Colllisions lcp 11: Part II spacewatch Fi

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Fig. 6 Col. Gen. Vladimir Popovkin, commander of the Russian Military Space Forces, told a news conference Friday that the national satellite network lacked a spacecraft capable of preventing an asteroid strike:

Over the last few decades there has been a great deal of debate about the level of danger posed by impacts from asteroids and comets. It appears the world needs to take the threat of asteroid strikes a lot more seriously.

Astronomers have already spotted about 800 asteroids, solid rocky celestial bodies, with a diameter of over 1,000 meters (3,250 feet) moving along circumsolar elliptical orbits. However, there may be as many as 2,000 large asteroids, and some 135,000 rocks with a diameter of 100 meters (325 feet) and more.

It should be noted that asteroid orbits are unstable and tend to change under the influence of gravitational fields of the terrestrial planets - Mercury, Venus, Earth and Mars.

An asteroid, which flashed past our planet at a distance of 5 million kilometers (3.1 million miles) in November 1996, returned in September 2004 and flew by just 1.5 million kilometers (930,000 miles) from Earth's surface. In March 1989, a 300 meter (975 foot) asteroid crossed the terrestrial orbit and missed the Earth by just six hours. Astronomers spotted the rock only when it was receding into space.

An asteroid measuring over 1,000 meters in diameter is potentially capable of destroying human civilization. Chances of a major asteroid impact in the 21st century are a mere 0.0002 percent, although there is a 2 percent probability of Earth colliding with a 100 meter asteroid before the year 2100.

The blast would equal to 100 Megatons in trinitrotoluol equivalent, and it would kill millions of people if it hit a populous industrial region harboring many hazardous enterprises.

Recent close approaches of comets and asteroids

Comet Lexell, discovered in 1770, was the closest approach of any comet so far. The comet was perturbed by large gravitational field of Jupiter, brought in close to the Earth (0.1 AU) and then thrown out by Jupiter into a long orbit never to be seen again.

Adonis, discovered in 1936, passed within 0.015 of the Earth, lost, and rediscovered in 1977.

Hermes, discovered in 1937, came within 0.006 AU, never to be seen again.

Icarus, discovered in 1949, passed within 0.04 AU in 1968. Every nineteen years the large asteroid Icarus swings by planet Earth, often coming within four million miles of the planet—mere spitting distance in astronomical terms. Icarus last passed by Earth in 1997. Before that, its previous approach was in June 1968. We now know that such near-Earth asteroids are not all that rare and in recent years Congress and NASA have shown greater interest in trying to track, and even visit them.

Apollo, discovered in 1932, when it came within 0.07 AU of the Earth, lost and rediscovered in 1973

Fig. 7 Comet Lexell Fig. 8 Asteroid Hermes
The most recent discovery: Asteroid 1991 BA, came within 0.0052 AU of the Earth when first seen, then its approach decreased to 0.0033 and as it passed the Earth-Moon system, it was only 0.0011 AU away.

The best way to get a sense of the distances involved is to convert these distances to lengths equal to the distance between the Earth and the Moon, 3.86x108 m, or 386, 000 km. Which one(s) of these close approaches are within 1 Earth - Moon distance?

Evidence of past impacts are clear and the presence of near-Earth asteroids show future impacts are possible, but the absolute probabilities are low. Nevertheless, the consequences of an impact could spell the end of civilization. For example, a repeat of a small Tunguska-type impact over a populated area would kill almost 100,000 people. On the basis that the risk of dying from an impact is roughly the same as dying in a plane crash a number of investigators suggest that the government (US) donate about $125 million a year toward the study and assessment of the asteroid hazard. The present allotment toward this goal is less than $2 million. The research in this area should be, of course, an international effort, since everyone on the globe is equally involved.
IL ** Move an asteroid 2008

In the last twenty years there has been an explosion of activity, observational reportage and media coverage of the dangers of asteroids or “interplanetary fugitives”. Although public awareness has been created of the high probability of a direct collision, most of the public does not seem to have the necessary scientific literacy to understand or assess this danger.

Well-made science fiction films benefit the general public in two ways: they can educate, increasing the scientific literacy and they can also promote a positive emotional response to support scientific and technological problems of common and global need and interest. Public concern and interest that is based on sound scientific information is able to have a great effect on funding of important research projects. The perceived awareness of the probability and the ultimate inevitability of a major collision with an asteroid or comet has already convinced the American government to finance groups of scientists such as the astronomers who are involved with Project Spaceguard. Our discussion below is partly based on the fine article “Sci-Fi in the Classroom”, published in Mercury, Dec., 1998, by Leroy Dubeck, a physicist and Rose Tatlow, a science educator, The following are two appropriate quotes for us, one is taken from the beginning and the other is the concluding sentence in their fine article:

The depiction of science - whether credible or incredible - coupled with an explanation of the relevant scientific principles, help make abstract ideas concrete.

Science fiction films can do more than any lecture or book to garner interest in and support of the sciences - both from the public and from students in our classrooms. Good luck in making a deep impact with your audience!

Fig. 9 In this scene from Deep Impact, people watch an asteroid zooming across the sky. It will strike the ocean and generate enormous waves.
Deep Impact”: The movie

The movie Deep Impact was released 1998. Steven Spielberg was co- producer, who is actually one of the directors of the Planetary Society. It should be noted that in order to “get the science right” the producers hired a number of consultants from space science and engineering”, including the late Gene Shoemaker his wife Carolyn and a NASA astronaut. It will be assumed that you have seen the movie at least once, so that we need not go into telling of the story and the describing the action in great detail. However, you should see Deep Impact again, soon after our discussion. Hopefully you will then see this movie with “different eyes”.

