<p>Yes, yes and yes.</p>

<p>Your main misconception is that satellites follow a sinusoidal path. That is just a distortion of the map.</p>

<p>The path of the satellite is in a single plane. When that plane meets the Earth produces a great circle (which is effectively straight, there are no straight lines on the curved surface of the Earth, great circles are the closest that you get to straight)</p>

<p>When you <em>project</em> that great circle onto a map you may get a sinusoidal path, it depends on the projection. Your statement "The ground track of a satellite follows a sinusoidal path" is not correct, that is just an effect of the transformation from a globe to a plane, not "real".</p>

<p>The path of the fragments is, compared to the size of the Earth is very short, it is on the same great circle that the satellite was orbiting in (except as the fragments get closer to the ground they start to move with the atmosphere). If projected they would be on the same track as the satellite. But a great circle is the closest thing to straight on the surface of the Earth.  The rotation of the Earth is much slower than the orbital speed. By the time the fragments have slowed to terminal velocity then they fall roughly in a parabolic curve to Earth, the same as if you threw a ball off a building.</p>

CC BY-SA 3.0
2018-04-05T08:55:30.673Z
Sorry I should have made clear that I realised the ground track is the result of map projection
Dan
CC BY-SA 3.0
2018-04-05T08:58:27.367Z
I'm still not clear tho what path the fragments follow. Surely the rotation of the planet effects the great circle route in some way. Twisting it slightly? Also for purposes of a thought experiment imagine there are very many fragments creating a long path , which I realise in reality would not hapen
Dan
CC BY-SA 3.0
2018-04-05T09:30:40.923Z
Simple answer "they go straight"
James K
CC BY-SA 3.0
2018-04-05T15:19:59.277Z
This is simply not correct. For example if a satellite impacted with many fragments on polar orbit it would form a curved collision path depending on the rotational speed of the planet. Admittedly with a slow rotation speed it would be a shallow curve. All I'm asking guys is there a recognised mathematics for any possible orbit and/or planetary rotation speed
Dan
CC BY-SA 3.0
2018-04-05T16:05:40.980Z
If you can write a better answer please write it and and allow the community to vote on it.
James K
CC BY-SA 3.0
2018-04-05T20:22:41.917Z
Ejecta from airless but more rapidly rotating bodies will follow more complicated, curving trajectories when plotted in the coordinate frame of the rotating planet or viewed as rays on the planetary surface.
Dan
CC BY-SA 3.0
2018-04-05T20:23:43.430Z
https://ntrs.nasa.gov/search.jsp?R=19940023792
Dan
CC BY-SA 3.0
2018-04-05T21:04:58.380Z
I now realise that this problem can be best approached by taking account of the Coriolos effect
Dan
CC BY-SA 3.0
2018-04-06T03:00:16.140Z
@Dan the article you posted above discusses ejecta, which is debris after impact.   Your question was on break-up in orbit and location of the various impacts.   Those aren't the same.
userLTK
CC BY-SA 3.0
2018-04-06T03:47:00.320Z
I realise it was about ejecta but the same principles apply  - its all about how about how impacting fragment interact with a rotating body. The Alvarez paper was the closest I could find
Dan
CC BY-SA 3.0
2018-04-07T09:15:23.303Z
@Dan saying the same principles apply is problematic.  Ejecta emanates from an impact shockwave, matterial is essentially thrown from (explodes from) the surface of the Earth.  Throw a ball from the surface while you are rotating along with the surface and that ball will essentially move along Coriolis rules.    A satellite impacting the Earth that breaks apart inside the roche limit is unaffected by the Earth's rotation, it follows a hyperboic (or is that parabolic?), I think it's parabolic, as it impacts.   Earth's curvature would affect the spread, the effect of rotation would be small.
userLTK
CC BY-SA 3.0
2018-04-07T09:17:49.303Z
Bottom line is, you've made an incorrect assessment and your answer is incorrect as well.   Coriolis effect would be secondary and perhaps even negligible.    I'm not trying to give you a hard time, but the spread of the two events would be considerably different regarding impact sites of something that breaks up in orbit prior to impact and ejecta.
userLTK
CC BY-SA 3.0
2018-04-07T11:49:38.367Z
Coriolis definition: ' objects attempting to move on a straight path relative to a non- rotating surface will end up travelling on a curved path with respect to a rotating surface'
Dan
CC BY-SA 3.0
2018-04-05T08:47:37.457Z
2018-04-05T08:47:37.457Z
James K
<p>Impact trajectories when plotted in the coordinate frame of a rotating body or planet can best be understood by taking into account the Coriolos effect</p>

