- Asteroids Terra For Saturn Mac Os X
- Asteroids Terra For Saturn Mac Os 10
- Asteroids Terra For Saturn Mac Os 11
- Asteroids Terra For Saturn Mac Os 8
Terraforming the planet Mars by means of asteroids
by Martin Brodeck (to be finished by February 6th 2000)
We would like to show you a description here but the site won’t allow us. In Mac OS X you can just double-click the zipped file. If you have an older operating system, you may need to use a third-party program to open zipped files. AsteroidData6000.csv.zip This file should work well on most computers and spreadsheet programs.
Index
Introduction
Terraforming using asteroids
What are asteroids?
What do asteroids consist of?
Where are the asteroids and how many are there?
Different approaches
Direct Impact
Propulsion System
Nuclear Explosives
The asteroid belt and the terrafactory
The ice asteroid in 'Red Mars'
Conclusion
Introduction
At the moment of the collision between the asteroid and the surface of Mars the heat will melt the ice carried with the asteroid. Water flows through the dry sand, gases can freely expand and the ground warms up for first time after an undisturbed period of thousands or rather millions of years. Such a scenario sounds wonderful, dramatic and perfect in the eye of any futurologist. But the realization of such a plan is not likely to take place in the next five hundred years. The problems and mistakes that can ruin such a mission are great, even greater than the chance of success. The maneuver of moving a structure with a diameter of several kilometers has never been tried before. Different technologies and systems have to be synchronized and if one of them fails, the whole project breaks down. Here are the steps that have to be completed to achieve a successful mission: 1. Find a good asteroid which is usable for terraforming 2. Find different propulsion systems useable on the asteroid 3. Find a path for the asteroid, using other planets as a slingshot in the process 4. Initiate an alternate ballistic course of the asteroid 5. Pick the perfect impact velocity: if it is too fast there will be considerable collateral damage, if it is too slow the effect will be insufficient 6. Bring the asteroid close to the perfect speed 7. Eventually break the asteroid using atomic bombs to prevent a complete surface destruction After these points are accomplished another great danger occurs: What if all the calculations are right by the book, but the reality looks completely different: The impact might ruin Mars's atmosphere forever ejecting part of it into space and leaving a toxic gas-layer above the planet's surface. Unknown gases might have been hidden inside the asteroid that are set free for the first time at the moment of impact. But the idea still remains fascinating. If things work out according to the plan, mankind could save thousands of years shaping the planet by just doing all the required changes in a few seconds.
Terraforming using asteroids
Asteroids Terra For Saturn Mac Os X
Before understanding the concept of terraforming using asteroids we have to understand what asteroids are, what they do, what they consist of and what kind of different asteroids are 'out there'.
What are asteroids?
Asteroids are large chunks of material floating in space at different velocities, until they hit another object. This other object could be another asteroid, a planet or anything floating in space. In the worst case the floating object would be a spacecraft. Bodies the size of several millimeters can shatter the hull of the spacecraft and endanger the life of the astronauts. That is the reason why even small loose particles like screws threaten space missions.
Usually asteroids are a danger to any living beings on a planet because of their very strong impact. An asteroid the size of a football-field and 50 meters in height (100x60x50) would have the energy of 39000 Terra Joule, which equals about 14 million tons of coal.
This can be calculated by means of the equation:
W = ½ * m * v2
W = Energy
m = mass
v = velocity
Taking the volume of the body and multiplying it by the density can determine the mass. The normal density of an asteroid (rock) is about 2600 kg/m3. That means that a cube of 1 meter weights about 2600 kg. If a body has the volume of 300000 m3 the mass would be 780000000 kg or 780000 tons.
m = V * prock = 300000 m3 * 2600 kg/m3 = 780000000 kg = 780000 t
V = volume
prock = density
Another required factor is the velocity of the flying object which ranges from 300 meters per second to more than 30000 meters per second. The average asteroid flying in the asteroid belt has an impact velocity of 10 km/s or 10000 meters per second. If we insert those figures in the equation we get the energy of the impact:
W = ½ * m * v2 = ½ * 780000000 kg * (10000 m/s)2
W = 3.9 * 1016 J = 39000TJ
V = volume
prock = density
J = Joule
TJ = Terra Joule = 1012 Joule
If we want to get the amount of coal needed to produce this energy we have to know how much energy one ton of coal contains: one ton of coal generates 0.79 kWh . One Joule equals 2.778 * 10-7 kWh, thus 39000 TJ are 1.08 * 1010 kWh. A power station like the one in Weisweiler in Northrhine Westfalia would burn 13,714,177 tons of coal to generate the amount of energy set free by the impact of such a rather small asteroid.
