, but a little slower now. To get to Venus, rather than just to its orbit, again requires that the spacecraft be inserted into its interplanetary trajectory at the correct time so it will arrive at the Venusian orbit when Venus is there. Venus launch opportunities occur about every 19 months.
Earth to Venus via Least Energy Orbit
Type I and II Trajectories
If the interplanetary trajectory carries the spacecraft less than 180 degrees around the sun, it's called a Type-I Trajectory. If the trajectory carries it 180 degrees or more around the sun, it's called a Type-II.
Gravity Assist Trajectories
Chapter 1 pointed out that the planets retain most of the solar system's angular momentum. This momentum can be tapped to accelerate spacecraft on so-called "gravity-assist" trajectories. It is commonly stated in the news media that spacecraft such as Voyager, Galileo, and Cassini use a planet's gravity during a flyby to slingshot it farther into space. How does this work? By using gravity to tap into the planet's tremendous angular momentum.
In a gravity-assist trajectory, angular momentum is transferred from the orbiting planet to a spacecraft approaching from behind the planet in its progress about the sun.
Note: experimenters and educators may be interested in the Gravity Assist Mechanical Simulator, a device you can build and operate to gain an intuitive understanding of how gravity assist trajectories work. The linked pages include an illustrated "primer" on gravity assist.
Consider Voyager 2, which toured the Jovian planets. The spacecraft was launched on a Type-II Hohmann transfer orbit to Jupiter. Had Jupiter not been there at the time of the spacecraft's arrival, the spacecraft would have fallen back toward the sun, and would have remained in elliptical orbit as long as no other forces acted upon it. Perihelion would have been at 1 AU, and aphelion at Jupiter's distance of about 5 AU.
However, Voyager's arrival at Jupiter was carefully timed so that it would pass behind Jupiter in its orbit around the sun. As the spacecraft came into Jupiter's gravitational influence, it fell toward Jupiter, increasing its speed toward maximum at closest approach to Jupiter. Since all masses in the universe attract each other, Jupiter sped up the spacecraft substantially, and the spacecraft tugged on Jupiter, causing the massive planet to actually lose some of its orbital energy.
The spacecraft passed on by Jupiter since Voyager's velocity was greater than Jupiter's escape velocity, and of course it slowed down again relative to Jupiter as it climbed out of the huge gravitational field. The speed component of its Jupiter-relative velocity outbound dropped to the same as that on its inbound leg.
But relative to the sun, it never slowed all the way to its initial Jupiter approach speed. It left the Jovian environs carrying an increase in angular momentum stolen from Jupiter. Jupiter's gravity served to connect the spacecraft with the planet's ample reserve of angular momentum. This technique was repeated at Saturn and Uranus.
Voyager 2 Gravity Assist Velocity Changes
Voyager 2 leaves Earth at about 36 km/s relative to the sun. Climbing out, it loses much of the initial velocity the launch vehicle provided. Nearing Jupiter, its speed is increased by the planet's gravity, and the spacecraft's velocity exceeds solar system escape velocity. Voyager departs Jupiter with more sun-relative velocity than it had on arrival. The same is seen at Saturn and Uranus. The Neptune flyby design put Voyager close by Neptune's moon Triton rather than attain more speed. Diagram courtesy Steve Matousek, JPL. |
|
The same can be said of a baseball's acceleration when hit by a bat: angular momentum is transferred from the bat to the slower-moving ball. The bat is slowed down in its "orbit" about the batter, accelerating the ball greatly. The bat connects to the ball not with the force of gravity from behind as was the case with a spacecraft, but with direct mechanical force (electrical force, on the molecular scale, if you prefer) at the front of the bat in its travel about the batter, translating angular momentum from the bat into a high velocity for the ball.
(Of course in the analogy a planet has an attractive force and the bat has a repulsive force, thus Voyager must approach Jupiter from a direction opposite Jupiter's trajectory and the ball approaches the bat from the direction of the bats trajectory.)
The vector diagram on the left shows the spacecraft's speed relative to Jupiter during a gravity-assist flyby. The spacecraft slows to the same velocity going away that it had coming in, relative to Jupiter, although its direction has changed. Note also the temporary increase in speed nearing closest approach.
When the same situation is viewed as sun-relative in the diagram below and to the right, we see that Jupiter's sun-relative orbital velocity is added to the spacecraft's velocity, and the spacecraft does not lose this component on its way out. Instead, the planet itself loses the energy. The massive planet's loss is too small to be measured, but the tiny spacecraft's gain can be very great. Imagine a gnat flying into the path of a speeding freight train.
