Stellar Oscillations across Spiral Arms

Jim, it's the total energy, we're concerned with. As a comet approaches the Sun, its potential energy is converted to kinetic energy; the opposite happens on the way out. If the orbit is unchanged, the total energy remains constant. For a comet to be captured, it must lose a big share of its total energy. The energy lost may be imparted to a third body, such as a planet. If there were only two bodies in the system, energy transfers would be limited to tidal friction and solar wind, neither of which is significant---unless tidal forces tear the comet apart, as they did with Schumacher-Levy 9.

PhilJ, Rather than getting bogged down in the physics of it all can we say the comet has to slow down relative to the excape velocity. By looking at the problem relative to the excape velocity details are focused better. The example being kicked around in this thread is a comet arriving from outside the gravity field of the sun at any speed relative to the sun and zero angular momentum. When the comet enters the solar gravity field it speeds up or slows down relative to the excape velocity. The reason this happens is because a slow moving comet spends more time being accelerated than a faster comet. SL9 is a good example of a capture event. SL9 was in orbit around the sun before Jupiter captured it. The capture required slowing its speed relative to Jupiter-right? But, there is another way capture can happen which is by a direct hit and that can be at any speed.

Jim, Why talk about complex interrelated variables when you could be talking about constants?

As a comet nears the Sun, it's present velocity and kinetic energy increase; so do the velocity and kinetic energy required to escape the Sun's gravity at the comet's present distance from the sun. The difference between present kinetic energy and escape kinetic energy remains constant---so long as no 3rd body interferes. Why talk about the difference between present kinetic energy and escape kinetic energy---both of which are constantly changing---when we could be talking about things that remain constant?

A comet's total energy relative to the Sun (kinetic + potential) is constant, as long as no 3rd body interferes. If we talk about a comet's total energy, there is ONE constant, Ex, for all comets under the Sun's influence. If a comet's total energy is less than Ex, it is an Oort Cloud comet, with an ellyptical orbit. If its total energy is exactly equal to Ex, it is an interstell comet with a parabolic trajectory. If it's total energy is greater than Ex, it is an interstellar comet with a hyperbolic trajectory.

<b>{EDIT, added 2005/07/16, 05:00 UT: </b>
Oops! I meant to say that that <b>Ex/M</b> is the same for all comets. Obviously, a more massive comet will need more energy to escape the Sun's gravity.

Also, I should note that potential energy is calculated relative to some arbitrary distance from the Sun; it could be relative to the Sun's center, the Sun's surface, the comet's perihelion or aphelion, or it could be relative to infinite distance from the Sun. We must choose a reference distance and stick with it.

Let's choose infinite distance. Then, a comet at the farthest extent of the Oort cloud has near zero potential energy. If it is an Oort Cloud comet at its aphelion, its kinetic energy and total energy are also near zero; we'll give it a gentle horizontal nudge to keep if from crashing into the Sun. As it falls toward the Sun, it's kinetic energy will increase and its potential energy will be go negative by the same amount, keeping the total energy at zero. At perihelion, the comet will have velocity Vx which is the escape velocity for that distance from the Sun. The kinetic energy will then be M(Vx squared) and potential energy will be -M(Vx squared).

If energy is lost to a planet on the way down, the perihelion may be higher or lower than it would have been. Kinetic energy at the new perihelion will less than M(Vx squared), where Vx is escape velocity at the new perihelion.

But why don't we forget about all that perihelion and escape velocity and kinetic and potential; let's just say, the comet lost energy, therefore its total energy is now less that zero, so the comet has too little energy to get back where it came from.
<b>END OF EDIT)</b>

Aside from collisions, the Sun, alone, cannot capture an interstellar comet. Only by passing on the left side of a planet (north being up) can in interstellar comet lose energy to the planet and possibly become an Oort Cloud comet. Conversely, an Oort Cloud comet might pass on the right side of a planet and be sling shotted out of the Sun's sphere of influence (much like the Voyager space capsules).

Yes, solar wind and tidal friction do influence comets very slightly. If a comet is about to pass almost close enough to a planet to be thrown into or out of the Oort Cloud, the solar wind and tidal effects might just tip the scale one way or the other. I believe these effects are statistically more likely to tip the scale in the direction of capture. These effects may slightly increase the total number of captured comets over a period of billions of years.

PhilJ, By focusing on the excape velocity all the other complicated details you are juggling here are avoided for the intent of modeling how a capture can be made. You have several interesting details in the above post that might prove something but anything entering the solar system will be at a speed relative to the solar excape velocity slightly or greatly above that velocity. Nothing else matters. How the process can proceed can be seen knowing only that fact.

Jim,
I take it that you are simply not comfortable working with energy. Velocity is something you can easily picture in your mind. I won't try to enlighten you in this forum, lest I insult your intelligence.

You are quite correct that a comet's velocity at any given moment is either greater or less than whatever the escape velocity is at the comet's present distance from the Sun. Just keep in mind that escape velocity is a not constant for a comet.

If you look up the "escape velocity of the Sun", you might read that it is 384 mi/sec; but that is true only for an object at the surface of the Sun. Likewise, the "escape velocity of the Earth" is 7 mi/sec---at the Earth's surface; but if you are starting from say <font color="yellow">L5</font id="yellow"> , you won't need nearly that much velocity to escape Earth's gravity.

PhilJ, Don't worry about insulting me I outgrew that problem:) and you may have more than review to offer in any comments. As to the picture on this thread about comet capture-the reason I am using the excape velocity is it is a constant all the way in or out from the gravity center. It is very simple to do all the math just from that starting point. Your example is very complex but enery is a very exciting topic and I always want to know more about that too.