CC BY-SA 3.0
2018-04-05T22:00:08.003Z
2018-04-05T22:00:08.003Z
Dan
<p>I'm not sure this is a good answer, but I'll give it a shot.</p>

<p>Starting with Shoemaker-levy 9, which isn't an ideal example because it broke apart during an orbit about 2 years prior to impact but it's interesting for comparison.</p>

<p><a href="https://i.sstatic.net/ev4bq.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/ev4bq.jpg" alt="enter image description here"></a></p>

<p><a href="http://ase.tufts.edu/cosmos/print_images.asp?id=16" rel="nofollow noreferrer">Source of picture</a>.</p>

<p>In the case of Shoemaker-levy, while the object broke apart somewhat randomly the pieces all had basically the same orbit, and the ones closer to Jupiter were accelerated a bit faster, so over time the fragments lined up in something close to a straight line.   This happens with debris that falls off a comet as well, the debris tends to orbit in a line, along the same basic orbital path.   Given enough time, and with an elliptical orbit (comets around the sun or Shoemaker around Jupiter), debris tends to line up and follow the same path.</p>

<p><a href="https://i.sstatic.net/4JkBu.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/4JkBu.jpg" alt="enter image description here"></a></p>

<p>Same thing happens with comets, though in the case of comets, the debris trail can over time encompass the entire orbit, in a kind of very narrow, elliptical, dusty/tiny asteroid belt.</p>

<p><a href="https://i.sstatic.net/8yFfr.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/8yFfr.jpg" alt="enter image description here"></a></p>

<p><a href="http://faculty.virginia.edu/skrutskie/ASTR1210/notes/mshower.html" rel="nofollow noreferrer">Source of picture</a>.</p>

<p>That's why meteor showers have radiants and they happen at the same time every year.  That's where the Earth intersects the orbiting dust trail from the comet. </p>

<p>In the case of meteor showers, any part of the Earth where the radiant is visible in the sky can get meteors.  The spread is significantly larger than the diameter of the Earth (but still a generally narrow band around the sun).</p>

<p>In the case of shoemaker levy, with the breakup happening only one orbit prior to impact, the spread was quite small, and, relative to Jupiter's plane or orbit, the shoemaker-levy objects all hit Jupiter in about the same place, but over a period of about a week, in which time Jupiter was able to make multiple rotations.  Jupiter has almost no axial tilt (3 degrees) so those impacts occurred nearly along a latitude line of Jupiter (3 degrees off).  That's a result of hitting in nearly the same place as the planet rotated below the impact.   In the case of Shoemaker-levy, where the fragments hit in single file over time, the impact site is on a plane perpendicular to the planet's rotation.</p>

<p><a href="https://i.sstatic.net/WOgaV.gif" rel="nofollow noreferrer"><img src="https://i.sstatic.net/WOgaV.gif" alt="enter image description here"></a></p>

<p><a href="http://www.ifa.hawaii.edu/images/sl9/" rel="nofollow noreferrer">Source of picture and more images</a></p>

<p>In your scenario</p>

<blockquote>
  <p>If a satellite in an inclined orbit breaks up in orbit and multiple
  fragments hit a planet</p>
</blockquote>

<p>It's a little unclear what you mean by orbit.  If the planet is Earth and the satellite is in orbit around the Earth, it mostly wouldn't hit the planet.  Orbits, by their very nature, rarely hit the object they are orbiting.   Now if the satellite was orbiting the sun, it could collide with the Earth if the orbits intersect.   That's the more common type of comet or asteroid impact.   In theory, such a comet or asteroid could break apart as it passed inside Earth's Roche limit.   Impacting objects rarely break apart just passing the Roche limit because they're usually too small and too well held together and if they do, the tidal forces that break them apart would give the pieces a relatively low radial velocity compared to the very high gravitational velocity towards the Earth so the spread would be tiny.  (this doesn't take into account super-heating and exploding in the atmosphere - which can spread out debris pretty far), but you said break apart in orbit, so lets ignore atmospheric spreading.</p>