1000 kg coal <=> 790 kWh
1266 kg coal <=> 1000 kWh
1 J = 2.778 * 10-7 kWh
39000 TJ = 1.08342 * 1010 kWh
1.08 * 1010 kWh <=> 1.371 * 1010 kg coal = 13714177 t coal
This computation shows the amounts of energy set free by asteroids at the moment of impact. This energy is important for the terraforming process, because part of the energy will heat up the atmosphere, which is a desirable side effect.
Asteroids Terra For Saturn Mac Os 10
What do asteroids consist of?
There are three categories asteroids can be placed in. The first ones are made of stony meteorites known as carbonaceous chondrites. These are considered the oldest material in the solar system. Being dark in color, they have never melted or cooled down since their creation. The asteroids of this material are called asteroids of the C-Type. Asteroids of the S-Type are made of iron and are related to the M-Type, which is an iron-nickel mix. Most scientists are not certain as to whether asteroids are parts of former planets. Most believe that there were some primeval asteroids in the solar system that collided with each other and formed smaller ones. The impact on the surface will heat up the atmosphere, but it will not procreate the essential gases so badly needed for a living environment. The most expedient asteroids are the ones containing a lot of water (ice), some CH4, CO2 and NH3. The CO2 is needed for the greenhouse effect, which heats up the atmosphere. Contrary to the earth, Mars has not enough CO2, so the rays of the sun just bounce off the surface and leave the planet, whereas CO2 would keep them within the atmosphere.
Where are the asteroids and how many are there?
Most asteroids are to be found in asteroid clouds. An example of those clouds are the Trojan asteroids. They follow the path of Jupiter, the first part of the cloud following Jupiter's path at 60°, the other one being 60° ahead. This arrangement has been originated by Jupiter's great mass. Any object coming close to this giant planet would be drawn into Jupiter's gravitational field and forced to hit the surface. Without Jupiter life on earth probably couldn't exist because many asteroids would have hit the earth if Jupiter's gravitational field had not pulled away them from their life-threatening course towards Earth. There are not that many asteroids in our solar system. If we added up the mass of all of them we would get much less than the moon's mass, so there is not that much of a choice when picking an asteroid for terraforming.
Different approaches
There are several possibilities of terraforming Mars with extra-planetary means. Asteroids aren't the only possibility, even though they are the most promising ones. Comets could be used and even small ice moons (like Hyperion, being a satellite of Saturn) are a good source of water and nitrogen. Comets and asteroids differ in density. An average comet has a density of 1000 kg per m3, whereas asteroids usually have 2600 kg per m3. This makes a very big difference at the moment of impact: Asteroids release more energy and vaporize the ground more than comets. Asteroids and comets also vary in their amount of volatile. Volatile is the amount of vaporized material clouding the object. Comets consist of 50% volatiles, asteroids of only 5% (comets mostly have a tail of these volatiles following them).
Left image: The asteroid 243 Ida in August 1993. Source : Microsoft Encarta 1997 Right image: The structure of a comet (no photography) showing the tail, the coma and the nucleus. Source: Microsoft Encarta 1997
Usually comets look much bigger than they are. Halley's Comet for example exceeds the size of Jupiter (including the hazy coma), but the solid part, the nucleus, is only 15 by 4 km in size. Usually comets contain lots of ice, so they are ideal for terraforming purposes. The problem lies in their distance from the target planet: Comets can most often be found in the outer solar system and are therefore much harder to move to Mars than most asteroids. Because of their long dust-tails comets are harder to approach and manoeuvre than asteroids.
When using an asteroid the problem of moving it to the target planet (in this case Mars) requires a lot of energy. To reduce the necessary energy one could divide the object into smaller parts and start moving these. This would not only reduce the amount of energy but also reduce the risk of a catastrophic impact because of the big size. Asteroids with a diameter of several kilometers would prove manageable when 'cut' into smaller pieces.
When elaborating the plan dividing asteroids into smaller pieces Freeman Dyson brought up the idea of a self-reproducing robotic factory. The material required for the follow-up factory would be taken from the asteroid itself. Only one of those factories would have to be built, which would then multiply, a process that might take about a month. After a year there would be a total of 212 machines, i.e. 4096. At the end of this process the machines start taking the moon apart. This would not take too long, considering a work force of 4096 machines. Just adding another month would have doubled the number. Freeman Dyson suggested taking one of Saturn's satellites, Enceladus, being 500 km in diameter. Equipped with a solar sail each piece would then change direction towards Mars.
