If Earth mass is doubled, what is the change of earth's orbit

I built a magic system that rationalized the creation of energy, so the energy source of magic is not a problem. (How this system works is not the main point of this question.)

When people keep using magic, the amount of energy on the earth increases.

Energy can also generates gravity. $$E=mc^2$$

If the amount of energy on the earth increases, the mass of the earth also increase. Will the change of mass affects earth's orbit? If yes, how would the orbit change? Assume $5.3748 \times 10^{41} J$ is generated by magic, that means the earth mass is doubled. (I need answers involve mathematics, thanks!)

What is the safe level of yearly generation of energy? Is it safe if the energy generation per year is below $8.694 \times 10^{24} J$ ? (equal to the mass loss per year) https://en.wikipedia.org/wiki/Earth_mass#Net_losses

Atmospheric escape of gases. About 3 kg/s of hydrogen or 95,000 tons per year[15] and 1,600 tons of helium per year[16] are lost through atmospheric escape.

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The mass created have the same velocity as the earth.

I need the math involved in accounting for the rest of the solar system and the eccentricity of the Earth's orbit,

@AlexandervonWernherr I designed the method to remove the same mass from the earth. (related to the rationalization of the creation of energy, it is pseudo science so I am not trying to mention it in a hard science post). I only put the problems of mass increase in an extreme way to see if there is problem.

• And what happens to the people? Double mass would imho also cause double gravity :/ – Alexander von Wernherr Jan 31 '17 at 14:12
• Related: physics.stackexchange.com/a/304652/104551 - hope this will be accounted for in answers. – Mołot Jan 31 '17 at 14:15
• @AlexandervonWernherr I designed the method to remove the same mass from the earth. (related to the rationalization of the creation of energy, it is pseudo science so I am not trying to mention it in a hard science post). I only put the problems of mass increase in an extreme way to see if there is problem. – fairytale Jan 31 '17 at 14:20
• I'm pretty sure that to a first order approximation, you could consider this a two-body problem between the Sun and Earth. I'm not up to doing the math right now, but the mass of the Earth is 5.97e24 kg, and the mass of the Sun is 1.99e30 kg. Increasing the Earth's mass to 1.19e25 kg is unlikely to appreciably alter the gravitational balance between the Sun and the Earth. – a CVn Jan 31 '17 at 14:26
• @fairytale yep. And that increase would be felt as global warming all right. Q&A I linked, and comments to both said Q and accepted A, seem to answer most of side questions you may have. – Mołot Jan 31 '17 at 15:37

When you break conservation of energy on this sort of scale, a lot of physics goes out the window. We also need to clarify some things about the mass-energy equivalence and what we mean when we say "mass".

Cooking The Earth Very Rapidly

Assume 5.3748e41 J is generated by magic

That's a lot of energy. Let's consult one of my favorite lists, Orders of Magnitude (Energy). Mandatory reading for sci-fi authors.

• That's nearly what you need to blow apart the Sun! (6.87e41 J)
• That's the total energy output of the Sun, not just what falls on the Earth, for 10 MILLION YEARS!
• You could blow up the Earth 1 BILLION TIMES!
• That's the energy the Earth receives from the Sun over 1e17 years, and the Sun won't even last that long!
• It's 1e19 times greater than all the energy of all our fossil fuel reserves. (4e22 J).

Even if it happened over 10,000 years (1e4 years), the whole of human civilization, that's still 5e37 J/year or 1000 times the total annual energy output of the Sun. You're dumping all this energy onto the Earth, it's going to very rapidly cook the planet.

"But we'll just turn it into inert matter with magic!" It can't all be turned into inert matter, you have to use some of it to do work (ie. magic) which will have waste heat and cook the planet. The amount of energy you're talking about is so enormous that if even 1 billionth is left as waste heat (an impossibly efficient system) you still have enough to literally blow the Earth apart.

Either you have to suspend more basic physics, or you cook the planet. What am I saying, you're going to cook the planet anyway...

Where Do You Put All That Mass?

Let's assume we can magically convert all that waste energy into mass with 0 loss to keep the Earth from cooking like flash paper dropped into a volcano. Where do you put it?

Let's assume we can magic this into a very dense, very stable substance like Platinum at roughly 20 g/cm3, the numbers here are so large we don't need to be exact. The density of the Earth is roughly 5 g/cm3. So we're talking about a volume of platinum a quarter the size of the Earth! 3e11 km3.

You don't just sweep this under the rug. Spread evenly over the Earth's surface, 5e8 km2, it would reach to a height of 600 km!

If living on a solid mass of platinum isn't bad enough, the crushing weight would drive the continents down into the mantle and utterly reshape the surface of the Earth. The overall increase in density will cause the less dense Earth below to collapse inward as the Earth tries to resort so the more dense material on the surface sinks to the core. It would cause enormous heating, tectonic upheavals, and make the Earth even more uninhabitable than a 600 km deep slab of platinum.

But How Would It Effect Our Scorching Ball Of Platinum's Orbit?

That depends. Does the created magical energy have the momentum of the Earth? If so, it has no effect. If not, it depends on what momentum it does have.

