Part B: The Mathilde flyby, or “Waltzing with Mathilde’

Note: Most of the information you will need for the following problems are found in Fig. 28.

` 1. From your sheet, showing the orbits of the SC and Mathilde:

a. Estimate the distance from the SC to the Earth and to the Sun, expressed

in AUs. Do your values agree with the NEAR report? Comment.
b. Show that the orbital velocity of Mathilde at this time is about 23.6 km/s and of the SC about 18.1 km/s

c. Estimate the angle between the two orbital velocities and work out the

relative velocity between the SC and Mathilde. How close is your value

to the one given in the NEAR report? Comment.

T he SC was then placed into an orbit that intercepted the orbit of Mathilde on June 27, 1997. According to the NEAR report, the distance to the Earth was about 2.2 AU at the time of the flyby, the distance to the sun about 2.0 AU and the flyby velocity (relative to the Mathilde) was estimated at about 10 km/s, in an “off-tangent” direction. The closest approach. was 1200 km.

2. From your sheet, showing the orbits of the SC and Mathilde:

a. Estimate the distance from the SC to the Earth and to the sun, expressed

in AUs. Do your values agree with the NEAR report? Comment.

c. Estimate the angle between the two orbital velocities and work out the

relative velocity between the SC and Mathilde. How close is your value

to the one given in the NEAR report? Comment.

3. Astronomers calculate the mass of an asteroid by direct observation of size,

then estimating the volume and the density; or by noting the perturbation

(deviation from the “normal” orbit) of the orbit when approaching a large body

like that of the earth. Would the close approach to a large asteroid, like that

Mathilde, significantly change the orbit, so that it is measurable?

a. Compare the force of the sun on Mathilde (expressed in N/ kg) with the force of Mathilde, using an estimated mass for the asteroid to be 1x10¹⁷ kg. Comment.

b. Estimate the gravity (m/s²) on the surface of the asteroid.

c. What would be the approximate escape velocity from the surface of the

asteroid?

4. About one week after the Mathilde flyby, a Deep Space Manoeuver (DSM) was

executed. This involved one of two major burns expected from the main thruster

of 450 N force The SC had to be slowed down by 279 m/s in order to lower the

perihelion distance of the trajectory from 0.99 AU to 0.95 AU.

a. How long did it take to slow down the SC by 279 m/s? Estimate the mass of the SC at this point.

b. Show that a Delta-V of 279 m/s is going to change the perihelion

distance from .99 AU to .95 AU.

5. Consider the orbit of Mathilde and calculate the velocity of the asteroid (relative

to the sun) at the time of the encounter with the spacecraft.

6. Now calculate the velocity (relative to the sun) of the NEAR spacecraft as it

passes Mathilde. From Fig. 28 you should be able to find that the spacecraft was

in an orbit whose semimajor axis was about 1.60 AU.

7. The Internet report of the Mathilde encounter claims that the fly-by speed of

NEAR was 9.92 km/s (velocity of spacecraft relative to the asteroid). The

encounter took place at a distance of 2.19 AU. Calculate the velocity (relative to

the sun) of the asteroid Mathilde and of the spacecraft. Does your calculation

agree with this value? Comment.

We have already encountered a swingby when discussing the strange case of comet Lexell. Gravity assist, variously known as swingby or slingshot effect, is used to change the direction and/or the inclination of a spacecraft orbit, to increase or decrease the speed of the SC. When the SC gets close to the planet, it enters the “the sphere of influence” of the planet’s gravity. This can be defined as the distance at which the planet’s gravitational pull on the SC becomes significant (see problem below). The speed of the SC then will increase to a maximum when it gets closest to the Earth. Clearly, the speed at this point must be greater than the escape velocity from this point. In effect, there is an exchange of angular momentum with the Earth and the Earth will slow down, or speed up, a very tiny bit. When the SC climbs out of the Earth’s gravitational field its speed is decreased. The conservation of energy principle requires that the incoming speed and the outgoing speed (at a point far away, beyond the “sphere of influence” of the Earth), relative to the planet, be equal. (To understand better what happens when a spacecraft flies by a large celestial body, a planet or the Moon, turn to the Supporting Text, or ST).