<p>The debris pattern in your scenario would be similar to a shot-gun blast.  Fairly closely concentrated simply because orbital break-ups are low velocity and gravitational impacts are high velocity, but the spread would look similar to, but not precisely the same as a shot-gun pellet spread.</p>

<p><a href="https://i.sstatic.net/o3sOr.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/o3sOr.jpg" alt="enter image description here"></a></p>

<p><a href="http://hammernews.blogspot.nl/2006/02/cool-hand-dick-questions-remain-in.html" rel="nofollow noreferrer">Source of image</a>.</p>

<p>There are some aspects to your question that are unclear such as when the object breaks apart and what it's orbiting, but mostly, comets or meteors don't break up on their way towards impacting a planet, or if they do, the breakup while in orbit is relatively insignificant.</p>

<p>There are cases where <a href="https://www.livescience.com/52154-ancient-meteors-struck-simultaneously.html" rel="nofollow noreferrer">earth has been hit by a pair of meteors</a>, that were orbiting each other at the time of impact.  We know that wasn't a "break-up" because the crash sites were too far apart, though in theory, it could have been a shoemaker-levy type of event that may have split the object as it flew past, and then the two objects hit during the next time around.   </p>

CC BY-SA 3.0
2018-04-06T02:20:13.290Z
2018-04-06T02:20:13.290Z
userLTK
<p>If a satellite in an inclined orbit breaks up in orbit and multiple fragments hit a planet, do the individual fragments follow
1 a sinusoidal path
2 a geodesic ( or great circle curve)
3 a straight line
I'm having real problems trying to visualise this!
I realise the ground track of an inclined orbit follows a sinusoidal path but what happens when impacting a rotating planet?</p>

CC BY-SA 3.0
2018-04-05T23:08:52.873Z
@Kozaky Shoemaker broke apart inside the roche limit and did another full highly elliptical orbit around Jupiter prior to impacting the planet.   That's a slightly different situation.
userLTK
CC BY-SA 3.0
2018-04-05T23:15:53.303Z
Small point, and this enters into some of your problems with James K's answer, but a straight line on a sphere is a bad term to use.   Many people think latitude lines are straight, because they look straight and don't intersect, like parallel lines.   Mathematicians think longitude lines are straight, because, on the surface of a sphere, straight lines are great circles, like longitude lines.    You should clarify  #3 where you say straight line.
userLTK
CC BY-SA 3.0
2018-04-06T02:57:31.163Z
It occurs to me, you might have been looking for a more mathematical answer and I gave a more practical one.   But it would help if you said specifically the approach of the satellite and what it orbited and how far away it broke up.  That would add some clarity to the question.
userLTK
CC BY-SA 3.0
2018-04-06T03:54:16.513Z
Yes I have no idea what the mathematics involved would be but it must surely take into the account the Coriolos effect  I admit my scenario is fairly obscure and it helps to think of it on a planet or body with no atmosphere so that drag dies not confuse the picture
Dan
<planet>
Impact of inclined orbit satellite on a planet
Dan
# Impact of inclined orbit satellite on a planet

(Asked by: username_0 on 2018-04-05)

If a satellite in an inclined orbit breaks up in orbit and multiple fragments hit a planet, do the individual fragments follow
1 a sinusoidal path
2 a geodesic ( or great circle curve)
3 a straight line
I'm having real problems trying to visualise this!
I realise the ground track of an inclined orbit follows a sinusoidal path but what happens when impacting a rotating planet?

**Tags:** planet
> **Comments:**
> **username_1**: @Kozaky Shoemaker broke apart inside the roche limit and did another full highly elliptical orbit around Jupiter prior to impacting the planet.   That's a slightly different situation.  
> **username_1**: Small point, and this enters into some of your problems with James K's answer, but a straight line on a sphere is a bad term to use.   Many people think latitude lines are straight, because they look straight and don't intersect, like parallel lines.   Mathematicians think longitude lines are straight, because, on the surface of a sphere, straight lines are great circles, like longitude lines.    You should clarify  #3 where you say straight line.  
> **username_1**: It occurs to me, you might have been looking for a more mathematical answer and I gave a more practical one.   But it would help if you said specifically the approach of the satellite and what it orbited and how far away it broke up.  That would add some clarity to the question.  
> **username_0**: Yes I have no idea what the mathematics involved would be but it must surely take into the account the Coriolos effect  I admit my scenario is fairly obscure and it helps to think of it on a planet or body with no atmosphere so that drag dies not confuse the picture  

---

## 3 answers

### Answer by username_2 on 2018-04-05. Score: 2

Yes, yes and yes.