E = mc2 is the mass-equivalence formula for rest mass or m0. Once things are moving relative to one another you have to factor in momentum. The full formula is E2 = m02c4 + p2c2 where p is the object's momentum. E = mc2 is what happens when the momentum is 0. Watch this Sixty Symbols video for more about that.

Momentum is important for the Earth's orbit, specifically angular momentum per unit mass. So long as the Earth's angular momentum per unit mass relative to the Sun remains constant, its orbit will not change.

If our energy is created with the same angular momentum as the Earth, there's no change in the Earth's angular momentum per unit mass relative to the Sun, and so there's no change in the Earth's orbit. This seems the most sensible way for magic to work, because you don't want to magic something into existence only to have it whizz off at 30km/s and burn up in the atmosphere or slam into the surface of the Earth (or the surface of the wizard).

...but this creates a new problem.

Changing The Earth's Spin?

In order to avoid changing the Earth's orbit, and having magicked items flying off in random directions at high velocities, they need to have the same angular momentum as the Earth relative to the Sun. But our wizard isn't just rotating around the Sun, they're also rotating around the Earth's poles at roughly 1600 km/h (ie. very fast)!

We want a world where wizards can wish whatever we want without worry it will whisk itself away because of the Earth's rotation and orbit. Put another way, we want the magicked energy to have an angular momentum relative to the wizard of 0.

But wait, there's more problems!

The wizard is rotating around the Earth at 1600 km/h, but he's doing so at an angle of 23 degrees relative to the orbit of the Earth around the Sun. The magicked mass-energy will pop into existence with the same momentum at an angle to the Earth's orbit around the Sun. What does that do?

This is where my knowledge of physics hits its limit.

• Can the magicked mass-energy both not change the Earth's angular momentum per unit mass relative to the Sun, AND not change the rotation of the Earth?
• This mass-energy will pop into existence on the surface of the Earth, not at its center of mass, how does that change things?

Open questions for whomever remains alive on your hellish ball of platinum. If you strip the magical elements out of this, it might make a good Physics.SE question.

How Much Would The Earth-Sun Barycenter Move?

Assuming the momentum balances out, it would double from about 450 km to 900 km. Not much, keeping in mind the Sun is 700,000 km in radius.

The Barycenter is defined like so: d / (1 + mS/mE). It's the distance between the two objects, divided by the mass ratio between the two objects (the Sun is so much more massive than the Earth that the 1 doesn't matter). If you double the mass of the Earth, you halve the Sun-Earth mass ratio, so you double the Barycenter distance.

To work it out... get the mass ratio for Sun and normal Earth.

• mS = 2e30 kg
• mE = 6e24 kg
• mS / mE = 3.33e5

(We can already see that doubling the Earth's mass isn't going to have much of an effect on such a large ratio.)

Divide the distance to the Sun, 1.5e8 km, by that and we get 450 km.

If you double the mass of the Earth to 1.2e25 kg you halve the mass-ratio to 1.66e5. 1.5e8 km / 1.66e5 is about 900 km.

A Closer, Wilder Moon

@Catalyst mentioned the Moon in the comments. There's gonna be some problems...

Orbital velocity is sqrt( Gm / h + r ) where...

• G = Gravitational Constant
• m = The mass of the Earth
• h = The height of the orbit
• r = Radius of the Earth

No need to plug in numbers, if we double the mass we increase the required orbital velocity by sqrt(2) or about 1.4 km/s. Another way to look at it is as if everything in orbit suddenly lost 30% of it's orbital velocity.

That's insufficient for the orbit's height, so the object will fall inwards gaining velocity and reducing its height. I can't do the orbital mechanics, maybe someone with Universe Sandbox can, but if the object is far enough out I reckon it will find a new orbit rather than scrape the Earth's atmosphere and crash. This new orbit would be far more eccentric than its original taking it very far out and then rather close.

At 400,000 km the Moon is probably fine, albeit in a wilder orbit. Anything in low Earth orbit, just 160 km to 2000 km, is probably going to scrape the atmosphere and come crashing down. Satellites in geosynchronous orbit at 35,000 km I'm going to guess will stay in orbit, but like the Moon will now be eccentric and no longer geostationary.

I'd say something about the tides, but nobody's going to survive on a flaming ball of liquid platinum and lava long enough to draw up a tide chart, and they won't have GPS to help them.

• Pretty good questions you pose in response to the posed concept! – kiltannen Feb 1 '17 at 3:02
• I concerned the first problem most (I worried that the energy will burn the whole planet), your answer really helps! Also, I didn't consider the problem of spin, your explanation is helpful. – fairytale Feb 1 '17 at 11:11
• Excellent answer. The magicked mass-energy can leave both rotation and orbit intact by distributing the mass equally on both sides of the planet. That way the rotational energy for the two halves is correct but the non-rotational part cancels out and leaves the orbit intact. – Tim B Feb 1 '17 at 12:18
• @TimB So when someone casts magic missile, some poor sap on the other side of the planet gets shot, too? Great. Or maybe it's like a recoilless rifle and the counterweight is high energy / low mass. What happens if magic use isn't evenly distributed around the planet (as it cannot be because of the way land is laid out)? You'll get clumps. – Schwern Feb 1 '17 at 20:07
• @MoĹ‚ot Easy, it doubles to 900 km. I'll put that into the answer. – Schwern Feb 1 '17 at 20:13