Move the Earth Away from the Sun

[christian everett]

I was interested in whether it would be possible to move the Earth away from the Sun by using two ion drives on the Moon.
One Ion drive on the far side of the Moon would operate briefly while the Moon was exactly between the Sun and the Earth.
The other ion drive on the side facing the Earth would operate briefly when the Earth was exactly between the Sun and the Moon .
The net effect would be to move the Earth further away from the Sun, provided the ions from the second drive were not captured by the Earth's gravitational field.
I calculate it would take 17,000 tons (probably potassium or sodium mined on the Moon) moving at the speed of light (without relativistic mass increase) to get the Earth moving one mile and hour in the opposite direction.

 

[jbriggs444]

As I understand this, the attempt is to apply a net outward (away from the sun) force on the moon, exerted once every two weeks for however long it takes.

Unfortunately, that is exactly the wrong way to go about changing an orbit. A single such outward force exerted in January will result in an orbital deflection -- an elliptical orbit. Come April, the Earth will be at aphelion -- the high point on the ellipse. In July, the Earth will be back where it started. In October the Earth will at perihelion, closer to the sun than when it started. When January rolls around again, the Earth will be back where it started again -- nothing gained.

A continuous outward thrust leaves you with a circular orbit slightly higher than you started, as long as you maintain the thrust. Once you discontinue the thrust, you are back to a slightly elliptical orbit at the same average orbital radius you started with. There is no other permanent change.

What you want instead is a net force that accelerates the Earth along its orbit. A single impulse in January still gives you an elliptical orbit. But this time the ellipse is always at or above the Earth's original orbit. A continuing thrust gives you an outward spiral orbit. Discontinue thrust and you end up in a permanently higher and slightly elliptical orbit.

That said, your math seems off. 17,000 tons sounds low by about eight orders of magnitude.

 

[russ_watters]

I don't see how knocking the moon out of its orbit around Earth would move the Earth to a higher orbit around the sun.

 

[willem2]

I don't see how knocking the moon out of its orbit around Earth would move the Earth to a higher orbit around the sun.

If you don't push very hard, the moon won't escape. If you push in the direction of the velocity of the earth/moon system around the sun, you will accelerate the moon half of the time and decelerate it the other half in a reference frame with the Earth as center, so you can give more and more energy to the earth, while keeping the velocity and distance of the moon relative to the Earth approximately the same. I think giving 1 km/s * M_moon of impulse to the moon each revolution around the Earth must be possible. I think that will do it in about a hundred years. Getting the reaction mass and energy you need in that time will be much harder.

 

[sophiecentaur]

I guess this is a Sci Fi scenario. A mad professor has a base on the Moon etc. etc.. ??

 

[christian everett]

Well, this isn't a reaction to global warming, rather it's to keep the Earth from getting burned up when the Sun expands into a red giant, something which is as necessary as deflecting or destroying asteriods which might collide with the Earth..

 

[DaveC426913]

That said, your math seems off. 17,000 tons sounds low by about eight orders of magnitude.

... moving at the speed of light...

 

[DaveC426913]

Well, this isn't a reaction to global warming, rather it's to keep the Earth from getting burned up when the Sun expands into a red giant,

5 billion years is a long time.

I think I'm OK with not risking extinction of all life on Earth in a deep freeze just yet.

 

[jbriggs444]

For what it is worth, here is the math behind the assertion that the original calculation was off by eight orders of magnitude.

Mass of Earth: 6.0 x 10²¹ tons.
Expected speed: 1 mph.
Momentum: 6.0 x 10²¹ ton miles/hour

Proposed reaction mass: 17900 tons.
Proposed exhaust velocity: 186,000 miles per second = 6.7 x 10⁸ miles per hour
Momentum: 1.2 x 10¹³ ton miles per hour

[We are asked to ignore corrections for special relativity]

21 minus 13 = 8 orders of magnitude.

 

[mfb]

To raise the orbit (with a measurable efficiency) you need thrust along the orbital path: Push Earth "forwards". You can do this via thrusters on the Moon but the energy requirements would be prohibitive. Many asteroid fly-bys are a more realistic option.

 

[Nugatory]

We are asked to ignore corrections for special relativity

We can even include them if necessary. They will be of the same order of magnitude as the gamma factor for the exhaust velocity, and even a factor of ten is far beyond what any realistic ion drive can do.

 

[mjc123]

The Sun isn't going to expand gradually over 5 billion years. It will stay relatively stable for that time, then expand pretty quickly (I'm not sure how quickly, or whether there'll be any warning signs in advance). We'd need to be beyond the orbit of Mars to be safe, which at 1 mph will take about 50000 years, so if we set off well before the expansion, we'll freeze.
Another scenario might be a bit more realistic. The Sun is very slowly getting warmer (much too slowly to account for current global warming, by the way), and the habitable zone is moving outwards; and long before it goes red giant (I believe in about a billion years?) it will be too hot for life on earth. Maybe a slow adjustment of the Earth's orbit might compensate for that?

 

[christian everett]

OK, I got the low mass of 17,000 tons from the Earth being 55 quintillion tons and Kinetic Energy = 1/2 mass x (velocity squared). So the speed of light 186,000 miles per second squared (34,596,000,000) times 3600 seconds in an hour squared (12,960,000).

