Expert: Courtney Seligman - 2/26/2011

Question
QUESTION: Dear expert,


The Earth has a sidereal rotation t=86164.09 seconds
and when Earth makes 360° around its axis, it also moves making an angle α = 0.9829° around its orbit around the Sun.


So, logically if Earth-Sun distance were increased to infinity, Earth's revolution period would be 0 because Sun's gravity would be 0, thus α would be 0 degrees,

and in this case, what would be Earth's NEW sideral rotation period t'?

Since Sun's gravity affects the lenght of the day because it increases Earth's kinetic energy, i propose the following formula:

t'= t/sqrt[cosine α]

in our case t'= 86164.09/sqrt[cosine 0.982956]= 86170.43s

THUS, according to my formula, Earth sidereal rotation period would increase to 6 seconds if it were outside Sun's influence.


Could you tell me if the result and/or my formula are accurate?



         Sincerely

ANSWER: "The Earth has a sidereal rotation t=86164.09 seconds
and when Earth makes 360° around its axis, it also moves making an angle α = 0.9829° around its orbit around the Sun."

That is correct.

"So, logically if Earth-Sun distance were increased to infinity, Earth's revolution period would be 0 because Sun's gravity would be 0, thus α would be 0 degrees,"

That is sort of right, but wrongly stated. The Earth's rate of revolution around the Sun would be zero degrees per (whatever period), but the period would become infinite (insofar as it still had any meaning at all).

"and in this case, what would be Earth's NEW sideral rotation period t'?"

There is nothing in the previous statements which would require any change in the rotation period. It is conceivable, depending upon HOW the change in our orbit was accomplished, that the rotation period would change; but there is no reason for it to change, simply because the orbital period changed. The two are completely unrelated.

"Since Sun's gravity affects the lenght of the day because it increases Earth's kinetic energy, i propose the following formula:

t'= t/sqrt[cosine α]

in our case t'= 86164.09/sqrt[cosine 0.982956]= 86170.43s

THUS, according to my formula, Earth sidereal rotation period would increase to 6 seconds if it were outside Sun's influence."

The Sun's gravity "increases" the length of the day not by changing the kinetic energy of the Earth but by making the Earth go around the Sun, which causes the 0.9-something degree per day orbital motion you mention early on. It does not directly affect the "energy of the rotation", or some similar term which might somehow affect our rate of rotation. I haven't investigated the formulae you used, because they aren't relevant to the problem you attack at the end -- namely, how does the length of the day change? As stated above, the rotation rate should stay the same, unless whatever force removed us from the Sun did so in a way which also affected the rotation rate (in which case, it would be impossible to say what the rotation rate would be). And as far as the length of the day is concerned, that would either become meaningless, since the Sun would just be a distant star, or if you still thought of once around in the sky for the Sun being a day, then the fact that we wouldn't be going AROUND it would eliminate the difference between the day and rotation period, so the length of the day would become the same as the rotation period (which is about 236 seconds less than the current day).

So to summarize, moving us away from the Sun wouldn't, in and of itself, change our rotation period at all; but it would make the day and rotation period the same, thereby shortening the day by nearly 4 minutes.

Although it's not directly related to the question as asked, you might find it useful to refer to my web page on Rotation Period and Day Length, at http://cseligman.com/text/sky/rotationvsday.htm. If you do, you'll see that for planets which are very far from the Sun, the rotation period and day length are so nearly identical that it is hard to tell any difference between them. The same thing would happen, but even more so, in the scenario you propose.

---------- FOLLOW-UP ----------

QUESTION: Thank you for your answer!

I thought since the Moon's gravity can affect Earth spinning rate by creating a torque, i deduced the Sun has the same effect on Earth.

Could you tell me why Sun can't create a torque on Earth's spin like Moon does?


         Sincerely

Answer
The Sun does create a torque like the Moon does, but the cause of the torque is the action of the tides on the body of the Earth. Since lunar tides are bigger than solar tides (by about a factor of two), if the Sun were very far away tides would be a little smaller, and tidal slowing would be, as well. Right now tidal slowing increases the length of the rotation period by a couple of thousandths of a second per day, each century. If the Sun were too far away to count, then this slow change would be closer to one thousandth of a second per day, per century.