Naked Science Forum

Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: geordief on 08/06/2019 00:46:35

Title: What if the Earth was twice as massive?
Post by: geordief on 08/06/2019 00:46:35
....but the same volume.....

I am wondering how many light minutes we would be from the Sun.

It is 8 light minutes now,isn't it?

Would this distance (Is it the same as the spacetime interval btw?) change (decrease?) as a result of the Earth's increased mass  ? Or would it only change when a new orbit was attained?
Title: Re: What if the Earth was twice as massive?
Post by: Halc on 08/06/2019 01:14:34
....but the same volume.....

I am wondering how many light minutes we would be from the Sun.
The mass of an object has nothing to do with its distance from another object.  Earth and Venus for instance are pretty similar in mass, yet orbit at quite different distances.
Maybe you didn't convey the entire question.

Quote
(Is [distance] the same as the spacetime interval btw?)
No, this is a spatial distance.  An interval is the separation between spacetime events.  Objects (sun, Earth) are not events.

Quote
Would this distance change (decrease?) as a result of the Earth's increased mass
If I read this correctly, you asking what would happen to our orbit if mass was added to Earth (fell down in the form of meteors for instance) in over some period of time.

That all depends on the momentum of the added mass.  If meteors strike Earth from behind, it would increase the momentum of Earth, just like the momentum of a moving truck increases if you drive onto it from behind while it is moving.
More likely, the meteors will come from all directions and add pretty much no net momentum to Earth, in which case it will drop to a lower orbit since a larger mass with the same momentum will move slower, and slowing makes an orbital thing drop to a lower orbit.
Title: Re: What if the Earth was twice as massive?
Post by: geordief on 08/06/2019 01:34:01
The 8 light minutes is not a spacetime interval?

Are you sure?

Is it  not a measure of the interval between the emission of the light from the Sun and it's  detection on the Earth?

I was actually wondering what would happen to that 8 light minute interval (whether it is a spacetime interval or just spacial) if the mass was doubled instantly (magically perhaps as a thought experiment)

What would the new distance be ?

Would it be stable ? Would it increase or decrease (I think you said it would increase as Earth would drop to a lower orbit)
Title: Re: What if the Earth was twice as massive?
Post by: evan_au on 08/06/2019 01:43:38
Quote from: Halc
More likely, the meteors will come from all directions and add pretty much no net momentum
The vast majority of objects in our Solar system orbit in the same direction, so any objects hitting the Earth would add their momentum to the Earth's momentum, travelling in the same direction around the Sun.

It is only the occasional object from the Oort cloud (or the even rarer interstellar visitor) that might have a retrograde orbit as it passes around the Sun.

Quote
in which case it will drop to a lower orbit since a larger mass with the same momentum will move slower
In this case, the increased mass of the Earth would carry the sum of Earth's momentum and the momentum of the colliding objects, which would have the Earth continue in pretty much its same orbit.

Of course, if there were a collision with a single really big object (like the hypothetical collision that formed the Moon), that could change Earth's orbit significantly.
Title: Re: What if the Earth was twice as massive?
Post by: Halc on 08/06/2019 02:55:41
The 8 light minutes is not a spacetime interval?
The distance between the sun and Earth is frame dependent.  An interval is not.

Quote
Is it  not a measure of the interval between the emission of the light from the Sun and it's  detection on the Earth?
That interval would be zero, but the distance would be ~150 million km.

Quote
I was actually wondering what would happen to that 8 light minute interval (whether it is a spacetime interval or just spacial) if the mass was doubled instantly (magically perhaps as a thought experiment)
Nothing directly, but the distance might change per my post above.

I didn't address the density thingy.  I don't see how you're likely to add mass to Earth without adding to its volume.  It can be done, but probably not with typical meteors.
Title: Re: What if the Earth was twice as massive?
Post by: Halc on 08/06/2019 03:05:49
The vast majority of objects in our Solar system orbit in the same direction, so any objects hitting the Earth would add their momentum to the Earth's momentum, travelling in the same direction around the Sun.
True, and if all this mass was present and vaguely in Earth's orbit already, the orbit of Earth would be fairly unchanged by it, but stuff from the Oort cloud as you say might have different effects.
We're positing the angular momentum of mass that doesn't exist, so it's conjecture to a point.