NEAR scientists used the swingby as an opportunity to test the performance and calibration of the SC’s six instruments and to practice co-ordinated multi-instrument observations of the type that will be used on Eros. The SC’s solar panels reflected the Sunlight to Earth so that it could be seen with the unaided eye. NEAR approached Earth at about 6.7 km/s and reached its reached its highest speed at about 13 km/s at its closest approach to Earth. It was then 478 km above South West Iran at 11:23 a.m. on January 23, 1998. NEAR then left the Earth at about 6.7 km/s.

On first thinking about gravity assisted flyby, you might ask: “How can this be a slingshot effect, if the speed does not change?” The clue to the answer is given by realizing that are looking for speed change relative to the Sun. See Fig. for a more detailed explanation.

After the engine failure, the NEAR spacesraft was allowed to return to earth, the earth-gravity assist swingby occurring on January 23 1998. The closest approach to earth was 540 km and the velocity of the spacecraft (relative to earth) was 13 km/s.

1. The spacecraft was travelling at about 6.7 km/s when it entered the Earth’s

“sphere of influence” .We can define the “sphere of influence” as the distance

from the Earth where the gravitational attraction of the Earth begins to exceed

that of the Sun.

a. Show the orbital velocity of the SC before entering the “sphere of

influence” of the Earth was about 36 km/s.

b. You can easily show that this happens when the SC is approaching

Earth at a distance of 2.6x10⁸ m. This is approximately 40 Earth radii

away, or about 2/3 the distance to the Moon.

c. At a distance of about 4 Earth radii, what is the gravitational influence

of the Earth, compared to that of the Sun?

You should have found that the gravitational attraction of the Earth at a distance of about 4 Earth radii is less than 1% of that of the Sun.. So, we can now assume that SC has passed from its helio-centric orbit to a geocentric orbit at about that time.

a. If the spacecraft passed overhead, approximately how long do you think

would it be visible?

b. Show that the velocity of the earth (relative to the sun) is about 30 km/s.

c Now calculate the velocity (relative to the sun) that the spacecraft

would have if the earth had no

influence on it. Show that this is about 36 km/s.

d. The earth’ gravity assist then changed the velocity from 36 to about

49 km/s.

e. What was the % increase in kinetic energy provided by the sling-shot

effect of the Earth?

2. The closest approach of the NEAR spacecraft was 478 km. The ground track was from Europe to Hawaii, changing the inclination from 0.5 degrees to 10.2 degrees, and reducing the aphelion distance from 2.17 AU to 1.77 AU, to enter an orbit that matches the aphelion distance of the orbit of Eros. This is our orbit V, described above.

a. We can now try to confirm the observation that the highest velocity of NEAR

was about 12.8 km/s, as it passed the point of closest approach. What iwould be

the velocity of the SC if it falls from a large distance (a distance of 60 Earth radii can be considered “infinity”) toward the Earth and then passes the Earth at a

close approach of about 500 km? Remember the SC already has a velocity of 6.7

km/s relative to the Earth.

b. When the SC reaches the closest approach, it is moving with a speed of almost 13 km/s.

We know that the escape velocity from the Earth surface is about 11 km/s.

What would have happened if the spacecraft had a velocity of, say, 9 km/s at the

point of closest approach?

c. If an Earth satellite were in a circular orbit at a height of 478 km (the height of

closest approach) , what would be its speed?

d. What additional velocity (what we have called Delta V) would be needed to have this satellite escape the earth?

e. Using a globe and a small sphere, like a marble, try to show the near-Earth

trajectory and explain to a fellow student i. the sling shot effect, ii. how you think

the change in inclination was produced.

d. Using the orbit parameters for the SC, the Earth and of Eros, estimate the

incoming velocity and the out-going velocity of the SC.

