Upload it! Just make sure either your upload the latex or pdf format of the mathematical formulas.

Dear Ada,

You say that you require an uncomplicated answer to this question.  There is a very involved solution to this problem using a second order ordinary differential equation (ODE) for elliptical orbits, which I can provide, and a more simple, high-school, version, each resulting in Kepler's Third Law.

If you would like to see the full worked solutions to both, then please do let me know.  The simpler method goes something like this.  We assume that the orbit of Earth about the Sun is more or less circular.  The force acting between Earth and the Sun is F(r) = (mv^2)/r = GMm/(r^2).  Simplifying this equation yields: v = Sqrt[GM/r], where r is the 'constant' radial vector joining Earth to the Sun.  Because G, M and r are all constants, effectively, then v is a constant and is equal to the velocity of Earth at any point in its orbit.  Substituting in the usual values for G, M and r, yields a velocity of around 29,800 metres per second.  Also, by rearranging the above formula for r = GM/(v^2), or r = GMv^-2, you can determine r, but this will be essentially constant (equivalent of 93,000,000 miles) for Earth-Sun distance.

For a generic planet-star (or even a comet-star) elliptical radial vector, r, at any time t, (whereby r is now variable in line with tracing out a more elliptical path) then I refer to what I said at the beginning of this message, in which you would need to derive a second order ODE in terms of r and t, and then convert it to its alternative form in terms of u (velocity) and theta (angle traced out by r), but this is quite involved.  The outcome is that you will obtain a radial vector of the form of:

u = a + bcos(theta)

where the velocity u = 1/r.  The radial vector is therefore:

r = 1/(a + bcos(theta))

You can then go further with the working, eventually resulting in Kepler's Third Law, again, but you need to make some assumptions about the (a + bcos(theta))^2 term that you get in the denomenator of the final integral.

Cheers,

Stephen

Hi Stephen thank you for your answer! Could you please go a little bit more in details with the ODE method? I would like a more precise equation in terms of r and t because my t value will be quite large (up to 6500 years). Also do you mind explaining why velocity u = 1/r? I don't quite get it. Again thank you so much! : )

Thank you Vikash for your reply! I will upload my workout later but it is very messy.

Ada

I guess this is simpler than thought. Earth's orbit is an ellipse with an eccentricity, e = 0.016708, semi-major axis, a = 149598023 km. One could use the formula to obtain semi-minor axis, b.  

e = sqrt {1 - (b^2/a^2)}.

And then substitute the values in the standard equation of an ellipse to obtain

y = +- (b/a) sqrt {a^2 - x^2}

So at any given reference time, if one knows x or y, the other coordinate can be calculated with little effort.

Hi Ada,

I shall upload my 4-page worked solution for you in a little while, after I have checked through it for the usual mistakes that I make, so do look out for it later today.  I also used to have a problem regarding the fact that u = 1/r, but this relationship simply means that the velocity of a body at any point on any elliptical orbit is simply inversely proportional to the distance, r, of the body from the main focus, S, of the ellipse.  It is similar to Newton's inverse square law, F = k/(r^2), except that the former refers to velocity.  I don't know if it matters that your time value is 6500 years, because Earth's orbit is so nearly circular that, after 6500 years, Earth will presumably be at more or less the same point in its orbit as it would be after 1 year, say.  There may be some slight discrepancy, however, due to the slight elliptical nature of our planet's orbit - and gravitational forces of course - but these effects are much more noticeable with bodies much closer to their parent star - such as Mercury orbiting the Sun - due to the more extreme gravitational forces there that are explained by Einstein's Theory of General Relativity.  In such a case, you would then notice a big difference, namely the precession in Mercury's orbit of 43 arcseconds per century.  I could also help you in this respect as well, as I do have my General Relativity notes from University.  Let me know if you would like further details along with the theory indicating Mercury's precession, as they might just be useful.

Stephen

 Thank you so much Stephen! It will help a lot. I would love to know more about general relativity and Mercury's precession if it is not too much trouble for you. The investigation I am doing at the moment is actually about the Earth's axial precession so it will be interesting to see another form of precession on a different planet. 

Ada 

Thank you Vikash! But the only problem with the function is that, t is the only variable that I know of. I am trying to find the x and y (therefore r) in terms of t. 

By the way, attached below is my workout. (I tried so many ways of uploading. Hope it works) I rewrote it so it is less messy now. Hopefully I didn't make any mistakes. 

Ada 

Dear Ada,

This is a stimulating question. In addition to the excellent answers already provided I would like to add that the actual orbital calculations are rather complex because the Earth's motion is perturbed by other planets in the Solar system. The problem was approached first by Le Verrier in 1856. Later on, Milankovitch used the corresponding solutions in the studies of the astronomical origin of of ice ages.

The best formal analysis of the orbits with perturbing forces was provided by Henri Poincare in his seminal work on celestial mechanics. The modern and highly accessible review of the subject is presented by F. Brechenmacher

https://arxiv.org/abs/1305.2689

For the details of calculations of the three body problem, I would also like to suggest the recent paper by Mandling

https://arxiv.org/abs/1308.0607

A problem of the calculation of the Earth's orbit gets even more complicated when we consider the very long time scale as discussed in the illuminating study of Laskar (1999)

http://www.astro.gla.ac.uk/honours/labs/solar_system/papers/laskar.pdf

In conclusion, a realistic evaluation of the orbit of our planet is still the area of an active research in the theory of dynamical systems as well as the in statistical mechanics.

Sincerely, Janusz

Thank you Janusz! I really appreciate the great resources and will defintely have a look at all three of them! : ) But I don't think it would be a good idea to include perturbation because it is way too complicated. Hopefully it wouldn't affect my results significantly.  