Quote
Of course, if there were a collision with a single really big object (like the hypothetical collision that formed the Moon), that could change Earth's orbit significantly.
Has anybody tried to guess as to where Earth orbited before that event?  Did Earth gain mass, or was more smashed away than the mass that it stole from Theia?  What happened to Theia?  Sure, the shrapnel formed the moon, but was there any main body remaining that went elsewhere?  It seems it would not have the energy required to leave the solar system after that hit.
Title: Re: What if the Earth was twice as massive?
Post by: Colin2B on 08/06/2019 09:50:39
The 8 light minutes is not a spacetime interval?

Are you sure?

Is it  not a measure of the interval between the emission of the light from the Sun and it's  detection on the Earth?
8 light minutes is a distance (a space interval) ie the distance light would travel in 8mins.
8mins is a time interval.
Spacetime interval is different and is special because all observers agree what it should be. It is related to Pythagorasís theorem and can be used to calculate a distance in different coordinates and get the same answer. This link shows how it works https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll1.html far better than I can try to explain here.
Remember, a major part of relativity is about how we view things from different coordinates.
Title: Re: What if the Earth was twice as massive?
Post by: jeffreyH on 08/06/2019 12:13:49
Quote from: Halc
More likely, the meteors will come from all directions and add pretty much no net momentum
The vast majority of objects in our Solar system orbit in the same direction, so any objects hitting the Earth would add their momentum to the Earth's momentum, travelling in the same direction around the Sun.

It is only the occasional object from the Oort cloud (or the even rarer interstellar visitor) that might have a retrograde orbit as it passes around the Sun.

Quote
in which case it will drop to a lower orbit since a larger mass with the same momentum will move slower
In this case, the increased mass of the Earth would carry the sum of Earth's momentum and the momentum of the colliding objects, which would have the Earth continue in pretty much its same orbit.

Of course, if there were a collision with a single really big object (like the hypothetical collision that formed the Moon), that could change Earth's orbit significantly.

In other words conservation of momentum. People should remember that. It is quite important.
Title: Re: What if the Earth was twice as massive?
Post by: jeffreyH on 08/06/2019 12:26:46
As an aside, and only indirectly related, what about conservation of momentum in free fall? Objects falling together change momentum in such a way that they fall at the same rate. It is as if they had identical mass. The gravitational field transfers just the right amount of energy to accomplish this. I find that fascinating. And quite important.
Title: Re: What if the Earth was twice as massive?
Post by: Halc on 08/06/2019 13:18:32
As an aside, and only indirectly related, what about conservation of momentum in free fall? Objects falling together change momentum in such a way that they fall at the same rate. It is as if they had identical mass.
You mean like a pebble and a big rock falling from the leaning tower?  The big one gains momentum a lot more than the smaller one, but total momentum (of the 3 bodies involved) is preserved.

If you're just talking about 2 bodies in freefall like Earth and moon, they don't 'fall at the same rate'.  The moon falls (accelerates) a lot more in inverse proportion to its smaller mass.  Momentum is again preserved.
Title: Re: What if the Earth was twice as massive?
Post by: geordief on 08/06/2019 13:35:48

8 light minutes is a distance (a space interval) ie the distance light would travel in 8mins.
8mins is a time interval.
Spacetime interval is different and is special because all observers agree what it should be. It is related to Pythagorasís theorem and can be used to calculate a distance in different coordinates and get the same answer. This link shows how it works https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll1.html far better than I can try to explain here.
Remember, a major part of relativity is about how we view things from different coordinates.

That is heavy going(I am getting confused about the double subscripts in (1.8 ) and can't really carry on further   with that fly in the ointment (I have come across that usage before but it is nothing like second nature and I can't see what Caroll's  "μ" refers to -seems a bit like it came out of the blue)

Just on that 8 light.minutes , what might be a figure  or formula for the emission of a photon  and its reception on the Earth that might  be expressed as a spacetime interval?

What would ,for example a rocket progressing along the Sun-Earth axis measure "our" 8 light.minutes as and how would they calculate it (I had assumed everyone might agree on that number but now I am starting to expect/suspect that they will not.
Title: Re: What if the Earth was twice as massive?
Post by: Halc on 08/06/2019 16:05:12
Just on that 8 light.minutes,
8 light minutes is an expression of spatial distance.  The sun is 8:20 light minutes away from Earth in the frame of the solar system.  This separation is the same as about 150m km.