At the start of the movie we meet a group of high school astronomy students are studying the sky and one of them sees an unusual ”star” and immediately sends a photograph of the object to an astronomer for identification. We soon learn that the “star” is a comet, seven miles (about 11 km) in diameter and on a collision course with Earth. The comet is supposed to be “the size of Mount Everest”, certainly large enough to destroy life as we know it. We are not informed of this impending disaster, instead we are given the code name E.L.E., for “Extinction Level Event”. This is probably done as a dramatic device of “foreshadowing”

We are told that the government has decided to try three plans to stop or to lessen the destructive power of the comet. The first one is called “the Messiah Project” and it would try to send a spacecraft to the comet and plant nuclear “mole” bombs within the surface of the comet in order to either destroy it or alter its course significantly. We are told that the spacecraft used for this important mission is propelled by the Orion propulsion system, a system originally designed for interstellar travel. This system uses a series of small controlled nuclear explosions to push against a massive metallic rear plate on the spaceship. There is, of course, a shock-absorbing system designed to reduce the impact of the explosions on the crew members. The push on the plate adds large increments of change in forward velocity. Unfortunately, we are not sure how these explosions are supposed to occur, one large one , several large ones, or very many small ones. The astronauts are seen moving about the ship in a weightless state and later they move about the comet surface the same way.

Once the astronauts are on the comet they try to plant the “mole” bombs in a race against time. There seems to be very little gravity and as soon as Sunlight strikes the surface, powerful jets of gases are emitted. They also find house-sized chunks of rocks surrounding the comet. The Messiah astronauts succeed in burying five nuclear explosives equivalent to megatons of TNT 100 meters below the surface. However, the explosions do not deflect the comet but manage to break the comet into two parts, both still on a collision course with the Earth! The first part is about 2.5 km across while the second about 8-9 km.

The second plan calls for the destruction of both comets by firing ICBM’s at them as they near Earth, but it also also fails. Unfortunately, we are not told or given any details as to why the mission failed or how many nuclear missiles were fired . It is difficult to comment here on the realism of the film.

The third plan was to build vast underground “arks” in various countries in the hope that those who are admitted to these places will survive long enough to emerge into a world that again is able to provide food and shelter. We are not shown the interior of these arks, but told that few people in the ark (except for some doctors, scientists, teachers, and artists) are over the age of fifty.

While all this is transpiring on Earth, the Messiah astronauts decide to embark on a suicide mission. As one astronaut says: “Look at the bright side; we’ll all have high schools named after us”. Their plan is to destroy the larger of the two newly formed comets by guiding the spaceship into a deep fissure and detonating a second set of four hydrogen bombs. Fortunately for the human race they are successful.

Other recent and major collision (doomsday) films

The movie Meteor, starring Sean Connery, treats the same theme: a five-mile (8 km) rock is on a collision course with Earth after an impact between an asteroid and a comet in the asteroid belt between Mars and Jupiter.

The human race is scrambling to find a way to stop the asteroid. The asteroid is finally destroyed by the co-operative efforts of USSR and the US. The nuclear rockets that destroy the asteroid are fired from both the American and Soviet missile platforms. These platforms were placed in space-orbits with weapon systems that were originally directed against each other’s countries. The destruction of the asterois, however, is not complete and splinters of the rock strike the northeastern coast of the United States, producing tidal waves that cause destruction not unlike that in “Deep Impact”. New York City did not fare well in either Deep Impact or in Meteor.

Another film which made an “impact” on the general public is Armageddon. The general opinion among scientifically literate people is that this film was a much less scientifically accurate than the other two. This is surprising because NASA allowed filming on one of its sites; the scene with the astronauts simulating “zero-gravity” (actually free fall) was filmed in a real training tank. The plot, however, was unrealistic and implausible.

The government is informed about an asteroid “the size of Texas” being on a collision course with Earth. What is implausible here is that this sighting was made only a week before the predicted collision! Clearly, most people would realize that a thousand mile wide asteroid would have been discovered a long time ago.

As you know, Ceres was the first to be discovered (1801) and it is still the largest asteroid we know and is only about 600 miles (1000km) across. Of course, a collision between such a large asteroid and the Earth would clearly spell the end of life on Earth and there would be no defence against such catastrophic event.

The film had even a greater box office appeal then “Deep Impact” which is attributable to the action and suspense created by the director as well as the universal popularity of the main actor, Bruce Willis. It is evident that despite the implausible scenario presented in the movie, the theme of an impending global disaster and how we would cope with such knowledge captures the imagination of the public.

Finally, there is a “Doomsday” episode in Star Trek: The Next Generation called In a Matter of Time (Episode 209, original airing date 11/91). In this episode the Star Trek crew tries to cope with the disaster created by a collision of a planet with an asteroid (in a different solar system, of course). A massive cloud dust is produced by the collision causing a cooling effect, quickly starting an ice age on the planet. The crew then attempts to create a “greenhouse effect” to counter the cooling effect of the dust.

1. Is it realistic (plausible) that a student using a simple 4 inch telescope would be the first on Earth to discover a comet of the size depicted in Deep Impact? In order to answer the question above you should answer the following

a. What percentage of the sky is being continually scanned by professional astronomers?

b. What nations are devoting resources to systematic searches for comets and asteroids ?

c. How much funding is provided for such research in the US? In Cananda?

d. Approximately how many potentially dangerous comets / asteroids are there?
2. Do you think that a potentially catastrophic event, involving a comet of the size of Halley’s comet, could be kept secret in our information age for so long? Discuss.

3 How realistic is the depiction of living in the weightless state of the spacecraft? On the comet? How were these effects achieved?

4. While the astronauts are drilling into the comet’s surface we see that as soon as the radiation of the Sun strikes the surface powerful jets of gas are emitted from within the surface. Is this what you would expect, given the composition of comets and the conditions of deep space?

