Mass, Gravity and Earth's Orbit
6 Nov 2008 3:27:47 UTC
Topic 194014
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I was watching the movie "Independence Day" recently and wondered about a question I've had for awhile now. Early in the movie it is mentioned that the mass of the alien spaceship is about 1/4 the mass of the moon, which would obviously add lots of mass and probably affect earth's orbit around the sun.
Now suppose an asteroid would hit earth, and it would have X mass to it. How much mass would have to be added to earh's mass in order for the earth to begin to spiral into the sun due strictly to the added mass? Ignore all other considerations, such as collision energies, etc. And how much mass would have to be added to earth for the moon to begin the same death spiral into its beloved neighbor?
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Mass, Gravity and Earth's Orbit
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It would be nice to simplify the question by ignoring other considerations, however one additional point must be considered to answer the question: What's the velocity of the Earth after the collision? The amount of total mass doesn't really enter into it – does this make sense? To put it another way, if you dropped a pound of lead and an ounce of plastic from shoulder-height, which object would hit the ground first, the heavier one or the lighter one?
So...do I understand this
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So...do I understand this correctly? The thought-experiment is basically that Earth would experience a spontaneous mass increase, and the question is what would happen then in terms of trajectories, depending on the amount of mass that is added.
So if we neglect the moon and the other planets for a while and assume that the new mass of the Earth would still be small compared to that of the Sun, it's basically a "one body problem" in the reference frame of the sun and nothing would change at all, right?
Only if the new mass of the Earth is no longer negligible compared to that of the sun, you have to look at it as a two body problem. Not only Earth would change its trajectory compared to todays situation, but you have to consider the new trajectory of the sun also, as both have orbits around their common barycenter. If anything, Earth and Sun would "crash" into each other then, not "Earth into Sun", right? Well, it all depends on the frame of reference :-).
All in all I think we are pretty save from a "Sun inspiral" as the damage from the impact of such a heavy mass would kill us all in the first place anyway. ;-)
EDIT: I also think that "inspiral" doesn't really match the situation here: in this two body problem, there are only a limited amount of solutions for trajectories, and what would usually called an "inspiral" trajectory is none of them (?). In this scenario the trajectories would be ellipses, maybe so close that earth and sun would touch.
CU
Bikeman
RE: So...do I understand
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Correctomundo on both counts! That is why I mentioned ignoring other considerations -- Life would end right quick, obviously, so just ignore the rest. One body, or two if you would like, how much mass could our planet spontaneously add before our orbit would be altered.
O that thou would fibbeth! :-D True that it would end up as an eliptical orbit, but in the end they would still collide I would think. Remember that the velocity would be the same as it is now, so that if there were an increase in mass such that it would irrevocably change the orbit, there would not be enough velocity to keep the earth from eventually kerplopping into the sun. An eliptical orbit assumes that the masses and velocities of the two objects are within a window that would keep them in orbit of each other.
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RE: It would be nice to
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I think Galileo answered this question. Without any air friction, they would hit the ground at the same time. I believe this experiment was performed on the Moon by Apollo XIV astronauts.
Tullio
RE: I believe this
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I remember seeing footage from that demonstration. If I recall my physics classes correctly, if you apply Newton's laws, all comes down to equations that have the term (m_1 + m_2) in it where the m_* are the masses of the two bodies involved (the moon and the object you drop), and as long as m_1 >> m_2, m_2 doesn't matter.
Yeah, ok, but when you say "eventually colliding" and "inspiral", I somehow get the association of a trajectory that has several orbits with decreasing radii and finally radius 0. Is that really a solution of the two-body-problem? Aren't there just a few like "fly-by" (objects will end up increasing their distance forever, hyperbolic trajectories????), head-on trajectories, and elliptical orbits? There can be collisions (Jupiter Shoemaker Levy like), but only when you consider the spacial radii of the objects. If you consider them point-masses, I guess Shoemaker Levy should still orbit an indefinitely small Jupiter, for example. No "inspiral". From an conservation-of-energy POV I don't think inspiral is right here.
Any physicists here ??
CU
Bikeman
RE: RE: It would be nice
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Yes sir, I believe so.
So why stop with the mass of one asteroid? Add the mass of all the asteroids in the entire solar system to the Earth, and then add Jupiter and the rest of the planets and all their mass to the Earth. Then add mass until the cows come home and the Earth eventually forms another star. The result is a new binary system in the Milky Way, with a couple stars in a nice circular orbit around a 'barycenter', as Bikeman mentioned, and ignoring things like general relativity, the new Earth/star would never spiral in to the sun ... sorry, it's hard to ignore a well mannered orbit :)
We may have some mixed
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We may have some mixed scenarios/assumptions here, so I'll attempt to cover ground ....
Inertial mass ( indicating resistance to alteration of state of motion ) is identical to gravitational masss ( indicating degree of forceful influence in another mass's presence ). This is true classically and also relativistically. So the Earth could be either half, or twice, it's current mass and its behaviour with respect one other body alone would not change. If :
f = force on the Earth
m = Earth's mass
a = Earth's acceleration
M = Sun's mass
G = proportionality constant
r = distance between the Earth and Sun
then :
f = ma = GMm/r^2
gives
a = GM/r^2
Actually it's a vector equation, but the important thing is that it is independent of 'm'. Now you can fully determine the orbit, by also supplying some initial conditions ( integrating once will get you velocity and yet again for the position ). One can restate the problem but this time to determine the Sun's path. If in addition :
F = force on the Sun
A = Sun's acceleration
then :
F = MA = GMm/r^2
gives
A = Gm/r^2
with the follow on as above. The reason I state it in this way is that both the Sun and the Earth follow ( elliptical ) paths around a common centre of mass. Because M is so much larger than m then the Sun's ellipse is so much smaller than the Earth's, and this common centre actually lies more or less within the Sun, but not at it's centre as is a usual statement/approximation. Anyhow there is no mechanism for 'spiralling' here due to gravity alone, or added or subtracted masses in situ. You'd need some punt from a third object to alter the direction of either.
