If 1/2 the Earth dissappeared what would happen to the Earth / Sun orbit?

[g__day (Matthew)]

I discovered this interesting question on another forum where folk where missing the correct answer.

The situation is simply if half the Earth just dissapeared, what would happen to the Earth / Sun orbit; would the Earth speed up or slow down, move closer or further away from the Sun?

It's actually a very good question to consider, because it requires you to think broadly.

To start at the basics the Earth is falling towards the Sun with a force proportional to G * Mass of Sun * Mass of Earth / average seperation distance ^2, and this is balanced by an outwards replusion equivalent to Mass of Earth * Velocity ^ 2 / average seperation distance.

Now half of the Mass of the Earth just vanishes in a moment of quantum pique, what happens and why?

[Starkler (Geoff)]

Nothing happens because on both sides of the equation mass is a direct multiplier, so a change in mass gives an equal and linear change to both the centripetal and gravitational forces, so they remain balanced.

Can you imagine an astronaut in the space station with a salami floating around which he then cuts in two and releases. The two halves will not shoot away from the earth to find a new orbit

[g__day (Matthew)]

Which was exactly the thinking that most folk on the other forum finally arrived at. Neither bodies velocity nor momentum changed.

Now can you think deeper and see why what you have just said isn't what actually happens!

A small hint - your salami or Astronaut model isn't the right framework. Image half the salami just popped out of existence - so matter and energy weren't necessarily conserved. Neither's velocity nor momentum is affected but how has your system changed?

Hint

Imagine in the Earth / Sun example its was the Sun, not the Earth that suddenly diminshed by 50% of its mass, would you still say your logic works or can you see where the gotcha is?

[Shawn]

I think it depends on which half, doesnt it,, If the top half dissapeared for example would anyone notice,,,.


S

[[1ponders] (Paul)]

When m1(sun) is much greater than m2(planet), nothing will happen because the speed of orbit is determined by the semi-major axis of orbit. Kepler determined that the Period of orbit squared is proportional to the semi-major axis cube. (In the case of the sun and planets P^2=a^3 )

For objects of similar mass then Newton modified Keplers law to


P^2= [4pi^2 / G(m1+m2) ] a^3

Where G is the universal gravitational constant (still not accurately known ) and a is the semi-major axis of the orbit and P is the orbital Period.

[Shawn]

Ah Paul thats a lovely equation, and accurate to boot, but forgive my prior humour, the question was half the planet dissapeared, not decreased in volume by 50%. now we have introduced something thats not in your equation.

Unbalanced centrifugal force. axis shift..and its influence on nearby objects.

[[1ponders] (Paul)]

If half the planet disappeared, it would eventually form an ovoid shape again under gravity anyway.

An easy way to prove this is to tie a sinker to a piece of string and swing it around your head at a velocity that just keeps the string horizonal. Do exactly the same thing with a ping pong ball with the same length of string. Different sizes and different weights but the period of orbit stays the same. (allowing for increased wind resistance of the ping pong ball)

Mass doesn't enter into it. Which falls faster; a hammer or a feather?

[Shawn]

eventually it sure would, but in the interim the centre of mass has changed by a few tens of miles to a thousand miles. depending on what bit dissapeared. the string length has changed.

S

[[1ponders] (Paul)]

Not appreciably thought. The earth is only about 6,380 km in radius, the earth/sun semi-major axis is 150,000,000 km. Or roughly a ration of 4.253e-5/1. The effect would be a very minor shift, if any.

If the near side disappeared the earth would slow down ever so very slightly, and if the farside disappeared the earth would speed up ever so very slightly. Too late in the night to work it out but I'll have a look at it in the day light

If the top or the bottom disappeared it would have no effect.

[Shawn]

Thats what I said, If the top half dissapeared, nobody would notice,,,.

Goodnite Paul..

Be Well..

S

[ballaratdragons (Ken)]

OK, I'll think outside the box!

If half the Earth suddenly dissappeared the world would break apart and disinigrate!

This would be caused by the unbalanced shape being thrust around it's axis causing severe stress on the odd shaped remaining half.

P.s. Geez I like the job Mojo and I did on my new Avatar

[g__day (Matthew)]

Guys that's pretty good thinking. I could keep hinting but maybe its better to inform.