5. In the movie we see house-size rocks on the surface of the comet. Is there any evidence for thinking that there are rocks of this size on comets? Discuss.

6. We have already found that the albedo of a comet is typically about .03, or 3%. That means that without the comet’s tail reflecting the Sunlight, we would have never discovered any comet. How then was it possible for the audience to see the scenes when the astronauts were drilling?

7. We are not shown the construction or the interiors of the “arks”. However, you can speculate about the construction of such a habitat and the supplies necessary for surviving for a long time.

10. How realistic is the description of the effect of the tidal wave that would result if a comet of the size (and at relative speed of about 50 km/s) landed in the Atlantic ocean? Would only New York and the immediate environment suffer? Discuss.

11. A recent (1999) Internet report on astronaut commented on the credibility of science in movies:

..There was a sensational movie 1983 on a man -made asteroid trajectory accidentally put on a collision course with Earth, but the Hollywood scenario was quite unrealistic. We can expect more of this, from Hollywood in the near future, which will influence the perspective laymen not scientifically inclined for a few sensational thrills and the monetary benefit of Hollywood elites. Nonetheless, one would expect that a quick response a rendezvous team would be set up to protect Earth in the long run against both man-made objects and naturally occurring asterois and big rocks that pass the Earth.

What movie is this statement referring to? Discuss this statement.


1. We have already calculated the approximate gravity (m/s2 ) on asterois and planets. What would be the gravity on a comet of the size and mass in Deep Impact, expressed as percentage of the Earth’s surface gravity?

2. The “Orion propulsion” system described in the movie uses a series of “small” nuclear explosions to impart incremental changes of velocity to the spaceship. In Chapter 2 we discussed the simple case of chemical explosions that were incremental, hurling equal masses away from the rocket at a given ejection velocity. Compare that analysis of rocket propulsion to the Orion propulsion system. Assuming that the mass of the spacecraft is a modest 10,000 tons and that you are in “gravity free” space, answer the following questions:

a. What incremental mass ejection is necessary to increase the velocity by 10 m/s? Assume that the ejection velocity (relative to the spacecraft) is 1000 m/s.

b. How many of these ejections would you need in order to change the velocity of the spacecraft by 1000 m/s?

c. Imagine one large nuclear explosion with the equivalent of 1000 tons of mass ejection in a time of 1/100 of a second. What acceleration would that produce?

d. Speculate on the probability of survival of the crew if an atomic detonation of that equivalent magnitude were to take place.

3. In the movie, the president of the US (played by Morgan Freeman) says, during a news conference,

“ This comet carries the equivalent of 500,000 mega tons of TNT.” Knowing the dimensions of the comet and assuming its density to be about 0.1 g/cm3 , comment on how realistic that number is.

4. We discussed the rotating space station of the type used in the famous film 2001: A space Odyssey” (1968) in Chapter II. Describe the “gravity” as an astronaut would feel it, moving from the center to the rim along one of the “spokes”.

5. The smaller comet does hit the Earth, just off the Atlantic coast of North America, creating a tidal wave, or tsunami. This wave is first about 30 meters high and moves with a velocity of 1100 miles per hour (or 1800 km per hour). On impact with the shore we see a tidal wave thousands of feet high (1000 meters?) that sweeps over New York City and moves inland to about 600 miles

(1000 km). Do you think the “scale of things” makes sense, in view of our previous discussions involving doomsday scenarios?

6. For the movie Apollo 13 (1995) the producers were actually allowed to use NASA’s anti-gravity simulator, the aircraft KC-135, to film the sequence in which the actors seem to be floating about the cabin. The aircraft first reaches an altitude of 30,000 feet (use 10,000 m as an approximation), with a speed of near Mach 1 (use 300 m/s as an approximation). The aircraft then descends, following roughly a parabolic curve, ascends again and is able to complete many cycles, as shown in Fig. 33. The people inside are actually in free fall for 23 seconds for each descending part of the cycle.

a. At the very top the people feel weightless and then remain weightless for 23 seconds

b. How far does the aircraft “fall” in a vertical direction the aircraft travel in the horizontal direction during this time?

c What is the radius of the arc of the circle necessary to ensure free fall?

d. Assume that free fall is maintained for 23 seconds. Describe the trajectory for this portion of the run.

e. If the bottom part of the trajectory has the same arc as the top, what will the weight of the astronauts be?

f. If the run is symmetrical what is the total vertical height through which the aircraft must manoeuvre during this time?

Fig. 10 The NASA aircraft KC-135 in a trajectory to simulate weightlessness.


Only about 150 impact craters have been recognized on Earth. Not until about 1930 was the impact theory for craters, such as the famous Barringer crater in the Arizona desert, accepted by geologists. It may come as a surprise to you that the origin of lunar craters was also a matter of dispute until about the 1950s. The astronomer Ralph Baldwin pointed out in 1949 that, because the Earth and the Moon were companions, the evidence of thousands of impacts on the Moon points to a similar impacting on Earth. In some sense then the Moon presents us with a better record of terrestrial impact cratering than the Earth does. Baldwin went on to describe the most famous impact scar on the Moon, the Tycho crater (named after the sixteenth century astronomer Tycho Brahe) : “The explosion that caused the crater Tycho would, anywhere on Earth, be a horrifying thing, almost inconceivable in its monstrosity”. The Australian astronomer Duncan Steel suggests that “With this compelling warning, the modern era of catastrophism began in earnest”.