3. Discuss the following claim: Voyager 2 toured the Jovian planets. The spacecraft was

launched on a standard Hohmann orbit (HOT) transfer. Had Jupiter not been there at the

time to give considerable boost to the spacecraft by the gravitational slingshot effect,

what would have happened to Voger 2?

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.

About 200 days before catching up with Eros, visual contact was made with the asteroid. From here it was possible to navigate “optically”. In addition, initial shape and rotation determination of Eros were attempted.

Unfortunately, on December, 20, 1998, just 21 days from its scheduled rendezvous with Eros, NEAR failed to complete a crucial engine burn, leaving scientists and engineers frustrated and scurrying to save the mission.

The burn was supposed to put the SC on track for an orbit insertion around Eros.. The SC defaulted into a safe mode, and waited for instructions from the NEAR operations center. Actually, the SC by now had lost 30 kg of fuel. Luckily at 7:30 p.m. contact was made with NEAR, much to the relief of the NEAR group. They now had to wait for the SC to make a preprogrammed 360 degree sweep, looking for a signal from Earth. The group had only a 10 minute window of opportunity to locate the signal and then upload crucial commands for the SC. Later, Tom Coughlin, the project manager remarked: “ They were the longest 27 hours of my life”.

The NEAR group now had to work round the clock to find out what went wrong and examine alternative options. Luckily, these were worked out before, in preparation for just such an emergency. But they were unable to save the mission but were faced with a new challenge: get as much as you can from a flyby of Eros.

New programs were hurriedly written that would direct the SC to take images of the asteroid and collect valuable data as it flew pas at a high relative speed. The group managed to get enough data to give mass, shape and composition estimates. There was no evidence of a Moon orbiting Eros, larger than 100 m.

On January 3, a 24-minute engine burn successfully increased the speed of the SC by 932 m/s in order to “catch” the asteroid. This was a critical maneuver since it used up 57% of the fuel, in order to closely match the speed of Eros. NEAR and Eros were now travelling in almost identical orbits around the Sun. NEAR was closer to the Sun at perihelion, on the “inside track”, and will catch up with Eros, hopefully on Valentine’s Day (February 14) 2000.

Questions

1. According to the NEAR research group, the scientists and engineers controlling the

mission turned a misfortune to their benefit. They point out that the chance of a close

encounter with an the asteroid before entering an orbit has actually been a lucky break.

The encounter gave them a chance to establish the dimensions and the mass of Eros more accurately and this knowledge will improve the orbit insertion manoeuver next year.

a. The new size of Eros is 38x13x13km and the mass was found to be about

2.12x10¹⁶ kg. Find the density of the asteroid.

b. The previous estimate of the size of Eros was 40x14x14 km. What is the %

difference between the two measurements?

c. Do you think the gravity around the asteroid is significantly different for the two? Discuss.

2. Discuss the following statement from the NEAR group:
Although 433 Eros does not represent a threat to the Earth at present, despite its occasional close approaches, this situation is slowly changing. Orbital calculations show that the perihelion distance is slowly decreasing and that a future impact is quite probable. However, this impact will not happen for another 100 million years. When it does, it will cause a crater of up to several hundred kilometers in diameter.
You could refer to the section that discusses “doomsday scenarios” and call this the ultimate disaster that would surely wipe out civilization and probably all of life on Earth.

Problems

1. Where was the SC and were was Eros when first visual contact was made?

2. Approximately how far was Eros from the SC when first visual contact was made?

3. On December 20, 1998, just 21 days before the planned rendezvous, engine failure

occurred. Where were the two bodies and was the approximate distance between them?

4. When the NEAR group tried to contact the SC how long did it take the signals to travel

from Earth to the SC?

5. The 24 minute burn by the main engine delivering 450 N force produced a Delta V of

932 m/s. Given this information, what was the approximate mass of the SC at this time?

Mining on Eros

Even a brief search on the Internet reveals a buzz of planning and organization aimed at investors daring enough to claim the potential riches that could result from space exploration, especially in asteroid mining. A typical advertisement is the following: “A very small asteroid, such as 3554 Amun at 2 km in diameter, contains material worth approximately US $20,000 billion!” The promise of great riches has attracted numerous companies, and new private commercial efforts are now being mounted to visit asterois and the Moon for mining purposes.

