In which Your Hero fails at orbital mechanics

All right, this is probably a simple question, but I haven't got the
maths even to know where to start. A planet somewhere between Saturn-
and Jupiter-mass is spiraling inward, on its way to becoming a hot
jovian. There's a roughly Earth-mass planet in its way. Which is
more likely: that the rocky planet impacts the jovian head-on, that
orbital decay makes it fall into its star, or that it's expelled from
the system?

Feel free to forward this to any other Usenet group that might help.

In rec.arts.sf.science, Damien Valentine wrote:
All right, this is probably a simple question, but I haven't got the
maths even to know where to start. A planet somewhere between Saturn-
and Jupiter-mass is spiraling inward, on its way to becoming a hot
jovian. There's a roughly Earth-mass planet in its way. Which is
more likely: that the rocky planet impacts the jovian head-on, that
orbital decay makes it fall into its star, or that it's expelled from
the system?

I have no formal training, but I can estimate with the best of 'em.

Well, no. The best are way better at estimating than I am. Never mind
that now.

It is well known (he says, after checking Wikipedia and verifying some
calculations) that the Earth is closer to solar-system escape velocity
than to falling into the Sun. (Going uphill from 1 AU requires about 42
km/s, of which our orbital velocity already supplies 30 km/s: delta
vee is 12 km/s. Going downhill requires cancelling that whole 30 km/s.
Okay, not the whole of it, because the Sun isn't a point, but most of it.)

So, per envelop dorsalis, a random perturbation of Earth's orbit is
more likely to throw out us than in. I'm not going to claim the odds
are 12-to-30, because I sense some square-or-square-root factors in
the probability calculation. But my money is on the cold dark.

A jovian planet cruising inward seems like a great example of random
perturbation, so this is my answer.

(This doesn't answer the question of whether the civilization on the
planet burns or freezes. That depends on how the perturbation
*starts*, which is a harder question to answer!)

I recall somebody doing a lot of simulation of our solar system --
which is chaotic in the very long term, even without Jupiter moving
south for the holiday season -- and I bet the answers there would be
enlightening. I don't remember the reference, though.

--Z

--
"And Aholibamah bare Jeush, and Jaalam, and Korah: these were the borogoves..."
*

On 10/01/2011 01:34 PM, Damien Valentine wrote:
All right, this is probably a simple question, but I haven't got the
maths even to know where to start. A planet somewhere between Saturn-
and Jupiter-mass is spiraling inward, on its way to becoming a hot
jovian. There's a roughly Earth-mass planet in its way. Which is
more likely: that the rocky planet impacts the jovian head-on, that
orbital decay makes it fall into its star, or that it's expelled from
the system?

Feel free to forward this to any other Usenet group that might help.

Most likely expelled from the system, though much depends on details.

Assuming the Jovian is *slowly* spiraling toward the star, its
instantaneous orbit at any given time can be approximated as circular,
with a larger semi-major axis and longer period than the Earth-mass
planet. So on each synodic period, the Earth-mass planet overtakes the
Jovian from below/behind.

As long as the Jovian is far enough for the Earth-mass planet to avoid
its sphere of influence on each pass, nothing happens (obviously a
simplification, since sphere of influence boundary is fuzzy and not
sharp). But once the Jovian has decayed to the point where the
Earth-mass planet passes through the sphere of influence, each pass will
simultaneously boost the Earth-mass planet (increasing semi-major axis,
so increasing period) and turn it (increasing eccentricity).

Once the Earth-mass planet's orbit is significantly elliptical, the
system becomes far less predictable since the effect of the Jovian will
vary depending on whether the Earth-mass planet is near perihelion (less
effect) or aphelion (more effect) at each encounter, and the two
planets' periods will not necessarily be resonant. Especially once the
Earth-mass planet's aphelion exceeds the semi-major axis of the Jovian,
since encounters may now occur at any point along the Jovian's sphere of
influence.

On Oct 1, 2:34*pm, Damien Valentine wrote:

A planet somewhere between Saturn- and Jupiter-mass is spiraling inward,
on its way to becoming a hot jovian. *There's a roughly Earth-mass planet
in its way.

If the jovian is spiraling in, why isn't the terrestrial? That might
be an important constraint.

*Which is more likely: that the rocky planet impacts the jovian head-on,
that orbital decay makes it fall into its star, or that it's expelled from
the system?

"Head on" isn't likely, as they would almost certainly both be
orbiting in the same direction (I'm guessing you mean "hits the
jovian"). From a random statistics approach, yes you are likely to end
up being scattered out rather than in, and if you are scattered out
then you have a longer orbital period, so orbital evolution is even
slower.

However... depending on how the jovian is evolving in, it might very
well sweep the terrestrial into a resonance and "push" it inward, like
Io sweeps Europa outward and Europa sweeps Ganymede outward. So I
suspect the answer is in detail...