By 1949 only half a dozen Earth-crossing asteroids were discovered, but astronomers now (1999) have identified about 160 such asteroids. Of these half are more than 1 km across. It is estimated that more than 2000 Earth-crossing asteroids exist that are larger than 1 km across. Examination of lunar rock samples from the Apollo mission (1970) finally convinced astronomers that the origin of lunar craters is not volcanic action but high velocity impact with comets and asteroids. In addition, there was good evidence for believing that there is a continuous bombardment by particles ranging from dust to basketball size. The Earth must undergo a similar bombardment from objects in space. A simple calculation (see Questions below) shows that the number of impacts per unit area on Earth should be about twice that we calculate for the Moon, because the larger gravity on Earth should pull in more. Moreover, the total number of impacts on Earth (because of its larger surface area) should be about 25 times as large as on the Moon. But where are these craters?

Clearly, the atmosphere protects us and many of the impactors burn up on the way down; the area of the ocean is much larger than the land area, and geological erosion and sedimentation obliterates most traces of impacts. The Moon, on the other hand, has no atmosphere or water (only traces, see Chapter VI) and erosion does not exist. The footprints of the astronauts are still on the Moon, and will be noticeable millions of years from now, barring, of course, a collision with a comet or asteroid where the footprints are.

Impact craters on Earth range in age from a few thousand to about 2 billion years. Impact craters are formed when a meteoroid or an asteroid collides with the surface of the Earth. A body whose mass is more than about 1000 metric tons and travels at typical speed of 20 km per second (relative to the Earth), would go through the atmosphere practically unhindered. A body less than about 100 tons, on the other hand, would decelerate through the atmosphere to about 50% of its original speed. The impact pressure would be enormous, about 100 gigapascals (one million times atmospheric pressure), and the atmosphere in the vicinity of the bolide would reach temperatures of several thousand degrees Celsius.

Fig. 11 Crater formation

Differences in formation of a simple crater and a complex crater. The central peak of the complex crater is formed as a result of uplift of material stratigraphically beneath the crater, which rebounds in response to compression caused by the impact. From Melosh (1989)

They are characterised by a simple bowl shape similar to that of the transient crater suggesting minor gravitational collapse following impact. Simple craters generally have depth / diameter ratios of between 1/5 (0.2) and 1/3 (0.33). One example of a simple crater is the Barringer Crater, Arizona (Figure 2), which is 1.186km in diameter today.

Much of the material ejected from the crater is deposited in the area surrounding the crater. Close to the crater, the ejecta typically forms a thick, continuous layer. At larger distances, the ejecta may occur as discontinuous clumps of material. Some material that is ejected is large enough to create a new crater when it comes back down. These new craters are termed secondary craters and frequently occur as lines of craters that point back to the original crater.

Material below the surface of the crater is significantly disrupted by the shock of the impact event. Near the surface is a layer of breccia (a type of rock composed of coarse, angular fragments of broken-up, older rocks). Rocks at deeper depths remain in place (and are termed bedrock) but are highly fractured by the impact. The amount of fracturing decreases as the depth below the surface increases. The energy of the impact typically causes some material to melt. In small craters, this impact melt occurs as small blobs of material within the breccia layer. In larger craters, the impact melt may occur as sheets of material.


1. Discuss briefly the difference between simple and complex craters. Find the diameter to depth ratio and explain how gravity influences the formation of craters. For example, on the Earth, the transitional diameter is 2 to 4 kilometers, whereas on the Moon it is about 15 - 20 km.

2. The Internet has a lot of information on craters and pictures of most of the craters on Earth. Can you clearly categorize them as either simple or complex? Explain.

3. Using sand in a box, throw metallic spheres at various angles and notice the shape of the crater made. Discuss.


1. Estimate the percentage of impacts by asteroids and comets that occur in the oceans. How is it then that the only underwater crater discovered is the 60 km wide and 50 million year old Montagnais structure on the coast Nova Scotia?

2. Discuss the following statement: The number of impacts per unit area occurring on Earth should be about twice that on the Moon. Therefore the Earth should encounter about 25 times as many impacts as the Moon.

3. It is estimated that about 500 tons of space material fall on the Earth surface every day. Assume that

this rate has not changed in 3.5 billion years. How many tons of space debris has each square meter accumulated? Comment.

4. For decades, maps of South Australia have shown a 35 kilometer wide dry salt pan called Lake Acraman. It was not until in 1986 that geologists realized that this is the residual basin surviving from a colossal impact about 600 million years ago. The true nature of the crater was not recognized for so long because all tell tale signs have been worn away. Now geologists believe that the collision of the crater also caused a shower of rocks that was ejected and rained down, more than 300 km away, in a mountainous region called Flinders Range. At the time, that region was occupied by a shallow sea, so that the ejected rock accumulated in the sediment that was laid down for eons, and eventually buckled up to form the mountain range.

5. Estimate the initial velocity of the ejected rocks and comment on and estimate the energy of the impact.

6. A simple analogue to the action involved when an impact crater is formed can be demonstrated by dropping a sugar cube in a cup of coffee. As in a coffee cup, many craters show a central uplift, or a spike. In fact, in large impacts, when huge pressured (mega and even gigapascals) and high temperatures (over 1000 degree of Celcius) are involved, the rock behaves like a fluid, but is then frozen into a characteristic rebound form. Sketch a picture of what you observe, or thought you observed, in the coffee cup.

7. Students can make models of craters in the classroom with a box, lined with a trash bag. Using flour at a depth of about 10 cm, some dry powdered tempera paint and various sizes of marbles. Students should look for crater features and then test the effect of different velocities, angles of impact and sizes of marbles

Jupiter and the mystery comet Lexell
Until recently, astronomers believed that the Earth is safe from impacts because the large gravitational field of Jupiter periodically “sweeps clean” the inner solar system. While it is true that Jupiter does “sweep clean” the inner solar system, Jupiter is also instrumental in occasionally sending us comets and asteroids into Earth-crossing orbits

This attitude of false security is partly due to a well-known series of events that involved comet Lexell, which was discovered in 1769. The famous French astronomer Charles Messier observed what seemed to be a verylarge comet and found that two weeks later it made the closest approach by a comet ever observed, missing the Earth by a mere 0.015 AU, or about two times the distance to the Moon. Messier was puzzled as to why a comet that had a tail of millions of miles long was not observed before 1769. The Russian astronomer Anders Lexell, however, soon showed that the comet had a short-period of 5.6 years. He also suggested that the small elliptical orbit had been produced by the close encounter with the high gravity of Jupiter. The comet made no reappearance in 1772, as was expected, and the Paris Academy of Sciences offered a prize in for the complete solution of this puzzle.