Quote
what might be a figure  or formula for the emission of a photon  and its reception on the Earth that might  be expressed as a spacetime interval?
That is a light-like separation of events, which always have an interval of zero.  Formula for interval is s≤ = c≤t≤ - d≤.  Time is 500 seconds, and d is 1.5e11 meters.
s≤ = 9e16 * 2.5e5  -  2.25e22 = 0

Quote
What would ,for example a rocket progressing along the Sun-Earth axis measure "our" 8 light.minutes as and how would they calculate it (I had assumed everyone might agree on that number but now I am starting to expect/suspect that they will not.
The interval as measured by the rocket depends entirely on how long it takes to make the trip.
In the frame of a rocket, they would calculate it with their clock since no space is traversed.  The rocket is stationary and it waits for Earth to get to it, so the interval would be ct.  Let's say the rocket gets there in a day on its clock, so the interval would be 3e8 * 86400 = 2.592e13 meters, considerably more than the 1.5e11 meter spatial distance from sun to Earth.
In the frame of the solar system, the rocket is moving so the interval would be
√(3e8≤ * 86414.4≤  -  1.5e11≤) - √(9e16 * 7.46746e9  - 2.25e23) = 2.592e13, same answer, different frame.
Note the 86414.4 seconds it takes for the trip as measured on the solar system clock, not on the ship clock.  There's a ~14.4 second difference between the two.

Choose a different time for the rocket to make the trip and the interval between departure and arrival events changes.
Title: Re: What if the Earth was twice as massive?
Post by: Colin2B on 08/06/2019 22:41:07
That is heavy going(I am getting confused about the double subscripts in (1.8 ) and can't really carry on further   
The important diagram is the one below (1.2) which shows why, in 2D, the distance s (or Δs) doesnít change with a change of coordinates. You really donít need to go beyond (1.3) which is the spacetime interval and is really only 1.2 in 4D
Title: Re: What if the Earth was twice as massive?
Post by: Janus on 08/06/2019 22:43:38
The vast majority of objects in our Solar system orbit in the same direction, so any objects hitting the Earth would add their momentum to the Earth's momentum, travelling in the same direction around the Sun.
True, and if all this mass was present and vaguely in Earth's orbit already, the orbit of Earth would be fairly unchanged by it, but stuff from the Oort cloud as you say might have different effects.
We're positing the angular momentum of mass that doesn't exist, so it's conjecture to a point.

Quote
Did Earth gain mass, or was more smashed away than the mass that it stole from Theia?  What happened to Theia?  Sure, the shrapnel formed the moon, but was there any main body remaining that went elsewhere?  It seems it would not have the energy required to leave the solar system after that hit.
The most recent model has the collision being more head on than glancing and that Theia's mass was absorbed into the Earth-Moon system. So there is no object that would be a significant remnant of Theia.
Title: Re: What if the Earth was twice as massive?
Post by: geordief on 08/06/2019 23:53:50

The important diagram is the one below (1.2) which shows why, in 2D, the distance s (or Δs) doesnít change with a change of coordinates. You really donít need to go beyond (1.3) which is the spacetime interval and is really only 1.2 in 4D
OK ,well I got that ,then .Yes that was interesting and perhaps a bit of an eye opener for me to see   those  2d rotations ( change of coordinate systems) which seem similar to those boosts in Minkowski diagrams.
Title: Re: What if the Earth was twice as massive?
Post by: geordief on 09/06/2019 10:03:30
More likely, the meteors will come from all directions and add pretty much no net momentum to Earth, in which case it will drop to a lower orbit since a larger mass with the same momentum will move slower, and slowing makes an orbital thing drop to a lower orbit.
Would that be a good science fiction  idea (to go along with warp drive etc)?

The "Mass Attenuator" would provide the means for accelerating massive objects up to light speed,wherupon the Uncertainty Shift would  move the craft into Tachyonic Overdrive. (Warp Drive would be for directional control perhaps)

Actually ,when I said the Earth could double in mass I meant instantly as a thought experiment.

You say it would move to a lower orbit . So the increase in spacetime curvature  would not counteract that to any degree?
 
Title: Re: What if the Earth was twice as massive?
Post by: Halc on 09/06/2019 14:53:19
Actually ,when I said the Earth could double in mass I meant instantly as a thought experiment.

You say it would move to a lower orbit . So the increase in spacetime curvature  would not counteract that to any degree?
I said it would move to a lower orbit because it would be colliding with all that mass, which might slow it down and cause it to fall.

As a nonsense thought experiment, (nonsense because the premise breaks all conservation laws), you're saying that suddenly there is this 2xEarth where Earth was, moving for the moment in the same trajectory.  For the most part, this is equivalent to dropping a large and small rock from the tower of Pisa.  They still fall at the same rate since on the grand scheme of things, they're both pebbles, as the Earth is a pebble in relation to the sun.

