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.

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

Hohmann transfer to Mars

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.

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

Hohmann transfer to Venus

 

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.

Voyager Trajectory 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. Voyager's gravity assists

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.)

vectorsThe 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.

vectors

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

DS-1All 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 mathe

 

Chapter 6. Electromagnetic Phenomena     CONTINUED

 


 

Radio Frequencies

Abbreviations such as kHz and GHz are all listed in the Glossary and are also treated under Units of Measure (see the menu bar below).

Electromagnetic radiation with frequencies between about 10 kHz and 100 GHz are referred to as radio frequencies (RF). Radio frequencies are divided into groups that have similar characteristics, called "bands," such as "S-band," "X-band," etc. The bands are further divided into small ranges of frequencies called "channels," some of which are allocated for the use of deep space telecommunications. Many deep-space vehicles use channels in the S-band and X-band range which are in the neighborhood of 2 to 10 GHz. These frequencies are among those referred to as microwaves, because their wavelength is short, on the order of centimeters. The microwave oven takes its name from this range of radio frequencies. Deep space telecommunications systems are being developed for use on the even higher frequency K-band.

This table lists some RF band definitions for illustration. Band definitions may vary among different sources and according to various users. These are "ballpark" values, intended to offer perspective, since band definitions have not evolved to follow any simple alphabetical sequence. For example, notice that while "L-Band" represents lower frequencies than "S-Band," "Q-Band" represents higher frequencies than "S-Band."

 

Band Approx. Range of
Wavelengths (cm)
Approximate
Frequencies
UHF 100 - 10 300 - 3000 MHz
L 30 - 15 1 - 2 GHz
S 15 - 7.5 2 - 4 GHz
C 7.5 - 3.75 4 - 8 GHz
X 3.75 - 2.4 8 - 12 GHz
K 2.4 - 0.75 12 - 40 GHz
Q 0.75 - 0.6 40 - 50 GHz
V 0.6 - 0.4 50 - 80 GHz
W 0.4 - 0.3 80 - 90 GHz
Within K-band, spacecraft may operate communications, radio science, or radar equipment at Ku-band in the neighborhood of 15 to 17 GHz and Ka-band around 20 to 30 GHz.

The Whole Spectrum

Bring up this page to study a table of the entire electromagnetic spectrum. The table shows frequency and wavelength, common names such as "light" and "gamma rays," size examples, and any attenuation effects in Earth's environment as discussed below.

 

Atmospheric Transparency

Because of the absorption phenomenon, observations are impossible at certain wavelengths from the surface of Earth, since they are absorbed by the Earth's atmosphere. There are a few "windows" in its absorption characteristics that make it possible to see visible light and receive many radio frequencies, for example, but the atmosphere presents an opaque barrier to much of the electromagnetic spectrum.

Even though the atmosphere is transparent at X-band frequencies, there is a problem when liquid or solid water is present. Water exhibits noise at X-band frequencies, so precipitation at a receiving site increases the system noise temperature, and this can drive the SNR too low to permit communications reception.

 

Radio Frequency Interference

In addition to the natural interference that comes from water at X-band, there may be other sources of noise, such as human-made radio interference. Welding operations on an antenna produce a wide spectrum of radio noise at close proximity to the receiver. Many Earth-orbiting spacecraft have strong downlinks near the frequency of signals from deep space. Goldstone Solar System Radar (described further in this chapter) uses a very powerful transmitter, which can interfere with reception at a nearby station. Whatever the source of radio frequency interference (RFI), its effect is to increase the noise, thereby decreasing the SNR and making it more difficult, or impossible, to receive valid data from a deep-space craft.

 

Spectroscopy

The study of the production, measurement, and interpretation of electromagnetic spectra is known as spectroscopy. This branch of science pertains to space exploration in many different ways. It can provide such diverse information as the chemical composition of an object, the speed of an object's travel, its temperature, and more -- information that cannot be gleaned from photographs or other means.

RainbowFor purposes of introduction, imagine sunlight passing through a glass prism, creating a rainbow, called the spectrum. Each band of color visible in this spectrum is actually composed of a very large number of individual wavelengths of light which cannot be individually discerned by the human eye, but which are detectable by sensitive instruments such as spectrometers and spectrographs.