The puzzle was solved by the astronomer Burckhardt, aided by the brilliant young mathematician Pierre Laplace. Tracing back the comet’s motion , Laplace found that it approached within 0.1 AU of Jupiter in 1767 and concluded that the high gravitational attraction threw the comet into a new orbit. He also showed that prior to this encounter, the comet had a period of about 52 years and was therefore too far to be seen earlier.

The new orbital period of the comet was found to be 5.6 years which happened to be almost exactly half that of Jupiter’s orbital period. In 1776, 5.6 years later, the comet should have arrived back to perihelion, but the

Sun was in the line of sight . In 1779, 5.5 years again, or about 11. 2 years later, both Jupiter and Lexell met almost in the same relative place as they did in 1767. However, Jupiter was now slightly ahead of Lexell pulled the comet into a longer orbit of about 20 years, never to be seen again.

So the mystery of the comet Lexell was solved, and in the words of the astronomer David Milne in a an article, written in 1828 (see References):”...the discovery may certainly be looked upon as having brought to light one of the most astonishing facts in the whole history of astronomy.”

Fig. 12 The mystery comet Lexell

Fig13. The story of Comet Lexell.

Fig. 14 Comet Lexell sighted in 1769, commemorated by this contemporary etching
1. Draw your own sketch to show and explain the close encounter of Lexell with Jupiter and describe how the trajectory of the comet changed twice.

2. Explain to a friend how Jupiter redirected the motion of the comet, once toward the Earth, and then away from the Earth.

3. The unexpected motions of comet Lexell were first brought to the attention of astronomers in general by an article written in 1828 by the astronomer David Milne. In this article he says: “At a time of their nearest approach, in August , Jupiter was distant from the Comet only 1/491 of its distance from the Sun, and hence exerted upon it a force of 225 greater .” Check this claim and try to confirm it or reject it.

4. Based on the story of the comet Lexell, discuss the role Jupiter plays in “controlling” the comet and asteroid distribution in the solar system.

5. After you have studied the section on “gravitational slingshot” and “gravitational breaking” on page you will be able explain how the gravitational pull of Jupiter was able to change the orbit of Lexell; once to pull it toward the Earth, and then to sweep it out into a very eccentric orbit, never to be seen again.

Fig. 15. Explanation of gravity slingshot effect
1. The celestial body is “infinitely” more massive than the object approaching it. Therefore, the celestial

body’s frame of reference can be considered an inertial frame. We consider the Earth as a good inertial frame (in spite of the fact that it rotates!). The celestial body can be considered to be moving at a constant velocity (constant speed in a straight line), even though it may be orbiting a larger body (the sun , in the case of the Earth, and the Moon, in the case of the Earth), as long as the time interval considered is small.

2 . Energy and angular momentum are conserved in a closed system. That means that when an object falls toward a large celestial body from very far away (but not influenced by an other large celestial body) with an initial speed of V will leave the large celestial body with the same speed V as measured at a very large distance from it.

3. If a an object approaches the large celestial body too rapidly, deflection decreases and if the object approaches too slowly, it will tend to crash into the large celestial body.

4. To increase its velocity the object must approach the large celestial body from behind; to decrease its velocity it must be approached from the front, that is from the direction in which the large body is travelling around the sun. the direction of about 45 degrees to the orbit, to 100-1000 x the original magnitude of that force, and 90 degrees to the motion, depending on where we choose place the orbit (See Fig. 29).

As before, we can find the approximate velocity of the object at the start of the Earth’s “sphere of influence” as we have defined that region.. From here the trip will only take about 8 hours to reach the closest approach to Earth.

Fig. 16 Gravitational assistance by the Moon to place the asteroid into the orbit of the Earth

Fig shows a typical lunar assist. The object comes in and catches up with the Moon, and swings around it, reducing the speed (relative to the Earth only!) by about 1 km/s. The object ( a payload form Eros in our case) arrives in the vicinity of the Moon, say at the height of 1 lunar radius (1.73x107.m). Using the same technique as

described in Fig. above.
Playing “orbital billiards: Capturing of the payload by using several Moon flybys.
In the game of “orbital billiards” we are tapping the gravitational energy source as our payload exchange orbital momentum with the Earth and the Moon: the payload slows down while the Moon “speeds up”. The effect on the Moon, as we have already mentioned, is very tiny indeed and cannot be noticed.

It is clear now that if we want to capture into a conveniently accessible orbit around the Earth a payload from Eros, we will have to use a double or even a triple “lunar assist”. We will look at only the first stage of this problem. One of the reasons for not going in to detail is the fact, that when discussing the lunar assist trajectories in detail we are dealing with a three-body problem. This is a notoriously difficult problem , as we have already indicated when discussing the libration points of Jupiter an the Moon. Essentially, Kepler’s laws and the vis viva equation do not apply anymore and we would need to use complex numerical methods of the type we applied when we discussed the calculations we used in determining the dynamics of bolides interacting with the atmosphere.

Doomsday reports

Fig. 17 An example of a “doomsday” prediction.
A example of recent ‘dooms day' reports:
IL ****

Astronomer David Tholen spotted it last year in the early evening of June 19, using the University of Arizona's Bok telescope. It was a new "near-Earth object," a fugitive asteroid wandering through space to pass close to Earth.