The greatest problem is to move the objects. In the above scenario, this problem has been solved by reducing the size of the objects. If such a machine as described above should ever be built, its construction will be a feat of a more distanced future. The only way to move a body from its orbit is to change its velocity. This change of velocity is called ?v, where ? means change. This ?v would be made perpendicular to the movement of the object, so the path would be changed. Three methods have been found to provide this deflecting impulse:
1. Direct impact
For smaller objects (with a radius of less than 100 m) a simple collision with an object, for example a rocket, would change the path radically. A rocket weighing 2000 tons, traveling at 500 m/s and hitting the object laterally would change the asteroid's velocity by 0.163 m/s.
Mrocket = 2000 t = 2000000 kg = 2 * 106 kg assuming a spherical object: V = ¾ * PI * (100m)3 = 2356194m3 m = V * 2600 kg / m3 = 6.126 * 109 kg ½ * masteroid * (vasteroid)2 = ½ * mrocket * (vrocket)2 (vasteroid)2 = mrocket * (vrocket)2 / masteroid = 2 * 106 kg * (500 m/s)2 / 6.126 * 109kg vasteroid = 9.03 m/s
Asteroids Terra For Saturn Mac Os 11
This is not very much considering the length of time required reach Mars (one year of flying time would change the course by 4183 km.
1 year = (60 * 60 * 24 * 356) s = 30758400 ( seconds per year ) 9.03 m/s * 30758400 s = 277748 km
This may be enough to change the planet's course, but will not be sufficient for the discussed purposes. The method seems to be feasible for smaller objects but it would require thousands of them to achieve a palpable change on Mars. When greater bodies have to be moved we need better and more efficient systems that can move greater objects.
2. Propulsion systems
Propulsion systems are used to move larger bodies (that exceed a radius of 100 m) around the solar system. One method is the attachment of large engines to one side of the asteroid, which will mine material from it and eject it into space. This matter will accelerate the asteroid in the opposite direction (the way a rocket works, just slightly larger). The higher the velocity the more thrust will be applied to the asteroid. Assuming that one of those machines could accelerate the mined material to the speed of vpush, and mining n kg of material per second we can calculate the time needed to achieve a ?v of vasteroid. t shall be the time the engines will run.
½ * masteroid * (?vasteroid)2 = ½ * mpush * (vpush)2 [n] = kg/s Total material minde: mpush = n * t ½ * masteroid * (?vasteroid)2 = ½ * n * t * (vpush)2 V = ¾ * PI * (rasteroid)3 masteroid = V * 2600 kg / m3
t = masteroid * (?vasteroid)2 / (n * (vpush)2 )
rasteroid is the radius of the asteroid (spherical) (e.g. 1000m) ?vasteroid is the change of the asteroids velocity (e.g. 5 m/s) n is the material mined per second (e.g. 100 kg/s) vpush is the speed of the accelerated and mined material (e.g. 500 m/s)
On the basis of those sample values the time needed to move the asteroid 5 m/s would be about 6000000 s or 70 days. The radius of this object is only 1000m, but the time needed cubes with the increase of the radius (e.g. if radius is 5 times as big the time will be eauivalent to 53 = 125). Taking a more realistic value for the radius (about 40 km) the time growths to 12746 years. This number shows how unrealistic the value for n is. Increasing the amount of material mined per second would be achieved by putting 100 of those machines on the asteroid. They would be able to mine 10000 kg of material per second. The time would decrease to 127 years, which seems to be more realistic. The question is how many of those machines are realistic. We have to bear in mind that each of them needs a great amount of energy to accelerate the material up to 500 m/s, so we can't just place as many engines on them as we want. The best and actually only method of winning the fuel would be by taking it from the asteroid itself. The material could also be ejected by using electrical energy. The mined material would then be electrically charged and accelerated by an electrical field. This method would not require any burning process but could operate on electrical power which could easily be generated by solar panels instead of extracting the power from the asteroid.
The basic problem of the propulsion system is the time. We cannot expect to use asteroids for the terraforming process within the next 100 years or so. But I think it is commonly known that we cannot change the face of a planet within 20 years.
3. Nuclear explosives
The last alternative and most practical one is the detonation of a nuclear missile. To use the largest amount of energy set free by the explosion the first possibility is the detonation right on the surface of the object. The explosion would then eject large amounts of matter away from the object, which gives it a thrust in the opposite direction. At the same time the pressure of the explosion pushes the asteroid to the opposite direction, creating an even greater ?v. The problem of this method is the risk of damaging the asteroid too heavily. If the ?v needed is too big, the explosives have to be more powerful, too. The impulse of the explosion could rip the small asteroid apart and the two pieces would leave the pre-calculated course. This would not be a disaster taking into account that there are other asteroids around, but the loss would cost dearly. Another way of using nuclear explosives is the stand-off detonation. This idea bases on the neutrons absorbed by the explosion. The crust of the asteroid will absorb the neutrons and therefore heat up. Because of the increase in heat the crust will than expand and in the end depart from the asteroid, which will create a ?v into the opposite direction. The thickness of the affected crust is only 20 cm, but the size of the area will be able to achieve a change. The problem is that once the radius of the asteroid expands the ?v is so small that thousands of nuclear explosions would be required to accomplish a change.