Greenless RainbowSuppose instead of green all you find is a dark "line" where green should be. You might assume something had absorbed all the "green" wavelengths out of the incoming light. This can happen. By studying the brightness of individual wavelengths from a natural source, and comparing them to the results of laboratory experiments, many substances can be identified that lie in the path from the light source to the observer, each absorbing particular wavelengths, in a characteristic manner.

Dark absorption lines in the sun's spectrum and that of other stars are called Fraunhofer lines after Joseph von Fraunhofer (1787-1826) who observed them in 1817. The image below shows a segment of the solar spectrum, in which many such lines can be seen. The prominent line above the arrow results from hydrogen in the sun's atmosphere absorbing energy at a wavelength of 6563 Angstroms. This is called the hydrogen alpha line.

 

Solar spectrum

On the other hand, bright lines in a spectrum (not illustrated here) represent a particularly strong emission of radiation produced by the source at a particular wavelength.

Spectroscopy is not limited to the band of visible light, but is commonly applied to infrared, ultraviolet, and many other parts of the whole spectrum of electromagnetic energy.

 

In 1859, Gustav Kirchhoff (1824-1887) described three laws of spectral analysis:

 

  1. A luminous (glowing) solid or liquid emits light of all wavelengths (white light), thus producing a continuous spectrum.

     

  2. A rarefied luminous gas emits light whose spectrum shows bright lines (indicating light at specific wavelengths), and sometimes a faint superimposed continuous spectrum.

     

  3. If the white light from a luminous source is passed through a gas, the gas may absorb certain wavelengths from the continuous spectrum so that those wavelengths will be missing or diminished in its spectrum, thus producing dark lines.

     

By studying emission and absorption features in the spectra of stars, in the spectra of sunlight reflected off the surfaces of planets, rings, and satellites, and in the spectra of starlight passing through planetary atmospheres, much can be learned about these bodies. This is why spectral instruments are flown on spacecraft.

Historically, spectral observations have taken the form of photographic prints showing spectral bands with light and dark lines. Modern instruments (discussed Spectral plot again under Chapter 12) produce their high-resolution results in the form of X-Y graphic plots, whose peaks and valleys reveal intensity (brightness) on the vertical axis versus wavelength along the horizontal. Peaks of high intensity on such a plot represent bright spectral lines (not seen in this illustration), and troughs of low intensity represent the dark lines.

This plot, reproduced courtesy of the Institut National des Sciences de l'Univers / Observatoire de Paris, shows details surrounding the dip in brightness centered at the hydrogen-alpha line of 6563 which is indicated by the dark line above the red arrow in the spectral image above. The whole plot spans 25 of wavelength horizontally. Click the image for a larger view.

Spectral observations of distant supernovae (exploding stars) provide data for astrophysicists to understand the supernova process, and to categorize the various supernova types. Supernovae can be occasionally found in extremely distant galaxies. Recognizing their spectral signature is an important step in measuring the size of the universe, based on knowing the original brightness of a supernova and comparing that with the observed brightness across the distance.


 

matical tutorials and example problems.

Chapter 6. Electromagnetic Phenomena     CONTINUED


 

The Doppler Effect

Christian Doppler

No Doppler

CLICK IMAGE TO
START / STOP ANIMATION
 

Regardless of the frequency of a source of electromagnetic waves, they are subject to the Doppler effect. The effect was discovered by the Austrian mathematician and physicist Christian Doppler (1803-1853). It causes the observed frequency of any source (sound, radio, light, etc.) to differ from the radiated frequency of the source if there is motion that is increasing or decreasing the distance between the source and the observer. The effect is readily observable as variation in the pitch of sound between a moving source and a stationary observer, or vice-versa.

 

Consider the following:

1. When the distance between the source and receiver of electromagnetic waves remains constant, the frequency of the source and received wave forms is the same.

This is illustrated at right. The waveform at the top represents the source, and the one at the bottom represents the received signal. Since the source and the receiver are not moving toward or away from each other, the received signal appears the same as the source.

2. When the distance between the source and receiver of electromagnetic waves is increasing, the frequency of the received wave forms appears to be lower than the actual frequency of the source wave form. Each time the source has completed a wave, it has also moved farther away from the receiver, so the waves arrive less frequently.