Tholen's team took three pictures that night and three the next night, but storm clouds and the moon blocked further observations. They reported their fixes to the Minor Planet Center in Cambridge, Mass., and moved on.

They had never measured anything as potentially dangerous to Earth. Impact would come on Friday the 13th in April 2029.

The holidays and the tsunami in South Asia pushed 2004 MN4 out of the news, and in the meantime additional observations showed that the asteroid would miss, but only by 15,000 to 25,000 miles -- about one-tenth the distance to the moon. Asteroid 2004 MN4 was no false alarm. Instead, it has provided the world with the best evidence yet that a catastrophic encounter with a rogue visitor from space is not only possible but probably inevitable.

It also demonstrated the tenacity of the small band of professionals and amateurs who track potential impact asteroids, and highlighted the shortcomings of an international system that pays scant attention to their work.

Doomsday calculations
The great detail and the data of the extensive research of the Barringer crater impact, Tunguska event and the Yucatan impact give us good background knowledge for the calculation of damage that could result from various sizes and kinds of impacts. We will look at these explosions and make some relatively simple calculations.
1. Studies of the damage caused by nuclear weapons shows that an airburst causes more destruction than a ground burst and that the area of the ground laid to waste varies as the energy of explosion raised to the power of 2/3. This is a purely empirical formula (based on observation and measurement), and is not connected to a clearly articulated theoretical background. Scientists often use such simple formulas as afirst “attack” in trying to understand a problem According to the astronomer Chapman, the Tunguska event, which occurred at an altitude of about 8 km, provides a good calibration for such events:

A = 400 Ex, x = 2/3

where A is the area of devastation in square kilometers, and E is the energy of the explosion in million tons of TNT equivalent.

Check this formula for the case of Tunguska. The area of devastation was about 2200 square kilometers. Remember, the bolide is thought to have been about 50 m across and the kinetic energy was supposed to have been somewhere between 10 and 20 Mt of TNT equivalent. Comment.
2. Moving up in size, say to about 100m, consider Tunguska-like bolide, coming in at 22 km/s.

a. Show that the energy of the impact is about 100 Megatons of TNT equivalent.

b. Show that, according to our empirical formula, the area of devastation would be about 10,000 km2.

c. Pick a large city, such as New York, London, or Toronto. From a city map estimate the area of the city you have chosen. How much of the city would be destroyed if such an impact took place?

Doomsday scenarios

We will now look at seven doomsday scenarios. For each case, estimate the destruction involved, considering

a. The number of casualties based on the idea of “global average”,

b. The number of casualties based on the population density of a large city of your choice

The average population density of the Earth is about 3 per square kilometer, from Hong Kong (about 5000) down to about 2 in Canada and 1.5 in Australia. Because approximately about 70% of the Earth’s surface is covered by water, the global average is about 10 people per square kilometer. Use our model, but you should ask questions that the model does not answer and then try to apply your knowledge of physics to answer them.
Scenario 1: A Barringer crater-like impact on the land. A 100 m bolide (iron-nickel) strikes the Earth surface.

Scenario 2: A Barringer crater-like impact in the ocean. !,000,000 people will die,

because of the effect of a tsunami on coastal population.

Scenario 3: A Tunguska-like explosion: A 100 m bolide (stone) exploding in the

atmosphere over land is expected to kill about 100,000 people. Greater

land devastation than in case 1.

Scenario 4: A Tunguska like explosion over the ocean. 1,000, 000 people will die because of

tsunami effects It is estimated that a tsunami is able to transport this energy at

very high speeds to coastlines, where the population density is usually high. A

small bolide impacting over the ocean then will probably cause more deaths tone

that falls over land. The estimate made by experts in the field is that in the first

case we can expect about 100,000 deaths and in the second about 1,000,000.

Scenario 5: A 1 km asteroid (stony), enters the atmosphere at 15 km/s. and falls on land.

A 1 km asteroid collides with the Earth with a kinetic energy of about 0.1 to 1.0

million Mt of TNT equivalent (1012 tons of TNT equivalent).This is considered the threshold energy for creating a “nuclear winter”, that is, enough dust (micron sized and smaller) is raised into the stratosphere to block out the Sunlight so that the surface temperature drops by several degrees. This amount of dust would take many months to settle out leading to a collapse of life in general. About 25%-50% of the human race would perish. Compare this to the potential destructive effect of the estimated combined nuclear warheads in the world.
Scenario 6: A 1 km asteroid (iron-nickel) falls into the ocean at 15 km’s and falls into the


Note: It turns out that this is about the smallest size of a space object we can

detect and tract with confidence. There are good reasons to believe that astronomers could detect and tract all of them in about 20 years.

Scenario 7: A Yucatan impact-type falls on land: A 10-20 km asteroid collides with the

Earth at about 15 km/s.

Scenario 8: A Yucatan impact-type falls in the ocean. A 10-20 km asteroid falls into the

ocean at about 15 km/s.

Table 7:
Impact Energy Altitude Environmental Casualties:

Scenario Involved of Stresses Heavily populated

(TNT equ.) Explosion ---------------------------

Global average

Scenario 1.

100 m asteroid

(15 km/s)

landing on ground


Scenario 2.

100 m


( 15 km/s)

landing in ocean


Scenario 3.

100 m


(15 km/s)

exploding in atmosphere


Scenario 4

100 m


(15 km/s)

exploding over ocean


Scenario 5

1 km


(15 km/s)

falling on land

Scenario 6

1 km


(15 km/s)

lands in ocean

Scenario 7.

10 km


(15 km/s)

lands on the ground


Scenario 8.

