Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.3 4.3bsd-beta 6/6/85; site ucbvax.BERKELEY.EDU Path: utzoo!watmath!clyde!burl!ulysses!ucbvax!space From: KFL@MC.LCS.MIT.EDU ("Keith F. Lynch") Newsgroups: net.space Subject: Re: Ulysses probe Message-ID: <[MC.LCS.MIT.EDU].837830.860304.KFL> Date: Tue, 4-Mar-86 01:06:20 EST Article-I.D.: <[MC.LCS.MIT.EDU].837830.860304.KFL> Posted: Tue Mar 4 01:06:20 1986 Date-Received: Wed, 5-Mar-86 04:39:55 EST Sender: daemon@ucbvax.BERKELEY.EDU Organization: The ARPA Internet Lines: 137 From: desj@brahms.berkeley.edu (David desJardins) The course desired is really only 1% or so out of the ecliptic; most of the velocity is obviously toward the Sun. Perhaps we are using words differently. I was refering to the angle between the plane of the ecliptic and the plane of Ulysses' orbit. If Ulysses is to pass over the Sun's north or south pole, this angle must be 90 degrees. But any course should be achievable by approaching Jupiter in the right orientation (i.e. slightly above or below and to the left or right), including 90% if this is what you want (except that turns near 180 degrees would require orbits intersecting the planet). I used to think so too, until I sat down and tried to analyze the situation. What does 26 degrees come from? Ok, let me explain my derivation. All units are in the Meters-Kilogram-Seconds system of units. All velocities are relative to the Sun except when stated otherwise. The symbols used are as follows. They represent the average values of these quantities if the quantities vary. RE = Radius of the Earth's orbit = 1.50E+11 meters RJ = Radius of Jupiter's orbit = 7.78E+11 meters VE = Velocity of Earth = 2.98E+4 meters/second VJ = Velocity of Jupiter = 1.31E+4 meters/second AE = Acceleration of Earth = 5.93E-3 meters/second/second AJ = Acceleration of Jupiter = 2.21E-4 meters/second/second VP = Perihelion velocity of Ulysses = 3.86E+4 meters/second VA = Aphelion velocity of Ulysses = 7.45E+3 meters/second Note that RJ/RE = (VJ/VE)**2 = (AE/AJ)**2 = 5.20 To get VP and VA, I assumed that Ulysses will follow an elliptical path whose perihelion and starting point is on the orbit of Earth, and whose aphelion is on the orbit of Jupiter. This is the least energy method of getting to Jupiter. And it explains why Ulysses can't be launched until June 1987 if it can't be launched in May 1986, i.e. when it reaches aphelion at Jupiter's orbit, Jupiter has to be there to meet it. So I used equations I derived (derivation on request) for the perihelion and aphelion velocity of an object in an elliptical orbit. Where the perihelion is RE and the aphelion is RJ, the perihelion velocity VP equals SQRT(2*AE*RE*RJ/(RE+RJ)) and the aphelion velocity VA equals SQRT(2*AJ*RJ*RE/(RJ+RE)). So Ulysses will approach Jupiter with a velocity of 7,450 meters per second. But note that Jupiter is moving in the same direction at a velocity of 13,100 meters per second. So Jupiter will actually overtake Ulysses. The two will come together at a relative velocity of VJ-VA or 5,650 meters per second. Of course as Ulysses approaches Jupiter it will move faster. It will swing by Jupiter in a hyperbola, and will then slow down relative to Jupiter and will leave the vicinity of Jupiter at 5,650 meters per second, the same speed as it arrived. Note, however that this is 5,650 meters per second relative to Jupiter. Not relative to the Sun. If Ulysses made a 180 degree turn around Jupiter, it would then be going at a speed of VJ+VA or 18,750 meters per second. The solar escape velocity in the vicinity of Jupiter is SQRT(2)*VJ or 18,500 meters per second. As such, Ulysses would have sufficient speed to escape from the solar system (just barely). But that will only happen if Ulysses makes a turn that is close to 180 degrees. Since the idea is to give it velocity in an out-of- ecliptic direction, not enough velocity component will be left in the forward direction for it to escape from the solar system. As such, whatever trajectory it gets into will have to be a closed ellipse about the sun. The aphelion of that ellipse will be in Jupiter's orbit. It will keep returning to that position in space. Fortunately, Jupiter will (I think) not be there, so its orbit will not be interfered with. The best way to see what velocities Ulysses can leave Jupiter at is to construct a vector diagram. From the origin (representing zero speed relative to the Sun) construct a horizontal line segment of length 13,100. The end of that line represents the velocity of Jupiter. Construct a sphere of radius 5,650 centered on the endpoint of the line. The point where the sphere intersects the line segment represents the velocity with which Ulysses approached Jupiter. The surface of the sphere represents the velocities with which it can leave Jupiter. The circle where the sphere intersects the horizontal plane represents the velocities available if it is to remain within the ecliptic plane. Solar escape velocity is represented by the exterior of a sphere of radius 18,500 centered on the origin. Only a small part of the original sphere is outside this sphere, and no part of that small part extends far from the horizontal (ecliptic) plane. What is wanted is a velocity represented by a point directly above or below the origin. Only a velocity like that will bring Ulysses over a solar pole. No such point exists on our sphere, however. The angle between the plane of the ecliptic and the plane of Ulysses orbit it equal to the angle between a line passing through the point on the sphere which represents the velocity of Ulysses leaving Jupiter and the horizontal plane. Imagine yourself sitting at the origin studying the sphere. It should be clear that the point representing the greatest such angle is on the top (or bottom) edge of the sphere, as visible from where you sit. Note that the topmost (or bottommost) point on the sphere has a lesser elevation as seen from your vantage point. So contruct a line tangent to the sphere at that point. The line segment between the origin and the point on the sphere will have length SQRT(2*VJ*VA-VA**2) or 11,800. Thus that is the aphelion velocity of Ulysses' new orbit. The angle to the ecliptic is ARCSIN((VJ-VA)/VJ) or 25.5 degrees. That will NOT bring Ulysses over either pole of the Sun. The only explanation I can think of is that fuel is expended during the close pass to Jupiter. This would have the effect of increasing the velocity with which Ulysses leaves Jupiter, i.e. increasing the radius of the sphere in the vector diagram. Since a point directly over (or under) the origin is needed, the radius must be greater than 13,100. Thus a delta-vee (change of velocity) of 7,450 meters per second is necessary. If the fuel is burned at or near the closest point to Jupiter, much less delta-vee is needed, due to the fact that you are gaining energy by expending the fuel in a deep gravity well. I haven't worked out just how much you would gain, but it would be considerable, and the smaller delta-vee needed is certainly within the realm of today's technology. After all, the Galileo probe has to do the same thing, in order to stay in orbit around Jupiter rather than flying right back out of the Jupiter system. The time it will take for Ulysses to rach Jupiter is Pi*(RE+RJ)**2/(4*RE*SQRT(AE*RJ)) or 6.64E+7 seconds or 2 years and 1 month. If Galileo and Ulysses are launched in June of 1987, they should arrive at Jupiter in July of 1989. This quite close to the time when Voyager will start closely approaching Neputne. I hope there are enough people to control all three probes at once! These equations are all of my own derivation (available at request) and there is a small chance they may not be correct. The numbers certainly suffer from roundoff errors, and only the first two digits should be trusted. Earth's and Jupiter's orbits are not quite circular, so these numbers might be off by as much as ten percent or so for the actual time of the Ulysses launch and approach to Jupiter. Can someone tell me how I can get technical information on these probes from JPL or NASA or wherever? All I have been able to get is very nontechnical publications. ...Keith