3. When the distance is decreasing, the frequency of the received wave form will be higher than the source wave form. Since the source is getting closer, the waves arrive more frequently.

Cases 2 and 3 are illustrated below. Notice that when the receiver is in motion toward or away from the source, the waveform at the receiver (the lower waveform) changes. It only changes, though, while there is actual motion toward or away; when it stops, the received waveform appears the same as the source.

 

Doppler Doppler
CLICK IMAGE TO START / STOP ANIMATION
 

 

The Doppler effect is routinely measured in the frequency of the signals received by ground receiving stations when tracking spacecraft. The increasing or decreasing distances between the spacecraft and the ground station may be caused by a combination of the spacecraft's trajectory, its orbit around a planet, Earth's revolution about the sun, and Earth's daily rotation on its axis. A spacecraft approaching Earth will add a positive frequency bias to the received signal. However, if it flies by Earth, the received Doppler bias will become zero as it passes Earth, and then become negative as the spacecraft moves away from Earth.

A spacecraft's revolutions around another planet such as Mars adds alternating positive and negative frequency biases to the received signal, as the spacecraft first moves toward and then away from Earth.

The Earth's daily rotation adds a positive frequency bias to the received signal as the spacecraft rises in the east at a particular tracking station, and it adds a negative frequency bias to the received signal as the spacecraft sets in the west.

The Earth's revolution about the sun adds a positive frequency bias to the received signal during that portion of the year when the Earth is moving toward the spacecraft, and it adds a negative frequency bias during the part of the year when the Earth is moving away. Differenced Doppler

Differenced Doppler

If two widely-separated tracking stations on Earth observe a single spacecraft in orbit about another planet, they will each have a slightly different view of the moving spacecraft, and there will be a slight difference in the amount of Doppler shift observed by each station. For example, if one station has a view exactly edge-on to the spacecraft's orbital plane, the other station would have a view slightly to one side of that plane. Information can be extracted from the differencing of the two received signals.

Data obtained from two stations in this way can be combined and interpreted to fully describe the spacecraft's arc through space in three dimensions, rather than just providing a single toward or away component. This data type, differenced Doppler, is a useful form of navigation data that can yield a very high degree of spatial resolution. It is further discussed in Chapter 13, Spacecraft Navigation.


 

Refraction

Refraction is the deflection or bending of electromagnetic waves when they pass from one kind of transparent medium into another. The index of refraction of a material is the ratio of the speed of light in a vacuum to the speed of light in the material. Electromagnetic waves passing from one medium into another of a differing index of refraction will be bent in their direction of travel. In 1621, Dutch physicist Willebrord Snell (1591-1626), determined the angular relationships of light passing from one transparent medium to another.

Air and glass have different indices of refraction. Therefore, the path of electromagnetic waves moving from air to glass at an angle will be bent toward the perpendicular as they travel into the glass. Likewise, the path will be bent to the same extent away from the perpendicular when they exit the other side of glass.

Refraction is responsible for many useful devices which bend light in carefully determined ways, from eyeglasses to refracting telescope lenses.

Refraction can cause illusions. This pencil appears to be discontinuous at the boundary of air and water. Spacecraft may appear to be in different locations in the sky than they really are.

Electromagnetic waves entering Earth's atmosphere from space are bent by refraction. Atmospheric refraction is greatest for signals near the horizon where they come in at the lowest angle. The apparent altitude of the signal source can be on the order of half a degree higher than its true height. As Earth rotates and the object gains altitude, the refraction effect reduces, becoming zero at the zenith (directly overhead). Refraction's effect on the Sun adds about 5 minutes of time to the daylight at equatorial latitudes, since it appears higher in the sky than it actually is.

 

Refraction in Earth's Atmosphere

Angles exaggerated for clarity. Low altitude refraction

If the signal from a spacecraft goes through the atmosphere of another planet, the signals leaving the spacecraft will be bent by the atmosphere of that planet. This bending will cause the apparent occultation, that is, going behind the planet, to occur later than otherwise expected, and to exit from occultation prior to when otherwise expected. Ground processing of the received signals reveals the extent of atmospheric bending, and also of absorption at specific frequencies and other modifications. These provide a basis for inferring the composition and structure of a planet's atmosphere.

 


 

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