If you put something actually big where Earth is (like another star rivaling the sun), then the orbit would change as the object is large enough to significantly pull the sun away from where it would otherwise be.  The orbital radius would drop.  Likewise, if I dropped a moon mass from the Pisa tower, it would get to the ground in a bit less time than would the rocks, not because its acceleration is higher, but because the moon is massive enough to pull the Earth up to meet it partway.
Title: Re: What if the Earth was twice as massive?
Post by: geordief on 09/06/2019 15:18:09
I said it would move to a lower orbit because it would be colliding with all that mass, which might slow it down and cause it to fall.

As a nonsense thought experiment, (nonsense because the premise breaks all conservation laws), you're saying that suddenly there is this 2xEarth where Earth was, moving for the moment in the same trajectory.  For the most part, this is equivalent to dropping a large and small rock from the tower of Pisa.  They still fall at the same rate since on the grand scheme of things, they're both pebbles, as the Earth is a pebble in relation to the sun.

If you put something actually big where Earth is (like another star rivaling the sun), then the orbit would change as the object is large enough to significantly pull the sun away from where it would otherwise be.  The orbital radius would drop.  Likewise, if I dropped a moon mass from the Pisa tower, it would get to the ground in a bit less time than would the rocks, not because its acceleration is higher, but because the moon is massive enough to pull the Earth up to meet it partway.
Thanks.  So ,if an object moving in space was to eject some of its more massive components  in a symmetrical (and instantaneous)  way so as not to disturb its direction of travel it would not speed up as a consequence  ?

I had (wrongly) understood that to be the case  from what you said at the end of your first reply to the OP.

("More likely, the meteors will come from all directions and add pretty much no net momentum to Earth, in which case it will drop to a lower orbit since a larger mass with the same momentum will move slower, and slowing makes an orbital thing drop to a lower orbit")

Your clarification  seems to show that I misunderstood that scenario.
Title: Re: What if the Earth was twice as massive?
Post by: Halc on 09/06/2019 17:14:47
Thanks.  So ,if an object moving in space was to eject some of its more massive components  in a symmetrical (and instantaneous)  way so as not to disturb its direction of travel it would not speed up as a consequence  ?
For a small object (Earth is pretty small) that is by itself (The other planets are, not so much Earth) would indeed keep going on its original path.

Given that we have a big moon, it would depend on the time of month you did all this.  You eject the momentum of Earth when it is slowest (full moon) and the orbit will increase (the Earth will speed up).  Do the same thing two weeks later (new moon) and the orbit drops due to loss of system speed.

If any of the other planets blew up like that, they'd not be dragged into different orbits by their insignificant moons. 

Quote
I had (wrongly) understood that to be the case  from what you said at the end of your first reply to the OP.
I had Earth picking up stationary mass, sort of like a barge plowing into and picking up a small beach, which would slow it down, even if there was no friction.

Quote
("More likely, the meteors will come from all directions and add pretty much no net momentum to Earth, in which case it will drop to a lower orbit since a larger mass with the same momentum will move slower, and slowing makes an orbital thing drop to a lower orbit")
No net momentum relative to the solar system, which slows it down.  If the new mass had no net momentum relative to Earth, then the speed would be unchanged.

Your exploding idea makes for all that ejected mass having no net momentum relative to Earth, so no significant change to the orbit.

Jupiter is big enough to make a difference.  If Jupiter wasn't there and Earth orbited on the exact same path, Earth would take a bit more time to orbit than does Jupiter.  Not a lot, since Jupiter isn't big enough to make a big difference, but it's big enough to cause a significant difference.
Title: Re: What if the Earth was twice as massive?
Post by: Janus on 09/06/2019 19:28:15
Thanks.  So ,if an object moving in space was to eject some of its more massive components  in a symmetrical (and instantaneous)  way so as not to disturb its direction of travel it would not speed up as a consequence  ?
For a small object (Earth is pretty small) that is by itself (The other planets are, not so much Earth) would indeed keep going on its original path.

Given that we have a big moon, it would depend on the time of month you did all this.  You eject the momentum of Earth when it is slowest (full moon) and the orbit will increase (the Earth will speed up).  Do the same thing two weeks later (new moon) and the orbit drops due to loss of system speed.

If any of the other planets blew up like that, they'd not be dragged into different orbits by their insignificant moons. 

Quote
I had (wrongly) understood that to be the case  from what you said at the end of your first reply to the OP.
I had Earth picking up stationary mass, sort of like a barge plowing into and picking up a small beach, which would slow it down, even if there was no friction.