Your main misconception is that satellites follow a sinusoidal path. That is just a distortion of the map.

The path of the satellite is in a single plane. When that plane meets the Earth produces a great circle (which is effectively straight, there are no straight lines on the curved surface of the Earth, great circles are the closest that you get to straight)

When you *project* that great circle onto a map you may get a sinusoidal path, it depends on the projection. Your statement "The ground track of a satellite follows a sinusoidal path" is not correct, that is just an effect of the transformation from a globe to a plane, not "real".

The path of the fragments is, compared to the size of the Earth is very short, it is on the same great circle that the satellite was orbiting in (except as the fragments get closer to the ground they start to move with the atmosphere). If projected they would be on the same track as the satellite. But a great circle is the closest thing to straight on the surface of the Earth. The rotation of the Earth is much slower than the orbital speed. By the time the fragments have slowed to terminal velocity then they fall roughly in a parabolic curve to Earth, the same as if you threw a ball off a building.

> **Comments:**
> **username_0**: Sorry I should have made clear that I realised the ground track is the result of map projection  
> **username_0**: I'm still not clear tho what path the fragments follow. Surely the rotation of the planet effects the great circle route in some way. Twisting it slightly? Also for purposes of a thought experiment imagine there are very many fragments creating a long path , which I realise in reality would not hapen  
> **username_2**: Simple answer "they go straight"  
> **username_0**: This is simply not correct. For example if a satellite impacted with many fragments on polar orbit it would form a curved collision path depending on the rotational speed of the planet. Admittedly with a slow rotation speed it would be a shallow curve. All I'm asking guys is there a recognised mathematics for any possible orbit and/or planetary rotation speed  
> **username_2**: If you can write a better answer please write it and and allow the community to vote on it.  
> **username_0**: Ejecta from airless but more rapidly rotating bodies will follow more complicated, curving trajectories when plotted in the coordinate frame of the rotating planet or viewed as rays on the planetary surface.  
> **username_0**: https://ntrs.nasa.gov/search.jsp?R=19940023792  
> **username_0**: I now realise that this problem can be best approached by taking account of the Coriolos effect  
> **username_1**: @Dan the article you posted above discusses ejecta, which is debris after impact.   Your question was on break-up in orbit and location of the various impacts.   Those aren't the same.  
> **username_0**: I realise it was about ejecta but the same principles apply  - its all about how about how impacting fragment interact with a rotating body. The Alvarez paper was the closest I could find  
> **username_1**: @Dan saying the same principles apply is problematic.  Ejecta emanates from an impact shockwave, matterial is essentially thrown from (explodes from) the surface of the Earth.  Throw a ball from the surface while you are rotating along with the surface and that ball will essentially move along Coriolis rules.    A satellite impacting the Earth that breaks apart inside the roche limit is unaffected by the Earth's rotation, it follows a hyperboic (or is that parabolic?), I think it's parabolic, as it impacts.   Earth's curvature would affect the spread, the effect of rotation would be small.  
> **username_1**: Bottom line is, you've made an incorrect assessment and your answer is incorrect as well.   Coriolis effect would be secondary and perhaps even negligible.    I'm not trying to give you a hard time, but the spread of the two events would be considerably different regarding impact sites of something that breaks up in orbit prior to impact and ejecta.  
> **username_0**: Coriolis definition: ' objects attempting to move on a straight path relative to a non- rotating surface will end up travelling on a curved path with respect to a rotating surface'  

---

### Answer by username_0 on 2018-04-05. Score: 0

Impact trajectories when plotted in the coordinate frame of a rotating body or planet can best be understood by taking into account the Coriolos effect


---

### Answer by username_1 on 2018-04-06. Score: 0

I'm not sure this is a good answer, but I'll give it a shot.

Starting with Shoemaker-levy 9, which isn't an ideal example because it broke apart during an orbit about 2 years prior to impact but it's interesting for comparison.