[ Under GR the system would radiate a smidgens of energy away from the system and this loss would eventually cause a lessening of the distance between the two. ]
With a many body solar system like our own there aren't any true perfect ellipses, but fairly close to it. The dominant effect on any given planet is still the Sun for sure, and there will be some 'base' orbital shape for each planet due to that. Each planet is hence altered by the influences of all the others, and as you'd expect the gas giants weigh in heavily there. Remember the laws are stated in terms of a two-body interaction, and it's the other body's mass that determines a given body's acceleration. To completely specify a body's orbit then requires a ( vector ) addition of all the accelerations from the all the others, then integrate etc for a full path definition.
This hasn't been exactly ( or 'analytically' ) solved for three or more participants. The mathematical horror increases rapidly with additional bodies. The 'Keplerian' parameters for orbits are quoted on the assumption of some closest fit pure ellipse, and often additional variables to relate some ( again approximate ) evolution of those in time.
When Mercury's orbit was deemed as abberant on Newtonian grounds ( which GR later answered pretty near perfectly ) it was due to so called abnormal 'precession' of the orbit. What this phrase means is that the real orbit is some looping flower petal type shape, that has been roughly described in terms of a single ellipse that is then rotated as a fixed shape ( which is not quite the same thing ). The orientation, with respect to distant stars, of it's point of closest approach to the Sun was altering quicker than expected on a pure Newtonian model - with other planet's included.
[ Does any one remember the Spirograph? Is it still about, this terrific kid's toy that produces these lovely patterns ( not actually spirals! ) using restrained cogs? ]
Just to be real obsessive about this explanation, I'll add the tidal factor. The Earth, or any similiarly shaped spheroid can be viewed for the orbital calculations above ( to a good first approximation ) as being replaced by a hypothetical point mass. But that only holds well if the radius of either body is small compared to their mutual separation. In that case the relative difference in gravitational force across one body due to the other is small-ish. Or alternatively the gradient of gravity is low from one side of the body to the other. But when they get close ( meaning their radii are now a significant fraction of the separation ) the closer sides will be pulled rather more than the distant sides. The point approximation breaks down and the bodies distort appreciably more than the quiet Earth/Moon system say. The eventual results here ( with 'solid' objects ) are to cause locking/resonance of rotations - which is why the same side of the moon always faces us ( rotational period around its own axis equals orbital period around the Earth ) - and an increase in Earth/Moon distance. This effect is rather complex to explain without a full treatment of angular momentum, but basically the Earth's oceans act essentially as 'brake shoes' upon the solid parts and slows Earth's rotation. Conservation of angular momentum in the Earth/Moon system then implies the Moon must move further out. Of course the Sun is a major player here too .....
Out at Jupiter, Io is subject to alot of tides as it is quite close to something really really massive and the same tidal stuff applies. Like kneading a lump of dough the internal substance of Io is heated by the internal friction from this action, hence volcanoes et al. The final inspirals ( GW induced now ) of neutron stars, black holes and whatnot have humungous tidal effects - the models of which are extremely challenging.
So back to answer the initial question. The only way to spiral in, under what I think is the initial scenario, would be to increase masses by many orders of magnitude to thus induce appreciable loss of energy away from the system by gravitational radiation.
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
Gerry, I think you asked a
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Gerry, I think you asked a wonderful question despite some initial confusion over the assumptions, and it's worthwhile pointing out that this scenario actually plays out all the time, in spectacular fashion, on a slightly larger scale: In some binary star systems having a gas giant and a white dwarf as the two stars, the white dwarf gains mass from the from the companion giant star - many, many asteroids worth of mass. The result is not any kind of drastic change in orbits leading to a collision, but rather the white dwarf keeps on gaining mass until it has enough for a Type Ia Supernova to occur, a very important tool for measuring distance for astronomers :)
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Unfortunately, the GOCE
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Unfortunately, the GOCE satellite built by ESA which should have mapped the Earth's gravitational field with great accuracy has been delayed in its launch because of problems with the launcher, which is a converted military Russian rocket. This reminds me of a forecast by which, in case of a war, about one half of the rockets launched would not launch or simply explode. Better not to make this experiment.
Tullio
I enjoyed the question and
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I enjoyed the question and appreciated Mike's explaination as well. I couldn't help to also think of Newton's first law of motion: "every body presists in a state of rest or uniform motion as long as it is not acted upon by an exterior force." The same equation applies F=ma. Gerry mentioned if only the mass of the Earth increased, I would think, m increases and that in proportion to the uniform motion changes due to the exterior force (which in this case didn't change but is the force of gravity from the Sun)
Nonetheless, as mass increases, a bell graph could be plotted until the uniform motion or obit changes. It would have to begin to spiral. m is proportational to F... as m increase, F increases.
f=ma.