Quote
("More likely, the meteors will come from all directions and add pretty much no net momentum to Earth, in which case it will drop to a lower orbit since a larger mass with the same momentum will move slower, and slowing makes an orbital thing drop to a lower orbit")
No net momentum relative to the solar system, which slows it down.  If the new mass had no net momentum relative to Earth, then the speed would be unchanged.

Your exploding idea makes for all that ejected mass having no net momentum relative to Earth, so no significant change to the orbit.

Jupiter is big enough to make a difference.  If Jupiter wasn't there and Earth orbited on the exact same path, Earth would take a bit more time to orbit than does Jupiter.  Not a lot, since Jupiter isn't big enough to make a big difference, but it's big enough to cause a significant difference.
Just to give you an idea of the difference;
The equation for orbital period is
T = 2 pi sqrt(a^3/(G(M+m)))
So assuming a ( distance between M and m) is constant.  The Earth has ~ 1/333,000 the mass of the sun and Jupiter has ~1/1000 the mass of the Sun.   
Ergo, the calculated orbital for the Earth when you consider the effect of its mass, is 0.999998499 of what it is ignoring the mass of the Earth,  which roughly works out to a difference of 47 secs per orbit.
For Jupiter, it comes out to 0.999500375, which works out to a difference of ~4.4 hrs per orbit.  The year* would be a little shorter.

*sidereal year.  The calendar year, which is based on the tropical year, is affected by the axial precession of the planet's axis. So for example, the Earth's present mean sidereal year is 365 days, 6 hrs, 9 min, 8 sec, while the tropical year is 365 days, 5 hrs, 48 min and 45 sec.  This is a greater difference than caused by ignoring the Earth's mass effect on the period.  Now if you replace Earth with Jupiter,  The tropical year would be result of Jupiter's axial precession.  It is estimated that Jupiter at present has a precession perod of 500,000 yrs, compared to the Earth's ~26,000.   However moving Jupiter closer to the Sun also increases the gravitational torque produced by the Sun, which drives the precession.   Jupiter is 5 times further away from the Sun, which makes the Sun's gravity 25 time stronger near the Earth. However this torque is more of a tidal force, which changes by the cube of the distance, so it should be ~ 125 time stronger at Earth distance, driving a faster precession.  This would drive Jupiter's precession down to a period of 4000 years, or roughly 1/6 of The Earth's precession.
Title: Re: What if the Earth was twice as massive?
Post by: evan_au on 09/06/2019 23:41:10
Quote from: Halc
Jupiter is big enough to make a difference.
You show the calculation of how orbital period changes when the mass of the orbiting body is not "insignificant".

From the viewpoint of curved space in General Relativity, could you say that the extra mass curves space a little more than the Sun does by itself, and that this reduces the orbital period?

Quote from: OP
What if the Earth was twice as massive?
I think the main difference is that the Earth would have held on to a lot more atmosphere and water, due to its stronger gravity.

Surface atmospheric pressure would be a lot higher, the atmosphere would extend a lot farther above the surface, and the Earth's crust might be entirely covered in water.

Speculating about Technology, I think it is less likely that water-dwelling species would develop fire. Without fire, development of rocketry and space probes is less likely. The stronger gravity and denser/deeper atmosphere means that rockets would need to be far more powerful to put space probes into a stable orbit than we have on the current Earth.
Title: Re: What if the Earth was twice as massive?
Post by: Halc on 10/06/2019 00:27:59
Quote from: Halc
Jupiter is big enough to make a difference.
You show the calculation of how orbital period changes when the mass of the orbiting body is not "insignificant".
Janus did the calculation, not me.

Quote
From the viewpoint of curved space in General Relativity, could you say that the extra mass curves space a little more than the Sun does by itself, and that this reduces the orbital period?
I didn't have to dig to GR to do it.  It just means Jupiter is big enough to move the sun significantly, and this shortens the radius of its orbit to less than its distance from the sun.

As for Earth (small and close), it's orbital radius is almost identical to its distance from the sun.
It is very much a function of distance, so Pluto, massing about a 23rd of Mercury, nevertheless displaces the sun over 4 times more than does Mercury.

Jupiter is the only planet massive and distant enough to separate the sun completely from the center of mass of the solar system, but only if the other planets cooperate.  If they're all on the other side, they have about as much effect combined as does Jupiter by itself, so the CoM of the solar system could be very close to the center of the sun given the right planetary alignment.