Gravity assists can be also used to decelerate a spacecraft, by flying in front of a body in its orbit, donating some of the spacecraft's angular momentum to the body. When the Galileo spacecraft arrived at Jupiter, passing close in front of Jupiter's moon Io in its orbit, Galileo lost energy in relation to Jupiter, helping it achieve Jupiter orbit insertion, reducing the propellant needed for orbit insertion by 90 kg.
The gravity assist technique was championed by Michael Minovitch in the early 1960s, while he was a UCLA graduate student working during the summers at JPL. Prior to the adoption of the gravity assist technique, it was believed that travel to the outer solar system would only be possible by developing extremely powerful launch vehicles using nuclear reactors to create tremendous thrust, and basically flying larger and larger Hohmann transfers.
An interesting fact to consider is that even though a spacecraft may double its speed as the result of a gravity assist, it feels no acceleration at all. If you were aboard Voyager 2 when it more than doubled its speed with gravity assists in the outer solar system, you would feel only a continuous sense of falling. No acceleration. This is due to the balanced tradeoff of angular momentum brokered by the planet's -- and the spacecraft's -- gravitation.
Enter the Ion Engine
All of the above discussion of interplanetary trajectories is based on the use of today's system of chemical rockets, in which a launch vehicle provides nearly all of the spacecraft's propulsive energy. A few times a year the spacecraft may fire short bursts from its chemical rocket thrusters for small adjustments in trajectory. Otherwise, the spacecraft is in free-fall, coasting all the way to its destination. Gravity assists may also provide short periods wherein the spacecraft's trajectory undergoes a change.
But ion electric propulsion, as demonstrated in interplanetary flight by Deep Space 1, works differently. Instead of short bursts of relatively powerful thrust, electric propulsion uses a more gentle thrust continuously over periods of months or even years. It offers a gain in efficiency of an order of magnitude over chemical propulsion for those missions of long enough duration to use the technology. Ion engines are discussed further under Propulsion in Chapter 11.
Click the image above for more information about Deep Space 1. The Japan Aerospace Exploration Agency's asteroid explorer HAYABUSA also employs an ion engine.
Even ion-electric propelled spacecraft need to launch using chemical rockets, but because of their efficiency they can be less massive, and require less powerful (and less expensive) launch vehicles. Initially, then, the trajectory of an ion-propelled craft may look like the Hohmann transfer orbit. But over long periods of continuously operating an electric engine, the trajectory will no longer be a purely ballistic arc.
For Further Study
Select the "Links" section below for additional references, including mathematical tutorials and example problems.
Chapter 6. Electromagnetic Phenomena
- Objectives:
- Upon completion of this chapter you will be able to describe in general terms characteristics of natural and artificial emitters of radiation. You will be able to describe bands of the spectrum from RF to gamma rays, and the particular usefulness radio frequencies have for deep-space communication. You will be able to describe the basic principles of spectroscopy, Doppler effect, reflection and refraction.
Electromagnetic Radiation
Electromagnetic radiation (radio waves, light, etc.) consists of interacting, self-sustaining electric and magnetic fields that propagate through empty space at 299,792 km per second (the speed of light, c), and slightly slower through air and other media. Thermonuclear reactions in the cores of stars (including the sun) provide the energy that eventually leaves stars, primarily in the form of electromagnetic radiation. These waves cover a wide spectrum of frequencies. Sunshine is a familiar example of electromagnetic radiation that is naturally emitted by the sun. Starlight is the same thing from "suns" much farther away.
When a direct current (DC) of electricity, for example from a flashlight battery, is applied to a wire or other conductor, the current flow builds an electromagnetic field around the wire, propagating a wave outward. When the current is removed the field collapses, again propagating a wave. If the current is applied and removed repeatedly over a period of time, or if the electrical current is made to alternate its polarity with a uniform period of time, a series of waves is propagated at a discrete frequency. This phenomenon is the basis of electromagnetic radiation.
Electromagnetic radiation normally propagates in straight lines at the speed of light and does not require a medium for transmission. It slows as it passes through a medium such as air, water, glass, etc.
The Inverse Square Law
Electromagnetic energy decreases as if it were dispersed over the area on an expanding sphere, expressed as 4pR2 where radius R is the distance the energy has travelled. The amount of energy received at a point on that sphere diminishes as 1/R2. This relationship is known as the inverse-square law of (electromagnetic) propagation. It accounts for loss of signal strength over space, called space loss.
The inverse-square law is significant to the exploration of the universe, because it means that the concentration of electromagnetic radiation decreases very rapidly with increasing distance from the emitter. Whether the emitter is a distant spacecraft with a low-power transmitter or an extremely powerful star, it will deliver only a small amount of electromagnetic energy to a detector on Earth because of the very great distances and the small area that Earth subtends on the huge imaginary sphere.
|