Nuclear explosives are not practical for moving asteroids as part of a terraforming scenario. The objects needed are so big that the impact of a nuclear explosion would be insufficient.
I personally think the best of these three methods is the second one. Right now it does not seem feasible to build thrusters on a relatively large body in space enabling it to move it around, but the criterion of feasibility applies to all of the above ideas. The second one though is the only one suited to move large bodies as the ones needed for terraforming.
The asteroid belt and the terrafactory
Another idea of terraforming Mars using asteroids was proposed by Lloyd E Vancil. He suggested building a terrafactory. This is basically a spaceship located at the inside of the outer asteroid belt which builds three types of different robots. The first one will seek out and search radioactive matter to be used as fuel. After finding such resources the robot would be positioned on the asteroid and start mining it. The second robot would search the asteroid belt for asteroids containing ice water, radioing their positions to the third kind of robots. The third robot, being a smart engine, would land on the asteroid and start burrowing into to the ice. The nuclear power plant on board the smart space-robot would then accelerate the ice creating a jet of steam. This would gradually accelerate the icetroid (i.e. an asteroid consisting of ice) propelling it towards its new home: Mars. The nuclear power plant would be fuelled by the radioactive material mined by the first type of robots. The terrafactory would make its way into the asteroid belt and start building the different kind of robots. Mars would be bombarded by thousands of small ice-asteroids all impacting the object and setting free large amounts of energy. The hope of Lloyd E Vancil is that the bombardment will also activate the volcanic activity of Mars thus increasing the temperature of the atmosphere. He thinks the first results could already been seen 100 years after the first impact on Mars.
The ice asteroid in 'Red Mars'
The novel 'Red Mars' by Kin Stanley Robinson is about the colonization of Mars, including terraforming. The main proponent of terraforming, Sax, has the idea of using an ice asteroid for the process of terraforming. Mars was already settled at this point of time, so the impact of the of an ice asteroid would have killed thousands of people and destroyed the surface of Mars. Anthropologist Ann is an important opponent of the terraforming process wanting to keep Mars as it is to find out if any life has ever existed on it. The impact of an asteroid would have destroyed important parts of the surface thus destroying any traces of life on Mars. More importantly, a direct impact is not feasible, because it could kill people and destroy human colonies on Mars. Therefore Sax decides to have an indirect impact. This means the asteroid enters the planet's atmosphere at a very low angle avoiding a direct hit and burning up in the atmosphere.
That leaves all the hydrogen, the oxygen and the water in the atmosphere, but nothing really hits the planet itself thus avoiding destruction of the surface and collateral damage.
Conclusion
Asteroids Terra For Saturn Mac Os 8
Terraforming Mars will definitely be a project for the future, not because we are not capable and do not have the technology of changing it right now, but because it isn't necessary right now. Terraforming will become a priority after the first Martian colonies have been built and the first humans start to live there over periods of more than 20 years. Why should we plan the process of terraforming in detail when we are not even able to land on Mars? Terraforming will involve close study of the planet, so the effects can be foreseen in detail and the process should not end in uncertainty. The basic ideas should be studied and examined, but facts and realistic results can only be made after close examinations of the planet. Therefore our first goal should be Mars itself. We have to focus on landing on the planet and building the first colonies. The advantage of using asteroids is the relatively low budget required for such a project. We can change everything with just one big crash. The whole environment can be changed to our advantage creating a green and blue planet full of life. The only problem is the enormous energy created by the impact of the asteroid. Just a small mistake could damage the planet forever, leaving a dead and destroyed place. If the angle of the incoming object is too steep the surface will be changed beyond recognition. In the process more matter could be ejected from the planet than it gained, which would result in a great loss rather than in a win. I am sure that once planet Mars is colonized terraforming will come, no matter what.
Martin Brodeck, February 6th, 2000
Used literature:
Terraforming by Martyn J. Fogg; 1995
Rheinbraun; Forschung und Entwicklung; 1994 Köln
“Asteroid,' Microsoft® Encarta® 97 Encyclopedia.