Before those great riches are harvested , however, a lot of preparatory work will have to be done. First, Eros will have to be studied for one or two years after the SC is placed in a stable orbit. The mass, density and composition will be studied and determined. We only have an idea what the surface of an asteroid would be like: silicate dirt, mixed with nickel-iron granules and volatiles, or pure metal and pure powder?

Landing on an asteroid and then later launching materials from an asteroid will be much different from landing or launching on the Earth or on the Moon. In addition, landing on an asteroid could serve as a training flight for future planetary missions.

The low gravity on the surface of an asteroid is good because it will take very little energy to remove the ore but it will also provide new challenges. Staying attached to and moving skilfully around on an asteroid might be done using harpoons, a net, or even magnets. A spinning asteroid would pose problems for placing the solar-powered processing equipment so that it faces the Sun for a long time. For example, how do you stop the rotation of a large rock with a mass of millions of tons?

The actual mining operation on an asteroid, however, would be much simpler than mining on Earth, or the Moon. For example, we do not need heavy mining and transport machinery and we don’t need complex processing equipment as we would on the Moon in order to get valuable materials.

What methods should be used for mining on an asteroid? Would strip mining work? Or tunnelling into the asteroid? One novel ideas is to place a canopy around a strip mining area to collect the ore which would be propelled by a “dust kicker’ into the canopy and taking advantage of the “centrifugal” force produced by the rotation of the asteroid. Another suggestion is to drill into carbonaceous asterois (25% of near-Earth asterois are probably dormant comets), much the way we do on Earth when we drill for oil and natural gas.

Processing

Asteroid material is expected to be exceptionally high quality and not needing much processing. Basic ore processing will yield for material and volatiles, usually stored as ices because of the extremely cold temperatures in the shade, as well as selected minerals such as glasses and ceramics. At the input chute, the ore will be ground up and sieved into different sizes as a first step in the basic ore processing system. Simple mechanical grinders, using a rocky jaw arrangement for course crushing and a series of rollers for fine crushing , will be arranged in a slowly rotating housing to provide centrifugal force-induced movement of the material. In addition, vibrating screens will be used to sift the grains for directing them to the properly sized grinders. Strong magnetic fields, generated by electromagnets that draw the electric current to produce these fields from solar powered batteries, will be used to separate the nickel-iron metal granules from the silicate grains. An alternate method might be to place the material into magnetic drums, so that the silicates and the weakly magnetic materila would deflect from the drum and the magnetic granules and pebbles would be attracted to the drum until the “scrape-off’ point is reached. Repeated cycling then, using a variable magnetic field,will yield high purity nickel and iron which then can be put into large bags.

After the mechanical-magnetic grinding, a so-called “ impact grinder”, or “centrifugal grinder” could be used. A very rapidly spinning wheel pushes the material along its spokes and hurls it against an impact block. Any silicate impurities that are still attached to the free metal are then shattered off. Drum speeds can be used that produce a centrifugal acceleration that will flatten the metal granules by impact. The small silicate particles can then be sieved out.

The non-magnetic material can then be placed into a solar oven where the volatiles are “cooked out ” Since we are in near zero gravity (but not zero!) and windless space, very large oven mirrors can be constructed from aluminum foil. The gas stream can then be piped to tanks that are located in the cold shadow. They can be placed in series, so that the furthest one is the coldest, thus achieving differential condensation.

To store the volatiles, thin and relatively lightweight spherical tanks could be sent and used for storing. To send materials back to Earth, tanks could be manufactured from the iron nickel metals mined on the asteroid, after the metals have been melted down in a solar oven.

Unused material can be cast into bricks (again, using a solar oven) and then used to shield the habitat from solar radiation. Of course, the waste material could all be bagged and then ejected from the asteroid. Remember, even on a large asteroid like Eros, the escape velocity is only about 10 m/s.