Ada 

Three body nonlinear dynamics and resulting chaos is an integral part of orbital mechanics, agreed. However earth's trajectory around the Sun as a function of time is itself an interesting problem as it offers to test the college level physics and mathematics for a real world problem. And when other perturbations and non-linearity are added it makes the problem difficult and challenging both.

I am working on it and will try to post something about it soon.

That's awesome Vikash! I am looking forward to your post : ) 

Dear Ada,

Please find a set of handwritten notes on calculating the orbital trajectory of a planet.  I hope that you can follow them.  Also, I attach a four-page solution of how the integral at the end of my afore mentioned document is achieved.  Please accept my apologies if you find my notes somewhat daunting - the mathematical working is very tedious and involved, and it took me some time to go back through my notes to try to eliminate errors.  Do look out for further mistakes, as I am sure that there will doubtlessly be a number of them.

Finally, I shall scan my General Relativity notes later this evening and send them on to you afterwards, as I am a little busy this afternoon.

I look forward to hearing from you in due course, and may you find my notes helpful.

Kind regards,

Stephen

Thank you very much Stephen! It is indeed very involved. I've learned a lot from it : ) Your equation solves this problem perfectly. Although the transformation to equation (3) and then from (3) to (4) are extremely difficult for me to understand. I am doing many external readings at the moment so hopefully I will be able to understand them soon. 

All your help is much appreciated. I will definitely get back to you when I have made some progress on this topic. Please just send your general relativity notes whenever you have time. 

Thank you so much,

Ada 

Thank you Ada for the response. Certainly at this stage you do not have to include perturbations. As far as the elementary aspects of the problem are concerned, I believe that one of the best expositions was provided by Richard Feynman in the famous "lost lecture", for the description please see

https://en.m.wikipedia.org/wiki/Feynman%27s_Lost_Lecture

You can find some further details in

http://calteches.library.caltech.edu/563/2/Goodstein.pdf

The method of Feynman is also summarized by Hall and Higson (1998)

http://www.cems.uvm.edu/~tlakoba/AppliedUGMath/planets_HallHigson.pdf

The approach proposed by Richard Feynman is useful as the primer before using the formal analytical methods.

Kind regards, Janusz

Thank you for your response, Ada.

I am glad that you found my notes useful, albeit a little daunting perhaps.  Just one more thing...I think there are indeed errors in my integral calculation within the typed notes, especially towards the end, which I sent to you earlier today.  Hopefully, you will be able to work through them and figure out what to do, as required.  I shall send you the scanned copies of the relevant section of my GR notes later on.

Kind regards,

Stephen

Dear Ada,

Please find attached around 16 pages from my General Relativity notes.  Of relevance to you will be section 3.2 starting on page 42, which refers to Equations Governing Orbits.  This also covers elliptic orbits in the Newtonian case, but then goes further in the relativistic sense.  Within this context, you will undoubtedly find the material relevant to your own work/studies, as orbital precession is also considered, with an example concerning Mercury's path as it travels around the Sun.

Hope this helps.

Regards,

Stephen

Dear Stephen, 

Thank you so much! Your notes are great help for me. I haven't got around to read the intergral calculation yet but thank you for reminding me. I will try to use the formula in the first document.  

Kind regards, 

Ada 

Thank You Janusz! Feyman's method is very intruiging. Trigonometry could be a good way of approching this problem. I wonder if I am able to derive an equation from it. 

Kind regards, 

Ada

Thank you for your response Ada. The method of Feynman was intended as an introduction to the subject without the need to use the advanced mathematical methods. For the analytical aspects of the problem I would suggest to consult the book "Mechanics" by Landau and Lifshitz.

The entire Landau and Lifshitz course is available at

https://www.reddit.com/r/Physics/comments/1dmxq7/our_beloved_landaulifshitz_books_are_available/

You can download the book Mechanics (position number 1 on the list) or go directly to the link

https://ia800307.us.archive.org/11/items/Mechanics_541/LandauLifshitz-Mechanics.pdf

In paragraph 14 on page 30 (Motion in a central field) you will find the derivation which is both elegant and powerful. Equations (14.6) and (14.7) are the general solution of your problem in the simplest case.

It is good starting point before going to the more complex problems including various perturbations of both Newtonian and relativistic character.

Kind regards,

Janusz

Thank you so much Janusz! This is a very well-explained book and I really like  its appraoches of the circular motions. It's great help to my understanding: ) All I need to do should be finding a suitable equation for (E-U(r)) in terms of r. 

Reagrds,

Ada

Thank you for the comments on the suggested book Ada. The answer to your question is in the next chapter (15. Kepler's problem). The final results are given in eq. (15.5), (15.10) and (15.11). I agree with your opinion concerning the clarity of the presentation in Landau & Lifshitz course.

If you are interested in the further details I can provide some additional references about the more complex problems mentioned by George.

Regards,

Janusz

Thank you George! Although I would love to consider the relativistic point of view, it is way to complex at my current level of understanding in mathematics and astrophysics. I wish I will be able to do it in the future.

Regards,

Ada 

Thank you Janusz! I am wondering if (E-U(r)) could possibly be considered as gravitational kinetic energy if I think of the sum of the Earth energy as (gravitational potential energy+gravitaional kinetic energy)?

Regards,

Ada

Thank you for this question Ada. You are correct, according to eq. (14.4) the quantity (E-U(r)) is the kinetic energy of a point in a central field. It is just sufficient to describe it as a "kinetic energy" instead of more wordy "gravitational kinetic energy".

Regards,

Janusz

Thank you Janusz! Now I know how this all works. 

Regards,

Ada