Chapter IV: Asteroids, Jupiter, Saturn,UranusA) Meteors and Asteroids
| 1. History 1.1 In 1766 Johann Titius noticed an interesting arithmeticregularity in the distances of the planets. This was presented inan astronomy book published in 1772 by Johann Bode, and has been calledBode'slaw ever since (although more careful citations now call it theTitius-Bodelaw). The law is presented in the table below:1.2 In fact the Titius-Bode law is really an exponential relation,and can be written as 0.3 x 2n + 0.4 n = -∞ 0, 1, 2, ... The question is whether this near exponential relation is real or coincidental. There are many cases of physical processes that give exponentials as aresult, and there have been attempts to see this relation as the resultof an actual physical process associated with the formation of the solarsystem. One example is the vortex theory of von Weizacker,another is the ring theory of Prentice. Similar relationshipscan be found for the orbits of the moons of the giant planets. |
| 2. The asteroid belt 2.1 There is a band at around 2.8 AU between Mars and Jupiterwhere asteroids are found. 2.2 They range in size from Ceres at 512 km to bodies much smaller. There are about 20 bodies larger than 125 km, some 200 larger than 50 km,and around 4000 larger than 5 km that have been discovered and named. The total mass of these asteroids is less than the mass of the Moon. 2.3 Measuring the size of an asteroid is tricky. Theyare too small for us to resolve a disk, so we must estimate theiralbedo. This is done by measuring the temperature in the infrared and assumingthat the albedo is independent of wavelength. In this case (and assumingan efficiency of 1) the temperature is related to the albedo via A measurement of the temperature will give the albedo, and this canthen be used to relate the observed brightness to the radius of the object. 2.7 S-type asteroids make up about 15% of the population, andappear to be related to stony-iron meteorites. 2.8 M-type asteroids may be related to nickel-iron meteorites,and may represent the core of a differentiated body. 2.9 Vesta (radius 277km) seems to show an igneous surface, and may have undergone melting inthe past. For such a small body to melt there must have been a ratherintense heat source available (26Al?). 2.10 Asteroids are no longer regarded as remains of a planetthat was destroyed, but rather as a planet that never got a chance to form. They collide with each other and most meteorites that hit the Earth arebelieved to originate in these collisions. |
| 3. Other asteroid populations 3.1Trojanasteroids orbit the Sun in Jupiter's orbit, 60° in front and 60° behind. These are the L4 and L5 Lagrangepoints. We can understand the nature of these points from thefollowing considerations. Think of a small body under the influenceof two much larger bodies in orbit around each other (i.e. asteroid, Jupiter,and Sun). The energy of the small body is simply the sum of its potentialenergy in the field of each larger body, plus its own kinetic energy. It turns out to be more convenient to do this computation in a system wherethe origin is fixed on the center of mass, and the x-axis connects thetwo massive bodies (i.e. the axes are rotating). The potential energyof the small body is then given by 3.2 The kinetic energy is mv2/2, but this is not all. We have to add the energy of the rotating system as well. If theframe is rotating with some mean angular velocity n, then the kinetic energyof an object at a distance R from the center will be mR2n2/2. Thus if a body is at rest in the inertial frame, it will appear to havethis energy in the rotating frame. We must therefore subtract itfrom the total. The total energy of the small body is then givenby This total energy has to be negative for a bound orbit. Definea positive constant C by C = -2E/m. The velocity is then given by v2 must be positive, but this will be true only if C is smallenough. For each value of C there are limits on r1, r2,and R where the small body is allowed to be. At the edges of theselimited areas, the velocity is zero. The five points of equilibriumare the five Lagrangian points. The other three are between the Sun andthe planet (L1), outside the planet on the planet-Sun line (L2), and outsidethe Sun on the same line (L3). Only L4 and L5 are stable and thisis where the Trojans are found. |
| 4. Meteorites 4.1 Most meteorites are believed to come from asteroids. 4.2 The most primitive meteorties arechondrites. These are stony meteorites containing millimeter-sized spheres called chondrules. These have a composition similar to that of the surrounding matrix, buthave undergone a brief intense heating event, followed by a rapid cooling. There is no good mechanism for producing the heating in a short time, ofthe right intensity, and affecting so much mass. Many not-so-goodmechanisms have been suggested. 4.3 Chondrites, because they contain many volatiles, are believedto represent the most primitive solar system material. The abundancesfound in these meteorites are used to help determine solar abundances. 4.4 There are several classes of chondrites (carbonaceous, ordinary,and enstatite). Carbonaceous chondrites are further divided intothree classes (CI, CM, and CV). The CI are the most primitive. Ordinary chondrites are classified by iron content (H, L, and LL). Enstatite chondrites are made almost entirely of enstatite. 4.5 Irons and stony-irons make up the remaining types. There are interesting variations in isotope ratios and minerology thatcontain clues to the conditions during their formation. 4.6 There are SNCmeteorites that are believed to come from Mars. Their compositiontells us something about conditions on the Martian surface. Thereare also meteorites that are believed to come from the moon. |
