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I’m wondering how the length of a day and year would change on Earth if it was twice as big, but the same mass (less density)?

Also, would such a difference cause it to orbit closer or further from the sun, or the same?

I found plenty of people asking this same question, though with a more massive Earth, but my curiosity has been piqued.

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    $\begingroup$ Do you mean in an case where our earth with present physical properties (mass, angular momentum, density profile) was inflated to twice its size? in this case you would have to take care of angular momentum conservation while changing the moment of inertia. Comparable to figure skating on ice, when you stretch your arms while spinning. $\endgroup$ – DomDoe Jul 26 '18 at 14:48
  • $\begingroup$ Obviously this isn't how our Earth is, but let's assume (for argument's sake) that our planet had its core, and then a layer of nothing (just empty space -- not even "air" so-to-speak, but just empty space), and then had the outer crust. Now, our current day is 24 hours (using whole numbers for simplicity) and takes 365 days to orbit the sun. If, for some reason, that empty space expanded, thus making the planet larger in size and causing a less dense crust, would that alter the length of the day OR the length of the year? Also, would it alter the distance Earth orbits from the sun? $\endgroup$ – Jonathan Plumb Jul 26 '18 at 15:13
  • $\begingroup$ What answers did the other questions receive? What did you learn from them? $\endgroup$ – sammy gerbil Jul 27 '18 at 20:25
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As mentioned in the previous answer, the length of a day could be anything for any mass/density/size of Earth. It only matters how much angular momentum Earth has, and then you can calculate the time period of one rotation based on its shape and composition. But historically, if the processes which gave Earth its angular momentum gave the same amount of angular momentum to a larger or more dense planet, that heavier planet would rotate more slowly due to a higher moment of inertia. If the density were simply doubled, then the moment of inertia would double, which would halve the angular velocity for the same angular momentum (doubling the length of a day).

As for the length of a year, the orbital period is to first order determined only by the distance to the sun and the mass of the sun (which would still be much larger than the new earth’s mass). Taken from the Wikipedia page, $$ T=2\pi\sqrt{\frac{r^3}{\mu}}. $$ Thus, the year would be the same duration, provided the radius of orbit remained the same. But for a twice as heavy Earth, this would again double the angular momentum of the system. If we once again assume a constant angular momentum, then doubling the Earth’s mass would require a reduction of the orbital radius by a factor of four (since angular momentum $L=I\omega=m_{\rm e}\sqrt{\mu r}$, where $I$ is moment of inertia, $m_{\rm e}$ is Earth mass, $\mu$ is gravitational parameter, and $r$ is radius). So in addition to us roasting to death, the length of a year would decrease by a factor of 8.

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  • $\begingroup$ Awesome. Thanks also for your feedback! So, a greater surface area of the planet would not really mean anything, if the mass remained the same? My buddy was saying the greater surface area (if Earth suddenly got larger, but the same mass) would give more matter for the sun's gravity to act upon, thus "pulling" Earth harder towards it creating a faster orbit. So, that's not true at all then? (I also suppose that if the mass remained constant, there wouldn't be "more" matter, right?) $\endgroup$ – Jonathan Plumb Jul 26 '18 at 16:01
  • $\begingroup$ @JonathanPlumb it all depends on what the constraints are on the angular momentum, L. If L stays constant when Earth’s mass suddenly increases, then as I mentioned above, the orbital radius and length of year both decrease. But if there are no constraints on L, the problem is under-specified, and the answer could be anything. Consider, for example, the gas giants in the solar system. They have a much larger mass and diameter compared to Earth, but they orbit farther away and with a longer year. This is totally fine, since orbital angular momentum is different in each case. $\endgroup$ – Gilbert Jul 27 '18 at 13:14
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The length of one day is just the amount of time it takes the earth to spin around once. Depending on what I want to do, I can spin a bowling ball as fast, slower, or faster than a tennis ball in my other hand. The masses and sizes don't particularly constrain anything.

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  • $\begingroup$ Thanks for the response. So, the length of the day would be unaltered, but what about the length of a year? A planet twice as large with the same mass would be less dense, but if the mass remains unchanged would the solar orbit remain the same? In looking at formulas on my own, it seems the calculation takes into account the gravitational constant, the mass of the satellite, and the distance between the center points of the two objects. I'm just not fluent enough to really understand these formulas as well as I want, but your answers are helping me better understand them. $\endgroup$ – Jonathan Plumb Jul 26 '18 at 15:09
  • $\begingroup$ @JonathanPlumb This answer is not saying it would be unaltered. It is saying the mass does not determine anything about the rate of rotation. You can have any mass and virtually any rotation speed (limited by other laws of physics like relativity or the material the object is made of). So your question is not answerable in general. $\endgroup$ – Aaron Stevens Jul 26 '18 at 15:15
  • $\begingroup$ @JonathanPlumb In terms of math, you could say that you have too many unknowns and too few equations. There are lots of ways to constrain the answer but you'd have to be more specific (i.e., add in some other facts) in order for the problem to have a unique answer. For example, you could say that the earth was changing shape by inflating like a balloon, in which case the day would get longer and the year would stay the same. (Conservation of angular momentum would slow the earth's rotation rate down as its moment of inertia increased.) $\endgroup$ – Display Name Jul 26 '18 at 15:27
  • $\begingroup$ Gotcha! Thanks for the feedback! Yeah, I was (in my head) picturing something like your balloon inflation example. I had someone telling me that if the earth expanded as such, it would increase the length of the year (over time -- so, our 365 day year could potentially end up being a 370 day year, etc.). I was trying to look into this on my own, plugging in numbers into formulas, but I definitely am not proficient enough to understand what variables I should even be looking at... $\endgroup$ – Jonathan Plumb Jul 26 '18 at 15:31

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