10 km

asteroid falling in ocean


Note: We have only considered asteroids in our scenarios. Most asteroids under about 100 m will explode high in the atmosphere. You can check this using our model. However, you may want to consider what would happen when a large asteroid like Halley’s comet would collide with the Earth. Remember that most asteroids orbit clockwise, that is, opposite to the motion of the Earth (comet Lexell was a notable exception). The entry velocities of comets, therefore, are much higher than those of asteroids

(about 30-35 km/s).

Use our model and find out what would happen if Halley’s comet collided with the Earth, say head-on. Consider the size of comet Halley to be about 10 km, with a density of about 0.1 kg/m3 . You should be able to estimate the velocity of impact from the data given in the first section.

Fig. 18. Size and frequency of collisions with Earth

Fig. 19 Travelling to Eros
Brief description of the NEAR trajectory:
I: Launch Phase
Feb. 27,1996: Launch took place at 20.43 UT (3:43 p.m. EST) from Cape Canaveral, using a Delta II-7925 expendable launch vehicle.

!The Delta II parking orbit (low Earth orbit (LEO) had an altitude of 183 km and an inclination of 28.74 degrees.

!The launch azimuth was fixed at 95 degrees.

!The coasting period in orbit was 13 minutes, allowing solar power to be used, one hour

after launch.

!The injection burn, powered by the third stage solid motor, lasted for 4 minutes. The space craft (SC) was entirely inside the Earth’s shadow.

!Approximately 22 minutes later the SC separated from the third stage.

!A yo-yo de-spin mechanism released the stowed position of 69 to 0 rpm and simultaneously released the solar panels.

!After the third stage, responsibility for attitude control was shifted to the SC guidance and control subsystems.

!The SC left the Earth’s shadow 37 minutes after launch. Up until now the space craft was battery powered.

II: Cruise Phase

(Remember: Eros orbits the Sun at an angle of 10.8 degrees to the ecliptic).

!The spacecraft followed the “Delta VEGA” trajectory, short for “change in Velocity, Earth Gravity Assist”. This was necessary to provide the extra energy needed to rendezvous with Eros.

!During the flight, low-level burns were performed to calibrate the propulsion system and to correct for any trajectory errors.

!The space craft was put into “hibernation mode”.

June/July 1997 period: NEAR maintained this course until preparations were made for the two upcoming critical events - the Mathilde flyby and the Deep Space Manoeuver, when encountering Eros.

Feb. 18, 1997: NEAR established record for the greatest distance from the Sun for a solar-powereed spacecraft at 327 million kilometers, or 2.18 AU.

June 27, 1997: The spacecraft flew within 1200 km of Mathilde at 12.56 UT with a relative velocity of 9.93 km/s, returning images and other data.

July 3. 1997: NEAR decreased the velocity in a two-part burn of the main 450 N


This action decreased the velocity by 279 m/s and lowered the perihelion from 0.99 AU to 0.95 AU.

January 23. 1998: Earth gravity assist swing-by took place at 7:23 UT (Universal Time) and placed the SC on its final approach with asteroid 433 Eros. Closest approach to Earth was about 500 km, above Ahvaz in South-Western Iran. This manoeuver changed the orbital inclination from 0.5 to 10.2 degrees and also changed the aphelion distance from 2.17 to 1.77 AU in order to match those of Eros.

April 1, 1998: NEAR sets the record for being the most distant manmade object detected by optical means. An amateur astronomer in Australia spotted the SC at a distance of 33. 65 million kilometers ( .22 AU). The previous record was the 1992 sighting of the Galileo spacecraft at a distance of 8. 0 million kilometers (.05 AU) from Earth.

Dec. 20, 1998: Attempted a 15 minute bi-propellant engine burn to align the SC with the position and velocity of Eros. However, burn had to be aborted and spacecraft entered Earth Safe Mode (ESM) orbit

About an hour later, a low voltage stage was detected and the SC was entered into the Sun Safe mode (SSM) orbit.

Dec 22, 1998: The SC was first detected while it was in the SSM; it was then recovered and guided int Dec 22, 1998: The SC was first detected while it was in the SSM; it was then recovered and guided into an ESM mode and finally put back into operation mode. The Multispectral Imager (MSI) was activated in preparation for the Eros flyby.
Dec. 23, 1998: The NEAR Infrared Spectrometer (NIS) and Magnetometer were turned on in preparation for the Eros flyby. The closest approach of the Eros flyby took place at 18:42 UTC.
III: Insertion phase
Jan. 3, 1999: Deep Space Manoeuver (DSM) was accomplished by a successful biprop engine burn that put NEAR on target for the Feb. 2000 rendezvous with Eros.
April 21. 1999: NEAR’s orbit brings it within 1.5 AU of the Sun. Reorientation of the spacecraft was necessary to reduce thermal stress on the solar panels.
Aug. 12. 1999: Trajectory Correction Maneuver (TCM) 19 will change the speed

(relative to the Sun) by about 21 m/s.

Oct. 10 1999: Re-start of instrument operations will begin.
Febr. 2, 2000: Orbit Insertion Maneuver (OIM) of about 10 m/s.
\Feb. 14, 2000: Orbit Insertion Maneuver (OIM) involving an approach speed of about 10 m/s will place the NEAR spacecraft in orbit around Eros.
Originally it was planned to begin a series of four rendezvous manoeuvers with the main thruster to slow NEAR down by 949 m/s in order to achieve a relative velocity of 5 m/s. It was then scheduled to fly by Eros on its sunward side at a distance of about 500 km. Studying the trajectory of the orbit will give scientists an opportunity to determine the mass to a 1% accuracy, allow the identification of several hundred surface landmarks, and determine the asteroid’s spin vector more accurately. A search will also be conducted for possible satellites of Eros. This investigation will have the capability of locating any bodies larger than 5 m.. After a couple of days NEAR will be parked in a circular orbit of about 200 km, finally tightening the radius to about 35 km.
Details about the NEAR spacecraft that will assist you in problem solving
Near spacecraft details:
On-orbit mass: 805 kg (including 318 kg propellent)

Power system: Four solar panels1800 Watts @ 1 AU

Shape: Octagonal prism shaped, 1.8m x1.2 m gallium arsenide solar panels

Fixed 1.5 m X-band high-gain radio antenna

Propulsion: A 450 N main thruster, a 20 N and a 5 N thrusters, total Delta V of 1450 m/s.