1. Data:
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| 2. How are these measured? a. Radius via parallax. b. Mass from moons. c. Rotation period from:
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| 3. What does the low density tell us about the composition? 3.1 The central pressure is greater than 10 Mbar (much greater)so that the effect of pressure will raise the density of any material considerably. You can show that there must be considerable amounts of hydrogen insidethe planet. 3.2 Spectroscopy does indeed show that there is a great dealof hydrogen in the atmosphere. 3.3 We also see some NH3 and some CH4as well as small amounts of many other compounds. 3.4 The beautiful colors we see on Jupiter are due to smallamounts of possibly organic material probably in aerosols. |
| 4. How do we get hydrogen into a planet? 4.1 One way is to gather up solids and have the hydrogen bein them like in H2O ice or NH3 or CH4 ice.The trouble is that bringing in so much heavy stuff with the hydrogen willraise the density above the observed value. 4.2 Another way is to attract gas gravitationally. Thismeans a fairly large gravitational field and it means that helium gas willbe attracted as well. This has two implications for Jupiter's structure. First there must be a core of heavy material. Second there shouldbe a solar ratio of helium to hydrogen. 4.3 Helium is hard to observe because it doesn't have any rotation- vibration bands, so its spectrum is mostly in the UV. |
| 5. Computing the internal structure: How can we tell if there is a heavy element core? We need to make theoretical models of the planet. This meanssolving the equations of planetary structure. 5.1 We have equations for the pressure and the mass, and wecan get an equation of state if we make an assumption about the composition. We still need to know something about the temperature distribution. 5.2 In principle, this can also be computed if we know the methodof heat transport in the interior.
From basic thermodynamics But for an adiabatic process, dS = 0, so dE = -PdV. If the heatcapacity of the material is C, then the internal energy can be relatedto the temperature by dE = CdT so The density is related to the volume and the mean molecular weight by V = μ/ρ For an ideal gas Substituting givesIntegrating givesWe can define a quantitywhich is related to the number of ways energy can be deposited in themolecule. Without taking quantum mechanical details into consideration,C = N0k/2 for each degree of freedom. For pure translationalmotion, there are 3 degrees of freedom, and a = 1.5, for rotation and vibrationthere are additional degrees of freedom and a can be larger. Complicatedmolecules can have a = 3.5. What we have done here is computed anadiabat through an ideal gas. In Jupiter the material is not ideal,so the computation is more complicated, but the idea is the same. Knowledge of the equation of state allows us to compute an adiabat forthe material, and then a variation of temperature with density. Wethus have four equations for the pressure, mass, density, and temperatureas a function of radius. With the appropriate boundary conditionswe can integrate these and get the internal structure. |
| 6. Computing the composition: The other problem is to decide on a composition. There are manycombinations of materials that will give the right total mass. Howcan we decide which composition is correct when our only measurements arefrom outside? 6.1 We can only measure integral properties of the density distribution. One such integral is the mass.6.2 Another possibility is the moment of inertia The only problem with this is that there is no good way to measure themoment of inertia. So it is not terribly useful. It turns outthat there is a set of quantities that are related to the moment of inertiathat can give more information: where Re is the equatorial radius of the planet, Pnis the n'th Legendre polynomial, q is the colatitude and Jnis the n'th gravitational moment. These moments are integrals ofthe density distribution and can be determined from measurements of theexternal gravitation field of the planet (using the orbits of its satellites,for example). In this way some additional limits can be put on thedensity distribution. In practice, with the Voyager flybys, we have goodmeasurements of J2 and J4, and some information onJ6. A mass sitting on the surface of the planet will notonly feel this gravitational potential, but will also feel a centrifugalforce. This can be written in terms of a potential as well and combinedwith the gravitational term to give Now the second Legendre polynomial is given by P2(cos θ)= (3 cos2 θ- 1)/2, so Furthermore, since the Jn's get smaller as n increases, let us limitourselves to the first approximation where n = 1. In this case thepotential becomes Now the planet will assume a shape such that the surface is an equipotential. In such a case, the shape of the planet can be approximated by where f is the flattening or oblateness, given by f = (Re- Rp)/Re. If we rewrite this in terms of Legendrepolynomials we get The potential on this surface is then given by substituting Rsfor r in the expression for the total potential. This gives a verymessy expression, but if we assume that any terms of the order of J2or f are small compared to 1, and terms of the order of J22 orf2 or J2f or higher are negligible, then we can geta much simpler (though approximate) expression which looks like But since Rs is an equipotential, it cannot depend on θso the second bracket must be exactly zero. This gives a relationbetween J2, f, and ω that must always be satisfied: Note that the second term is basically the ratio of the centrifugalto gravitational forces. So long as this term is small our approximationworks. In any case, if we can measure any two of these three quantities,we can compute the third. |