You didn´t factor in the 12,960,000.

Before the Earth acutally gets engulfged by the expanding Sun it will get too hot for life as we know it, like 500 millioon year.

 

[mfb]

The Sun isn't going to expand gradually over 5 billion years. It will stay relatively stable for that time, then expand pretty quickly (I'm not sure how quickly, or whether there'll be any warning signs in advance). We'd need to be beyond the orbit of Mars to be safe, which at 1 mph will take about 50000 years, so if we set off well before the expansion, we'll freeze.
Another scenario might be a bit more realistic. The Sun is very slowly getting warmer (much too slowly to account for current global warming, by the way), and the habitable zone is moving outwards; and long before it goes red giant (I believe in about a billion years?) it will be too hot for life on earth. Maybe a slow adjustment of the Earth's orbit might compensate for that?

The luminosity will slowly double over the next 6 billion years. The rapid expansion afterwards is still a process that happens over tens of millions of years. 50,000 years is nothing for the Sun.

OK, I got the low mass of 17,000 tons from the Earth being 55 quintillion tons and Kinetic Energy = 1/2 mass x (velocity squared). So the speed of light 186,000 miles per second squared (34,596,000,000) times 3600 seconds in an hour squared (12,960,000).

You didn´t factor in the 12,960,000.

Before the Earth acutally gets engulfged by the expanding Sun it will get too hot for life as we know it, like 500 millioon year.

You have to match momentum not energy. Velocity, not velocity squared. Momentum is conserved while you have to put in a lot of energy to make this work.

 

[christian everett]

If it's momentum not kinetic energy then it's a no go with current technology.
The fundamental premise of moving the Earth/Moon system by applying force only to the Moon, which was highly speculative, was sound though.
So how much energy, in kilograms or metric tons of pure energy would it take to get the Earth and Moon, which is 1/81 the mas of the Earth, moving one mile an hour away from the Sun?

 

[jbriggs444]

So how much energy, in kilograms or metric tons of pure energy would it take to get the Earth and Moon, which is 1/81 the mas of the Earth, moving one mile an hour away from the Sun?

That question cannot be answered unless we know what you are planning to push on. Momentum conservation still applies.

There is no such thing as "pure energy". Energy is an attribute (a frame-variant one at that), not a physical thing.

 

[mfb]

Let M be the mass of Sun, m the mass of Earth, v is its the current orbital velocity. G is the gravitational constant, r is the current distance (1 AU, approximating the orbit as circle).

First we have to find the necessary acceleration. The orbit of Earth stays approximately a circle, so kinetic energy is half the negative gravitational potential energy. We have to supply half the change in potential energy (the other half comes from the kinetic energy). v = 1mph = 0.447 m/s. The rate of change in gravitational potential energy is P=vGMm/r², plugging in variables we get P=1.58*10²² W.

To increase Earth's total energy with half this power we need to accelerate it by a = P/(2mv) = 4.41*10^-8 m/s². We need a force of F=P/(2v) = 2.63*10¹⁷ N. To reach this with the lowest power we should eject mass from the Moon with a velocity of about 3.4 km/s, fast enough to leave the Moon and Earth/Moon system with significant velocity but not too fast to keep the energy to momentum ratio reasonable. The mass will leave the system with about v_t=2.4 km/s. Faster if we shoot forwards in the Moon's orbit, slower if we shoot behind, but we have to find some average to not make the Moon deorbit anyway. This needs a power of P=Fv_t/2 = 3.2*10²⁰ W to shoot F/v_t=1.1*10¹⁴ kg of mass per second away from the Moon. At this rate the Moon will be gone in 21 years, after we raised the orbit of Earth by merely 300,000 km. It is just coincidence that this is roughly the Earth/Moon distance. We can make the Moon last longer if we increase the velocity of the material shot away, but this will also increase the necessary power.

As comparison: Humans use about 2*10¹³ W today, and Earth receives 2*10¹⁷ W of power from the Sun.

 

[Janus]

If it's momentum not kinetic energy then it's a no go with current technology.
The fundamental premise of moving the Earth/Moon system by applying force only to the Moon, which was highly speculative, was sound though.
So how much energy, in kilograms or metric tons of pure energy would it take to get the Earth and Moon, which is 1/81 the mas of the Earth, moving one mile an hour away from the Sun?

Others have addressed the issue of how much mass it would take to produce such a change in the Earth's motion. But what I don't think you understand is that such a change would result in only an very small change in the size of the Earth's orbit. It's not a matter of giving the Earth a 1 mph push away from the Sun and it will keep drifting away at that speed.
A 1 mph push, applied in the most effective direction (which as already mentioned is in the direction of the Earth's orbital motion), will increase the Earth's average orbital distance by ~ 4500 km. This would be its new orbital distance unless you supplied another push to change it further. This is also just about the same distance the Earth moves in and out from the Sun due to its monthly orbit around the Earth-Moon barycenter.

One push and done would not produce the desired result of slowly moving the Earth away from the Sun over time. It would take a constant thrust or a continuing series of thrusts to accomplish this.