Finally, after “consuming” the asteroid, the equipment could be moved to the next asteroid to begin a new mining engagement. Rocket fuel for the delivery trip back to Earth could be produced by separating oxygen and hydrogen, mostly using the water available as ice. It may be possible to have hydrogen chemically bond with carbon to produce methane and use this a fuel.

The equipment should be sent to an asteroid in advance of the asteroid mining crew, on a slower and more fuel-efficient trajectory. Once in place and all its vital systems functioning, the crew will be sent. Their first task, of course, will be to set up a radiation-protected environment, followed by a secure commuting system that protects the space miner from “escaping” the asteroid by just making a too large a jump by mistake.

1. Travelling, then living and working on an asteroid, and finally going back to Earth will take a very long time. How long?

2. Gravity on any point on the surface of a spherical asteroid would be easy to determine. But what would the gravity on the surface of an asteroid that is shaped like a peanut? Discuss.

Problems:

2. Would it be a practical proposition to try to stop a small asteroid rotating? To answer that question, we must first answer the following: How much energy would it take to stop a small asteroid from spinning? To calculate this energy, consider a cylindrical asteroid, a 1000 m long with a diameter of 300 m and a density of 3 g/ cm³. The cylinder is rotating along its axis once every 10 hours. The energy required to rotate a cylinder is given by the rotational analogue to (linear) kinetic energy,

where I is the moment of inertia (the analogue to mass) and  (radians per second) the rotational speed. The moment of inertia of a cylinder is ½ M r², where M is the mass (kg) of the cylinder and r the radius. To get a sense of how much energy this is, convert into TNT equivalent. Comment.

After the mining has been accomplished, several very large bags of goods mined on Eros will be well packaged and fitted with retroactive rockets. Next, we must escape the gravity of Eros and then slowly pull away from the large rock, while staying roughly in the same orbit.

The most direct reasoning would go like this. We could approach the Earth at a place where the orbital distance separation is the smallest. It so happens that the orbital velocities here are almost equal, in other words, the relative velocity is the lowest. Referring to your sketch of the orbit of Eros and the Earth, you can easily find where the closest approach can occurs. When you have found this, you should determine the approximate separation between the payload and the Earth. This distance will be at least .2 AU, which is about 3x10¹¹ m. To get a better sense of this distance, we can express it in terms of the distance to the Moon, 3.86x10⁸ m, or about 780 times that distance. The payload must be redirected, using retroactive rockets, to fall into the Earth’s “sphere of influence”. This still a long distance from where we start our transition orbit. It will take a long tome for the payload to “fall into the Earth’s sphere of influence”. A quick calculation of the energy required to redirect the object to go from Eros’ orbit to the Earth’s orbit ,when the Earth and Eros orbits are about .2 AU apart, is equivalent to retroactive burning to produce a Delta -V of 17 km/s ! To accomplish a Delta-Vee of that magnitude would require too much energy.

You will remember that the first orbit of the NEAR spacecraft crossed the orbit of Eros. After a little reflection, you will realize that at this crossing we could redirect the payload (and our spacecraft) and insert it into the first orbit of the NEAR spacecraft. Using the vis-viva equation you can quickly find that, here, the relative velocity between the two orbits is only about 1 km/s. An orbit insertion, going from the orbit of Eros to that of the first orbit of the NEAR spacecraft.

This manouvre will put the payload and the our own SC , into an orbit that connects with tthat of the Earth at perigee. As before , when we investigated the flyby of the NEAR spacecraft, we can decide how to capture the payload as well as our own returning SC containing the “asteroid miners”.

Capturing the payload, using an Earth flyby

As before, when we discussed the NEAR spacecraft flyby, the returning SC and the payload come closer to the Earth, and enter the Earth’s “sphere of influence” at a distance of about 2.6x10⁸ m. As we have seen before, this is very close to the Earth, only about 2/3 the distance to the Moon.

From here on, the force acting on the payload and the SC will vary from 8.3 x10^-3 N / kg, in 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.

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.

Fig 34 ? 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 above: