Stellar Oscillations across Spiral Arms

Tom, I'm not actually ready to respond to all of your comments in you're previous post, but since you're getting ahead of me, I'll respond to some of your latest comments, now.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">If a small planet perturbation enters the picture,... <hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">I assume we're both thinking of a planet many times larger than the comet.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">Technically, the primary effect of the planet is usually on the comet's angular momentum. <hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">If you are talking about angular momentum about to the center of the Sun, that's obvious; the comet's <u>foreward</u> momentum <u>is</u> its angular momentum. But if you mean relative to the comet's own center of gravity (CG), I find that tough to swallow. Can you imagine a comet ten meters in diameter whose surface is spinning one kilometer per second? However, I don't doubt that the angular momentum is significant, and certainly necessary to conserve both the momentum and the energy. I think we have both been ignoring angular energy when we talk of total energy. The spin of the comet accounts for part of its kinetic energy, but does not affect its trajectory---unless of course centrifugal force tears it to pieces.

Now, I <u>can</u> see how a significant portion of the impulse imparted to the <u>planet</u> would be off center, and therefore impart angular momentum to the planet about the planet's CG. But the planet's gravity should affect both sides of the comet almost equally, due to the comet's small size. When we talk about conserving angular momentum, we must not change our reference axis in the middle of the problem. We don't say, for example, that the comets angular impulse about its CG is equal and opposite to the planet's angular impulse about the it's CG.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">If a small planet perturbation enters the picture, the whole passage of the comet past the planet can be approximated as a single impulse at the point of closest approach. <hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">I assume you mean the point where the comet's original trajectory would have had its closest approach to the planet if not for the perturbation. If you mean the actual point of closest approach, that would require successive approximations.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">That impulse must be directed toward the planet.... Any impulse that adds to the comet's net speed as it passes the planet will increase the comet's energy (ability to escape), and any impulse that decreases the comet's speed will decrease its energy. <hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">If the impulse vector (impulse = force x time = momentum) is directed exactly toward the planet at the point of closest approach, then it is perpendicular to the comet's momentum vector. How, then, can the impulse possibly decrease the comet's momentum? The sum of two perpendicular vectors is always greater.
[EDITED a few minutes after posting: Forget I said that; I think it is not true that the velocity is perpendicular at closest approach because of the planets foreward motion. I need to sleep on it. Next paragraph deleted. End of edit.]

This discussion, so far, reminds me of the Greek philosophers; we'll never prove anything this way. Assuming that Keppler's laws are true, you have two options: calculus or numerical analysis. The safest way to generalize a method of approximating by one impulse at one point is to run the virial equations, via computer, on every type of 3-body problem and work your way backward to the approximating method. Would I be overly presumptive in guessing that you have already done that? Or perhaps you have read the report of someone who has.

In 1996 comet Hale-Bopp past near Jupiter over the north pole. How did that passby effect the speed of Hale-Bopp? It seems to me it caused the comet to speed up and also changed its parahelion but in looking at the tables that were generated at the time I don't see any information about this detail.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">[tvf]: If a small planet perturbation enters the picture,...

[PhilJ]: I assume we're both thinking of a planet many times larger than the comet.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">Yes. I should have said "a small perturbation by any planet". These simplified generalizations may not work for large perturbation cases.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">[tvf]: Technically, the primary effect of the planet is usually on the comet's angular momentum.

[PhilJ]: If you are talking about angular momentum about to the center of the Sun, that's obvious; the comet's <u>foreward</u> momentum <u>is</u> its angular momentum.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">Yes, that is what I meant. Orbit perturbations do not affect spin unless the approach is so close that tidal forces come into play.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">[tvf]: If a small planet perturbation enters the picture, the whole passage of the comet past the planet can be approximated as a single impulse at the point of closest approach.

[PhilJ]: I assume you mean the point where the comet's original trajectory would have had its closest approach to the planet if not for the perturbation. If you mean the actual point of closest approach, that would require successive approximations.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">For small perturbations, these two cases are the same and no iteration is needed.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">[tvf]: That impulse must be directed toward the planet.... Any impulse that adds to the comet's net speed as it passes the planet will increase the comet's energy (ability to escape), and any impulse that decreases the comet's speed will decrease its energy.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">Your subsequent confusion is because this was also badly worded. To simplify again, consider the case where the comet's motion is perpendicular to a planet's orbital motion, with the planet assumed to be on a circular orbit. (Nevermind that the comet must then be headed for collision with the Sun. We are making approximations here.)

Whether the comet passes in front of or behind the planet, its path will be bent from perpendicular, and acquire a new forward or backward component along the planet's orbit. That "path-bending" part changes the orbital angular momentum, which changes mainly the comet's perihelion distance. Approximately, the total energy is unchanged by the path-bending itself.

But there is another important change, which (as you say) is occasioned by the planet's own motion. If the comet passes behind the planet, its path will be bent forward, and it will spend more time near the planet after the closest approach than it did before the closest approach because the two bodies then have a motion component in the same direction. The result will be a net slowing of the comet's forward speed -- a loss of total energy.

If the comet passes in front of the planet, it spends less time near the planet after closest approach, with a corresponding net gain in speed and energy.

A similar kind of reasoning can predict what will happen if the initial comet orbit is not perpendicular to the planet's orbit.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">[PhilJ]: The safest way to generalize a method of approximating by one impulse at one point is to run the virial equations, via computer, on every type of 3-body problem and work your way backward to the approximating method. Would I be overly presumptive in guessing that you have already done that? Or perhaps you have read the report of someone who has.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">I have computed lots of orbits and developed a "feel" for them. However, I am not testing what I am saying here with numerical simulations. But I recommend that you get an orbit integrator and play with some cases to see how orbital mechanics works. I have outlined the basic principles involved with 2-body orbits in chapter six of my book. -|Tom|-

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by Jim</i>
<br />In 1996 comet Hale-Bopp past near Jupiter over the north pole. How did that passby effect the speed of Hale-Bopp? It seems to me it caused the comet to speed up and also changed its parahelion but in looking at the tables that were generated at the time I don't see any information about this detail.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">The orbital period dropped from 4200 years to 2400 years. So the comet lost energy. -|Tom|-

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">(PhilJ)...If the impulse vector (impulse = force x time = momentum) is directed exactly toward the planet at the point of closest approach, then it is perpendicular to the comet's momentum vector.... I need to sleep on it.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote"> <blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">(TVF)Your subsequent confusion is because this was also badly worded.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">Actually, my confusion and self-doubt resulted from mental fatigue. I was right to begin with; the comet’s momentum at the point of closest approach is perpendicular to the direction from the comet to the planet—regardless of whether you chose the sun or planet as the origin of your coordinate system. So any vector approximating the total impulse of the encounter must either be placed elsewhere and/or it must point in a different direction; otherwise, it cannot decrease the comet’s momentum and energy.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">(TVF)...To simplify again, consider the case where the comet's motion is perpendicular to a planet's orbital motion, with the planet assumed to be on a circular orbit. (Nevermind that the comet must then be headed for collision with the Sun. We are making approximations here.)<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">Actually, the comet’s trajectory could be perpendicular to the <u>ecliptic</u>, rather than radial from the Sun; but then we would have to think in three dimensions, so let’s avoid that scenario for the time being.

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote">(TVF)But there is another important change, which (as you say) is occasioned by the planet's own motion. If the comet passes behind the planet, its path will be bent forward, and it will spend more time near the planet after the closest approach than it did before the closest approach because the two bodies then have a motion component in the same direction. The result will be a net slowing of the comet's forward speed -- a loss of total energy.

If the comet passes in front of the planet, it spends less time near the planet after closest approach, with a corresponding net gain in speed and energy.<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">
Your conclusion may be correct, but your reasoning is flawed.

When you say “forward” , you apparently mean “in a prograde direction”, i.e., counterclockwise as we look down from north of the ecliptic. A comet approaching radially has no such “forward” speed; so it cannot decrease.

If the comet is approaching radially toward the Sun, as you have postulated, any small prograde or retrograde impulse would be perpendicular to the comet’s initial momentum, so the total energy would not be significantly affected. If however, the impulse does have a radial component, relative to the Sun, that <u>would</u> affect the comet’s total energy. If, as you say, the comet spends more time close to the planet after closest approach, it seems reasonable to apply the approximating impulse vector a point after closest approach; there it would have a component away from the Sun, thus decreasing the comets momentum and energy. If the comet were moving radially away from the Sun—ignoring the obvious impossibility of that scenario—then the impulse vector applied after closest approach would have a radial component toward the Sun, again decreasing the comet’s momentum and energy.

However, I’m not convinced that the comet does, as you say, spend more time close to the planet after closest approach. Since the comet is headed in toward the Sun, its speed must be increasing. The speed can decrease only if the closest approach is so near the planet that the planet’s gravity is stronger than the Suns gravity. So, for “small planet perturbations”, the approximating vector logically should be applied at a point farther from the Sun than the point of closest approach—regardless of whether the comet is inbound or outbound.

[EDIT moments after posting]P.S.: Okay, I do see how a prograde impulse would result in more time close to the planet after closest approach---but only compared to a retrograde impulse.

To improve the mental picture I was trying to draw, let's exaggerate it. Suppose a comet's inbound trajectory passes behind Jupiter and is bent 90 degrees so that the comet then moves away in the same direction as Jupiter's orbital motion.

The post-encounter separation speed will be relatively slow because the comet speed is about 19 km/s and Jupiter's is 13 km/s, so they separate at just 6 km/s. The approach relative speed [sqrt(19^2+13^2)] is 23 km/s, so the post-encounter relative speed is less than the pre-encounter relative speed.

That means Jupiter's gravity has less time to speed the comet up pre-ensounter and more time to slow it down post-encounter. The net is a loss of energy.

Clear yet? -|Tom|-