[![enter image description here](https://i.sstatic.net/ev4bq.jpg)](https://i.sstatic.net/ev4bq.jpg)

[Source of picture](http://ase.tufts.edu/cosmos/print_images.asp?id=16).

In the case of Shoemaker-levy, while the object broke apart somewhat randomly the pieces all had basically the same orbit, and the ones closer to Jupiter were accelerated a bit faster, so over time the fragments lined up in something close to a straight line. This happens with debris that falls off a comet as well, the debris tends to orbit in a line, along the same basic orbital path. Given enough time, and with an elliptical orbit (comets around the sun or Shoemaker around Jupiter), debris tends to line up and follow the same path.

[![enter image description here](https://i.sstatic.net/4JkBu.jpg)](https://i.sstatic.net/4JkBu.jpg)

Same thing happens with comets, though in the case of comets, the debris trail can over time encompass the entire orbit, in a kind of very narrow, elliptical, dusty/tiny asteroid belt.

[![enter image description here](https://i.sstatic.net/8yFfr.jpg)](https://i.sstatic.net/8yFfr.jpg)

[Source of picture](http://faculty.virginia.edu/skrutskie/ASTR1210/notes/mshower.html).

That's why meteor showers have radiants and they happen at the same time every year. That's where the Earth intersects the orbiting dust trail from the comet.

In the case of meteor showers, any part of the Earth where the radiant is visible in the sky can get meteors. The spread is significantly larger than the diameter of the Earth (but still a generally narrow band around the sun).

In the case of shoemaker levy, with the breakup happening only one orbit prior to impact, the spread was quite small, and, relative to Jupiter's plane or orbit, the shoemaker-levy objects all hit Jupiter in about the same place, but over a period of about a week, in which time Jupiter was able to make multiple rotations. Jupiter has almost no axial tilt (3 degrees) so those impacts occurred nearly along a latitude line of Jupiter (3 degrees off). That's a result of hitting in nearly the same place as the planet rotated below the impact. In the case of Shoemaker-levy, where the fragments hit in single file over time, the impact site is on a plane perpendicular to the planet's rotation.

[![enter image description here](https://i.sstatic.net/WOgaV.gif)](https://i.sstatic.net/WOgaV.gif)

[Source of picture and more images](http://www.ifa.hawaii.edu/images/sl9/)

In your scenario

> If a satellite in an inclined orbit breaks up in orbit and multiple
> fragments hit a planet

It's a little unclear what you mean by orbit. If the planet is Earth and the satellite is in orbit around the Earth, it mostly wouldn't hit the planet. Orbits, by their very nature, rarely hit the object they are orbiting. Now if the satellite was orbiting the sun, it could collide with the Earth if the orbits intersect. That's the more common type of comet or asteroid impact. In theory, such a comet or asteroid could break apart as it passed inside Earth's Roche limit. Impacting objects rarely break apart just passing the Roche limit because they're usually too small and too well held together and if they do, the tidal forces that break them apart would give the pieces a relatively low radial velocity compared to the very high gravitational velocity towards the Earth so the spread would be tiny. (this doesn't take into account super-heating and exploding in the atmosphere - which can spread out debris pretty far), but you said break apart in orbit, so lets ignore atmospheric spreading.

The debris pattern in your scenario would be similar to a shot-gun blast. Fairly closely concentrated simply because orbital break-ups are low velocity and gravitational impacts are high velocity, but the spread would look similar to, but not precisely the same as a shot-gun pellet spread.

[![enter image description here](https://i.sstatic.net/o3sOr.jpg)](https://i.sstatic.net/o3sOr.jpg)

[Source of image](http://hammernews.blogspot.nl/2006/02/cool-hand-dick-questions-remain-in.html).

There are some aspects to your question that are unclear such as when the object breaks apart and what it's orbiting, but mostly, comets or meteors don't break up on their way towards impacting a planet, or if they do, the breakup while in orbit is relatively insignificant.

There are cases where [earth has been hit by a pair of meteors](https://www.livescience.com/52154-ancient-meteors-struck-simultaneously.html), that were orbiting each other at the time of impact. We know that wasn't a "break-up" because the crash sites were too far apart, though in theory, it could have been a shoemaker-levy type of event that may have split the object as it flew past, and then the two objects hit during the next time around.


---