From a Newtonian point of view the system is unchanged. Both bodies have no net change on the other as determined above.

But we don't live in a Newtonian world...

The system has lost rest mass, so the systems gravity well has diminished. So whilst the planets motion through spacetime doesn't change the curvature of spacetime itself does change under relativity when mass or energy is suddenly lost!

So you would see within this frame of reference the orbit slightly, but discernible change (expand and become more eccentric).

So whilst Kepler and Newton where giants, Einstein does come into play in some cosmic games!

[xelasnave]

Gravity rain predicts as there is less shielding available the gravitational influence is less and the orbit will seek a larger radius .. but GR will always agree with space time so whatever Dr A says is fine with me . GR is the stuff that does the actual bending you see .
alex

[[1ponders] (Paul)]

I can agree that the orbit will be ever so lightly larger. My previous change in center of mass was wrong. It wouldn't be 6380km, it would be less than half that. Making a guestimate of an increase of 1300 km (for various reasons and ignoring the fact that this is way smaller than the difference between the earths semi-major axis and semi-minor axis AND the earths orbit isn't circular) I get an increase of about 30 min added to the standard earth year of 365 days.

I would be interested in how the change in the earths orbital eccentricty with change, considering is it continually changing anyway (over a 40000 year period that is)

[xelasnave]

We are presuming that when half disappears that half the mass goes with it...
What if the halving is confined to volume and not mass... mmmm If mass is not halved it seems nothing should change.
alex

[ving (David)]

Id have thought one of the major changes would be the gravitation point where the earth and moon rotate around? maybe this point would now be outside the earth body... either way it would change the way the moon rotates around us and in so changing the orbit around the sun...

i am no astro physicist but this make sense to me

and when you say half, i automatically thought layers from the outside. that would leave the molten core exposed to space cause i dont think there would be any ozone layer or atmosphere or anything....
gah! my head hurts

[[1ponders] (Paul)]

You are probably right ving. The problem is trying to work out the results for a three body system. With a two body system (sun/earth) it's relatively ( ) easy to solve. A three body system is an entirely different kettle of fish, though with the earth and moon being much less massive than the sun the overall average orbit probably wouldn't change that much. Different story for trying to determine the orbit of a planet around two similar mass suns

[g__day (Matthew)]

Mass (or Energy, which is its equivalent) have some relativistic effects on any frame of reference. Increasing mass slows time and increases the curvature / geometry / bendiness of space.

Think of a large 3d volume of space, say a sphere, that would readily contain the Earth and the Sun. Say a sphere of radius 10 light minutes centred on the Sun's point of suspension. Well that volume isn't just a mathematical volume regardless of if its empty or not. Under relavity the space itself actually changes at the physical level, it curves as it becomes more massive. The opposite is also true, lose mass and the curve straightens. Whilst you can have a straight line in maths easily, in space there is no abstract straight, straight depends on where your mass and energy densities are localed and their magnitude!

Spacetime itself changes and so do all the paths through this volume. Minutely but certainly detectably at the supposed mass loss in this experiment.

Space tells matter how to move and matter tells space how to curve...

PS

And yes as mentioned above the Bary centre of the Earth / Sun pair would shift towards the Sun further, meaning the Earth would be orbiting at a slightly increased distance from the Sun too.

[xelasnave]

G day you no doubt are aware of the use of the ball on the rubber sheet being used to demonstrate the concept of how mass bends space. I contend that the example is misleading and humbly seek your input if you are prepared to comment.
I feel the ball on the sheet example has the ball on the wrong side of the sheet... if one left the sheet and the distortion of the sheet in place as if the ball were upon it but moved the ball "under" the sheet would this not be a fairer representation of the way space is bent? Or another way of putting this.. take a "straight" grid line I think it bends towards the mass point not as the example demonstrates "bent around it". I see a lot of representations of gravitational lensing that has the grid line bending around the mass point as opposed to being bent towards it.
A very curious alex

[g__day (Matthew)]

Hi Alex, nice ask I will try and answer.

First think of gravity as an aspect of geometry rather than a force. Second realise that the Universe is always trying to be more chaotic and dis-orgainsed (increase its entropy levels - which generally means not pack things into a smaller volume of space in a more ordered fashion).