The system carries 209 kg of hydrozine and 109 kg of TNO axisizer in two oxidizer and three fuel tanks.

Space craft guidance is achieved through the use of a sensor suite consisting of five digital solar attitude

detectors, an inertial measurement unit AMU) that contains contains hemispherical resonator gyros as well as accelerometers. There is also a star tracker camera, pointed opposite to the instrument that establishes a direction.

Fig. 20 Science instruments used on flyby.

NEAR’s six science instruments:
The magnetometer (MAG) will determine the strength of a magnetic field around Eros, if one exists.

A magnetic field would be strong evidence for the asteroid containing abundant metallic iron. Many of the recovered meteorites contain iron.

The X-ray/gamma-ray spectrometer (XGRS) will determine the existence and the quantity of elements such as iron, uranium, thorium, and potassium.
The near-infrared spectrometer (NIS), will analyze the reflected light and map the mineral composition of Eros.
The laser rangefinder (NLR), actually a “laser radar”that is an altimeter, that will determine the shape of the asteroid with an accuracy to a few meters.
The multispectral imager (MSI) will map the shape , landforms, and color properties of Eros in order to determine the configuration of its different types of rocks and determine the processes that have shaped the surface.A radio science experiment (RS) will track the tiny changes in NEAR’s radio frequency caused by the changes in velocity (using the Doppler effect) as the spacecraft responds to the gravity of Eros. This response will determine the mass of the asteroid. Establishing the volume, as determined by imaging, will allow scientists to find the density of Eros.


You are expected to find the information about NEAR EARTH ASTEROID RENDEZVOUS (NEAR) on the Internet. There is a lot of detailed information available. Read the information with the following guiding questions in mind:

1. Why was Eros chosen as the “ideal” asteroid to study? Give several reasons for

this choice.

2. What are the specific goals of the NEAR mission?

3. Astronomers in the early 1800 thought they were observing an atmosphere on the large asteroids, especially on Ceres. Why do astronomers now believe that

neither Ceres nor even the much larger Moon could support an atmosphere?

4. Using the Internet, find the NEAR education page and describe one of the

instruments in detail and discuss what its function is for the NEAR.

1. Estimate the mass of Eros, based on the data given above.

2. Assume that Eros is a sphere and calculate the surface gravity. What percentage

of the Earth gravity is it?

3. Calculate the escape velocity on Eros. Could you throw a ball into orbit?

4. The closest approach of Eros to Earth in the twentieth century was on January 23, 1975, at about 0.15 AU.

a. How many ‘Earth radii’ is that?

b. How many ‘Moon distances’ is it?

5. It is interesting to find out what the attraction of the Earth on the asteroid was at \

the distance of 0.1 AU and compare that to the attraction the Sun provided.

Show that the gravitational attraction of the Sun is still much larger, in fact about 10,000 times as large.

6. The Moon’s distance from the Earth is about 3.84 x 10 8 m, or about 60 Earth

radii. Imagine an asteroid

Coming as close as 120 Earth radii, or two Earth-Moon distances. This would be

about 0.0051 AU, a very close approach, indeed. Calculate the ratio of the

Earth’s and the Sun’s attraction. We can disregard the Moon’s influence at this

distance. Why?

7. You can now calculate the Earth’s sphere of influence, that is, the distance

between the Earth and the Sun where the attraction of the Earth on an asteroid or

comet is equally strong. At this distance an asteroid like Eros would feel equally

attracted to the Earth and the Sun.Compare this distance to the distance from the

Earth to the Moon.

8. In the above problem we disregarded the effect of the Moon. To show that we

really are not allowed to do that, consider the hypothetical situation below. Here, we have a complicated situation where four bodies are involved. A problem like

this can only be solved by numerical methods; no closed analytical solutions are


a. In the position shown, what is the instantaneous force (stated as a vector) acting on Eros?

b. Find the instantaneous acceleration on Eros.

c. Do Kepler’s laws apply here? If not, why not?

Orbit calculations for NEAR spacecraft mission: A guided problem discussion

(Note: For all these problems consult Fig. , The NEAR trajectory. In addition, it is recommended that you accurately sketch the orbits to scale on a large sheet of paper (60x40 cm is a good size), and mark your progress as you go along. Use two pins and a string to trace your ellipses) Also assume that the orbits of Eros, Mathilde, and the Earth are coplanar (we know that, actually, Eros has an inclination of 10.8 degrees and Mathilde 6.7 degrees with the eclectic, or the plane of the Earth’s orbit).

We will describe four orbits for our problem: The orbit of: the spacecraft, of asteroid Mathilde, the asteroid Eros, and, of course the orbit of the Earth. Use 1 AU = 1.5 x 1011 m.

We will assume that

a) all orbits are moving in the plane of the ecliptic, and

b) that the orbit of the Earth is circular.

The first assumption leads to considerable errors, because the inclination of Eros is about 10 degrees and of Mathilde a little more than 6 degrees. The second assumption will lead to very small errors. However, for purposes of getting a rough idea of the motion, this is a pretty good approach. Of course, you would not be hired by the NEAR research group if your calculations were based on this elementary approach!

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