It depends

--
Brian Davis

In article
,
Damien Valentine wrote:

All right, this is probably a simple question, but I haven't got the
maths even to know where to start. A planet somewhere between Saturn-
and Jupiter-mass is spiraling inward, on its way to becoming a hot
jovian. There's a roughly Earth-mass planet in its way. Which is
more likely: that the rocky planet impacts the jovian head-on, that
orbital decay makes it fall into its star, or that it's expelled from
the system?

Feel free to forward this to any other Usenet group that might help.

What would cause such a "spiraling?" If one planet is spiraling in, then
how is the disturbance that caused the spiraling limited to that planet?

All orbits are conics -- whether hyperbolas (exo-Solar origin),
parabolas (indefinite origin), ellipses (Solar capture) or cirles
(ellipses with zero eccentricity). Unless the planet is shedding energy
in a retrograde direction, (exercising propulsion), its orbit is never a
spiral.

On Oct 1, 8:58*pm, Orval Fairbairn wrote:

All orbits are conics...

Well... not quite. Orbits are conics if the are generated in a
universe with only two objects, both of which are point-like, under
the influence of only one force, which varies as one over the distance
squared. Add other objects, non-point-like objects, other forces, or
forces that don't have a 1/r^2 dependance, and orbits are no longer
simply conics.

Unless the planet is shedding energy in a retrograde direction...

Energy doesn't have a direction, but your point is valid. There are a
number of mechanisms that can cause an orbit to "spiral in". For
*extremely* small objects, the poynting-robertson effect indeed causes
dust particle to spiral in. For larger objects like planets, you have
to look elsewhere, but it can certainly happen due to gas drag
effects, slowly increasing mass of the central star, and tidal
effects.

The real question, as you mention, is what is causing this
perturbation to the orbit, and how is it effecting *all* the objects.

--
Brian Davis


, (exercising propulsion), its orbit is never a
spiral.

In rec.arts.sf.science, Brian Davis wrote:
On Oct 1, 8:58*pm, Orval Fairbairn wrote:

All orbits are conics...

Well... not quite.

The pedantry is admirable, but I think we're talking about:

Current models predict that giant-planet cores will form within a
protoplanetary disk before the disk gas disperses provided the disk
is a few times more massive than the minimum-mass solar nebula.
Gravitational interaction between a planetary core and the disk
cause rapid inward type-I migration of the core. [...] Giant planets
themselves open an annular gap in the disk and undergo slower
type-II migration...

( http://adsabs.harvard.edu/abs/2007DDA....38.0604C , from Wikipedia
link. The abstract does imply that we don't expect Earthlike planets
to form at all in such systems, so the original question seems
unlikely.)

--Z

--
"And Aholibamah bare Jeush, and Jaalam, and Korah: these were the borogoves..."
*

Damien Valentine wrote:
All right, this is probably a simple question, but I haven't got the
maths even to know where to start. A planet somewhere between Saturn-
and Jupiter-mass is spiraling inward, on its way to becoming a hot
jovian. There's a roughly Earth-mass planet in its way. Which is
more likely: that the rocky planet impacts the jovian head-on, that
orbital decay makes it fall into its star, or that it's expelled from
the system?

Feel free to forward this to any other Usenet group that might help.

Spiraling inward already implies that the planet is exchanging energy of
position (potential energy of the orbit) with other bodies, which gain
energy and thus end up with higher orbits. So the most likely fate of the
Earth-sized planet is to get tossed outwards, though probably not out of the
system altogether.

However, it is possible that the transfer could go the other way, or there
could be a collision. There is some complicated stuff that happened
(celestial mechanics theoreticians say) early in our own solar system that
moved the orbits of the giant planets around, through encounters with
smaller asteroidal bodies.

--
Mike Dworetsky

(Remove pants sp*mbl*ck to reply)

In rec.arts.sf.science message , Sat, 1
Oct 2011 19:13:56, Andrew Plotkin posted:


It is well known (he says, after checking Wikipedia and verifying some
calculations) that the Earth is closer to solar-system escape velocity
than to falling into the Sun.

In Newtonian physics with spherical smeary, escape speed from a distance
is always root 2 times circular orbit speed at that distance. Escape
energy is twice circular orbit energy.

Low-orbit speed is attained by a horizontal acceleration of one local
gee for a distance of half a radian; escape speed takes twice the
distance.

See http://www.merlyn.demon.co.uk/gravity2.htm.

--
(c) John Stockton, nr London, UK. Turnpike v6.05 MIME.
Web http://www.merlyn.demon.co.uk/ - FAQqish topics, acronyms and links;
Astro stuff via astron-1.htm, gravity0.htm ; quotings.htm, pascal.htm, etc.
No Encoding. Quotes before replies. Snip well. Write clearly. Don't Mail News.

On Oct 1, 4:53*pm, "Jorge R. Frank" wrote:
Assuming the Jovian is *slowly* spiraling toward the star...

I'm assuming it takes around a million years, plus or minus an order
of magnitude. Your call whether that counts as "slow".