| 7. How do we determine the compositions of the envelope and core? 7.1 We have already seen that for high pressures, where thematerial behaves like a dense fluid, the density of a mixture is givenby Now, suppose I assume that the envelope is composed of a mixture ofhydrogen, helium, and water, and that the ratio of hydrogen to helium issolar. The solar ratio of hydrogen to helium is 2.7 by mass, so XH=2.7 XHe. If there is an additional mass fraction XH2Oof water, then 3.7XHe + XH2O = 1, and we get XH= 2.7 XHe = (1 - .73 XH2O). The density of the mixture then becomes Now I can compute models with different values of XH2O inorder to find the best fit to observations. This will give the watermass in Jupiter's envelope. |
| 8. Jupiter's magneticfield. 8.1 The transformation to atomic hydrogen occurs for the followingreason. The H2 molecule is held together by the electronsthat the two atoms share. When the density is high enough, an electroncan no longer tell which pair of atoms it is associated with, and movesbetween adjacent pairs. As a result the pairs are no longer bound,and the molecules break up. In addition, the electrons move morefreely and the material behaves like a metal. 8.2 This metallic hydrogen has important consequences for theplanet because it conducts electricity. This is one of the componentsof the dynamo needed to generate a magnetic field. Another componentis rotation, and Jupiter certainly rotates ... about once in 10 hours. The third component is convection in the conducting region. We havealready seen that Jupiter convects, so all the conditions are there forproducing a magnetic field. Such a field is, indeed, observed. |
1. Data:
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| 2. Internal structure: 2.1 In general terms the internal structure of Saturn is verysimilar to that of Jupiter. There is a dense central core of some10 Earth masses, surrounded by an envelope of hydrogen and helium withan additional 20 Earth masses or so of heavy material mixed in. 2.2 There are some differences as well. The H2/Heratio measured for Saturn is much higher than the solar value. Heis present as only 6% of the atmosphere, compared to 24% in Jupiter. This has been explained by the fact that at high pressures hydrogen andhelium are immiscible. It is expected that the heavier helium sankout, leaving an excess of hydrogen in the envelope. This argumentis strengthened by calculations of the cooling rate for the planet. It is observed to be emitting more heat than the models predict. The additional heat source provided by helium rainout would explain thedifference. There is some evidence that the H2/He ratioon Jupiter is somewhat higher than solar. Possibly rain-out has begunthere as well, but because of Saturn's lower temperatures it has progressedfurther there. 2.3 Saturn shows fewer features on its disk. This toois because of the lower temperatures. The temperature in the interioris high, and as you go up towards the surface, you hit regions where thetemperature is low enough so that clouds can form. Since the temperaturesare lower in Saturn, these points occur deeper in the planet, so the cloudsare harder to see. 2.4 In other respects the interior structure of Saturn is similarto that of Jupiter. The core is estimated to be about 20 Earth masses,and there are about 20 to 40 Earth masses of heavy material mixed intothe envelope. Since Saturn's envelope is only 1/3 the mass of Jupiter's,the percent of additional material is higher. |
| 3. The system of rings -themost impressive feature of Saturn. 3.1Galileo discoveredthe rings in 1610, but it was only in 1656 that ChristianHuygens realized that they were actually rings. In 1857, J.C. Maxwell showed that they could not be solid, since tidal effectswould cause them to break up. If they were stationary, they wouldbe unstable since a small perturbation would bring one side closer to theplanet and the additional pull would cause it to continue in that directionuntil it eventually crashed into the planet. Giving it enough spinto stay in orbit doesn't help, since different parts of the ring have tomove at different speeds to stay in orbit. This is not possible fora solid ring. In fact they must be made up of many smaller bodies. 3.2 As a body approaches a planet, the part nearer to the planetfeels a slightly stronger gravitational pull than the further side. If the body is small, the difference is not large, and the material strengthof the body is enough to keep it together. For larger bodies, thedifference in force is larger, but the body's own gravity helps to holdit together. When a moon-sized body gets too close to the parent,the tidal force can overcome the moon's gravity, and rip the moon apart. This critical distance is called theRochelimit, and depends on the sizes of the interacting bodies, buta good rough guess is about 2.5 times the radius of the planet. 3.3 The rings are very thin: Only about 2 km thick witha width of some 20,000 km. They are optically thin, and stars canbe seen through them. Spectra show that they consist mostly of waterice particles with sizes ranging from centimeters to meters. 