Now the sheet analogy is a way of trying to visualise a 4D piece of geometry in a 3D world - something which is really hard to do! You are replacing the invisible force with a visual element of geometry meant to represent what really underlies the nature of our reality. Throw a ball or fall of a cliff and you are seeing geometry of spacetime, not a force in action - all very hard to come to grips with.

If I had to answer how do we visualise gravity as a 3d geometry, I'd start by saying remember 2d circular log paper - which looks like an old fashion radar screen? Well imagine that in 3d with a mass being at the epi-centre. So to a distant observer the lines showing how far 1 centimetre of distance is are smaller as you get closer to the point source of mass.

So spacetime is more curved the closer you get to a point of mass. A lightyear of distance could look like a micron to a distant observer of spacetime very close to a black hole's event horizon.

Similarily when you get very, very, very close to the baryonic matter that comprises the nucleus of any atom (protons and neutrons) you'd expect the gravitational curvature of spacetime at distance less than 10 ^ -20 metres to be rather extreme given G is inversely proportional to a 1/d^2 effect.

We don't really undertstand theoretically how the four forces interact at these very close distances to atomic matter. Say that again - our best scientific models can't apply and describe what happens very, very close to the nucleus of an atom. In a certain way a proton may have its own gravitational event horizon - if its density (volume / mass) is above a certain level. We can't model these happenings yet, at best we move from relativist physic (which is exotic at time) to quantum mechanics - which is the equivalent of relativity on LSD. Under QM particles can be in two places at the same time, very often (called tunnelling), and interfere (differaction) with each other, and appear and dissappear etc...

So after my long set up rant, my responce to your question - the volume of space has to be geometrically more distorted closer to the concentrated centre of mass (or energy). So yes you could use a bowling ball on a sheet of foam or you could consider helium balloons released inside a large circus tent. All will want to travel towards the apex.

Rather than lines to or around a point soure of mass, consider space and distance shrink as you get closer to mass.

* * *

PS - where this thought experiment was heading and how it ties into dark energy / dark matter

The loss of mass in a closed system leading to no change in a Newtonian / Kepler framework, but a significant change to a relativistic framework is where I wanted folk to head. Throughtout school and most of our life we see space as a fixed entity that only Captain Kirk (or Piccard) moves through easily, as if he was going down to the corner store, or driving across country.

But space isn't always like that. If on a cosmic scale matter can be created and destoyed without apparently conserving energy or momentum, in finite spots for a finite time, then there is a different framework for studying cosmic expansion other than dark energy or dark matter (or perhaps offering a candidate for them).

Hawking has showed its possible for matter or energy to jump from the cosmic foam (alah the world of the really tiny quantum mechanics that our universe appears to float in) and not necessarily spell doom and gloom. Hawkings initial postulate was a pair of virtual particles (pro and anti to conserve mass and energy) tunnel into our reality then immediately zap each other and annihilate... except if they appear near the event horizon of a black hole, so one escapes and one is sucked in. In this situation you've just added mass and energy to the Universe.

My thought experiment was a simple two body test of what are the effects of a sudden large extract of mass from the universe - answer spacetime uncurves and we can detect this change in an an otherwise unexplained shift in the orbit of planets. The sun is losing mass every day as its converted into energy and it radiates past us. So spacetime around our solar system is uncurving (very, very, very slowly) every second of our lives.

But what if around galactic centres matter is coming into existence? Physicists studying MOND have shown you only need a very small adjustment to explain why galaxies don't fly apart. Calculated as if this force was attributed to mass gain you need about 1 hydrogen atom per cubic metre of space per year. Theoretical physics has shown that empty space should contain about 10 ^ 120 Joules of energy per cubic metre, the greatest whoopsie in the framework of our thinking - or is it? What if that energy is there - but it manifests very, very slowly over all time - not just all at once? If there is that much potential energy, but in only leaks in in the form of 1 Hydrogen atom per year, maybe all parties could reconcile some pretty wierd theories.

But all that at the moment is just pure theoretical spectulation with no solid grounding - yet. But as said the thought experiement was to get folk thinking and have them realise space, the final frontier, is not a constant, but a variable dependent on localised mass and energy.