3.4 Originally the rings were labeled A, B, C, and D, with theA ring being furthest out. Voyagerdiscovered an additional E ring, beyond the A ring, and then an F and Grings between the A and the E. Theorder is thus (going outward from the planet) D, C, B, A, F, G, E. 3.5 It was originally thought that mutual collisions betweenrings particles would keep them spread out, but in 1675 GiovanniCassini discovered an empty region between the B and the A rings, nowcalled the Cassini division. This empty region is at a distance of about 120,000 km (about 2 Saturnradii) from the center of the planet. FromKepler'slaws, the period of an orbit varies like the distance to the 3/2 power,so the ratio of the periods of two bodies at distances R1 andR2, should vary like (R1/R2)3/2. Saturn's moon Mimas sits at a distanceof 186,000 km from the planet, so it circles Saturn in almost exactly twicethe time a Cassini division particle would complete an orbit. Sincesuch a particle would see Mimas in the same place every second orbit, itwould eventually get pulled out of the region by this 2:1 resonance. 3.6 A similar empty region, the Enckegap, was discovered in the A ring itself by Johann Encke in the19th century. This would put it into nearly a 5:3resonancewith Mimas. Until the Voyager flyby of Saturn, this was thought tobe the basic structure of the rings: a thin disk with two gaps caused byresonances with Mimas. Voyager showed that the rings had much morestructure, however, consisting of a very large number of fine ringletswith gaps between them. It was soon realized that one needed a muchmore sophisticated approach to understand their structure. |
| 4. Some ring dynamics: 4.1 Ring particles collide with each other, not just in actualphysical collisions, but also through longer range gravitational interactions.Such collisions redistribute energy and momentum between the particles,and tend to bring all the particles in any particular region closer tosome average value of energy. This tends to circularize the orbits. 4.2 If an orbit is inclined to the equator, this averaging tendsto reduce the inclination to zero. Even those particles that arein orbits that avoid collisions don't do so for long. The oblatenessof Saturn (i.e. J2) causes the orbits to precess, so that theyeventually encounter other particles and lose their momentum perpendicularto the equatorial plane. That is why the rings are flat. 4.3 The presence of small moonlets (called shepherdsatellites) inside the rings also affects the motion ofthe ring particles, and causes them to remain in tightly restricted areas. This is a part of the explanation of the many ringlets that are seen. The rest of the explanation lies in the behavior of groups of particleswhose mutual interactions influence the motion of the group. Thisspiralwave theory, originally developed to explain the structure of spiralgalaxies, has been successfully applied to explain much of the structureof Saturn's ring system. 4.4 There are still some things not completely understood. Thebraided structure of the F-ring is one example. Electric chargingeffects on the dust, due to UV radiation or interactions with the magnetosphereis another example. |
1. Statistics
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| 2. Atmosphere 2.1 Uranus atmospheredisplays very few features, and it has an essentially featureless disk. As a result it is very difficult to measure a rotation rate. We havealready seen three methods for measuring the rotation rate. Timingfeatures as they cross the disk is not practical for Uranus, since we cannotresolve any features from the ground. Instead one can use photometryto measure the overall brightness of the disk. If there are any features,even though we cannot resolve them, they will affect the total brightness,and by monitoring this over time we can hope to measure a periodic changefrom which the rotation period can be derived. Early attempts atsuch measurements around 60 years ago yielded a period of about 11h. Spectroscopic observations by Moore and Menzel in 1929 also gave a periodof 10.8h, and this was the number quoted in the literature for many years. 2.2 Above, we see still another method for measuring the rotationperiod based on measuring the flattening and J2. The difficultyis that the flattening is also difficult to measure, since it involvesthe difference between two numbers that are quite close to each other. Observations of the oblateness by stratoscope balloon-bearing telescopesindicated a period of about 18h. A number of attempts were made tomeasure the period after 1974, but the results varied widely (12h to 24h). The current value comes from the magnetospheric measurements of Voyager. 2.3 Spectroscopy has revealed that the atmosphere consists ofhydrogen and helium in solar proportions, with a rich admixture of methaneand ammonia. |
| 3. Interior 3.1 There are no completely satisfactory models of Uranus' interior,but most models agree that there is a deep atmosphere overlying a planetmade up mostly of a mixture of compounds of C, N, O, Si, Fe, and otherelements common in the solar system. These may be roughly classifiedas ice and rock. 3.2 Like Jupiter and Saturn, we would expect Uranus and Neptuneto have internal heat sources. Neptune does, but Uranus doesn't. The reason is not completely clear, but it may be due to the fact thatUranus has cooled enough so that it is now in equilibrium with the Sun. 3.3 Uranus has a magnetic field that is tilted from the rotationaxis by 60°. The reason for this is not clear either, but may beconnected with the fact that the field-forming region is not at the centerof the planet, but in a conducting shell. The conducting materialmight be something like H3O+, NH4+,and similar ions. 3.4 Uranus has its own systemof rings (as does Jupiter). They too are narrow and mediatedby shepherd satellites. They are also of extremely low reflectivity;some tens of centimeters in radius and on average a few meters apart fromeach other. |