Yeah, but that means its orbit is now hyperbolic, so it won't be coming back.

I figured. But you can see it now with a high magnification telescope with the clearest optics. (PS) Nice going with the "mercury's perihelion" document, it gave me a lot of great understandings on why mercury shifts. Although I already knew about this because gravity is one of my favorite things to learn about, but it was still great!

Now I have the code do variable time steps, such that the distance moved in each step is always some set fraction (typically less than 10-6) of the distance from whatever it's orbiting, which turned what would be a several weeks long computation for the comet into something just over an hour.  I ran it a few times and as I turn up the precision the solution appears to converge to around 3.5 arcseconds per orbit!  Much more than Mercury (about 0.1 arcsecond per orbit), but still so small that you wouldn't really be able to see it in an orbital overview, and I imagine other effects like outgassing would mask it.

Beautiful. Thank you for examining this. Indeed is a small angle (even if one order of magnitude above that of Mercury's). That means the aphelion of the comet moves nearly one million kilometers due to the effects of relativity! but also when you try to draw it is just absurdly small.
It's interesting to note that it is still a very bad laboratory for relativity when compared to Mercury. Even if for each revolution the effect is stronger for Comet Lovejoy, the fact that Mercury has a shorter orbital period means that for each arcminute of perihelion precession we need to wait just 140 years for Mercury compared to the 10.663 years we would need for Comet Lovejoy to see the same (it would not even survive the 17 revolutions needed to do that and also the effect would surely be masked by outgassing, planetary perturbations and YORP). I'm starting to think that Mercury is in fact the perfect place in the entire Solar System to observe this effect or at least the place were it is more evident. Vulcanoids should be great examples but since they do not exist... well we don't have the example at all. Maybe 2007 EB₂₆ could be a better candidate than Mercury since it has the smallest semi-major axis (after Mercury) and its perihelion is inside Mercury's orbit. 2005 HC₄ should also be a nice place to easily observe this effect since it has the smallest perihelion of any asteroid (but has a relatively large orbital period compared to 2007 EB₂₆).


By the way. The planet Vulcan hypothesis was completely debunked because of this relativistic effect. People thought that an unseen planet was the cause of the anomalous precession of Mercury. So I'm going to add that to the list just to make clear how important is this explanation for several historical aspects of Solar System dynamics.

Wow, very good posrt.
Created accout to say thanks

What tool did you use to plot?
(Ignore prev post, forum bugged and printed only special char)

Thanks, HarmonicGF2!  I used Matplotlib to create the plots.

FastFourierTransform, one of the things that really struck me is how precise astronomical observations are, to be able to notice the anomalous precession of Mercury by the mid 1800s.  0.1 arsec per orbit amounts to shifting the aphelion by only 34km!  Of course it helps that it adds up over time, but that's still only 6 Mercury radii per century.

 My personal motivation is that even if I'm learning a lot I lack the most basic intuition for this part of astrophysics and would like to learn from you (ahem, Watsisname please), very smart and educated people that I know I can learn so much from.

I'm pretty sure you can lock this topic from anyone except watsisname if you want to.
But if you do want to, you should ask for his permission first!

It's FFT's thread so he can do as he likes.   

As for me there are many topics in the list that I know little about and would need to do a lot of research to understand them.  There are others I understand fairly well but still must put a lot of thought into how to explain them.  And I don't think you all would want to read me talking about everything anyway.  Actually I think it would be neat if other people grapple with a few and "present" what they learned, as it were.  I learned a great deal myself while working through the details for Mercury's precession!

It's FFT's thread so he can do as he likes.  

Yep, that's why I said "if he wants to" and ONLY if he wants to.

Guys, Is not about Watsisname nor me. You are all welcome to contribute. The great thing of SE's community is that even if we are all ignorants in one way or another, there's a lot of people that embraces the method, wants to learn, is able to make clever self-criticism, search for sources and wants to learn toghether. I mean, there are a lot of places where people want to learn toghether, but it can end up been a giant confusing nonesense with anti-educational potential. Here I feel there's enought people that is able to pinpoint the errors in reasoning of the others as to truly help in the overall learning.

Watsisname is a total reference here in terms of knowledge and I'm always glad to hear his opinion in every subject. But this thread is open to all of you. Even if we don't make perfect and correct contributions I know we as a community can correct that and make sense of the ideas while learning about our preconceived notions without reassuring them. Monolgues are great, I would totally read a physics book written by Watsisname, but we are in a forum and the idea is to bring questions (even if they sound stupid) so that some can learn where the focus of explanation should be placed and others can learn the subject. Also I know a lot of people here that can explain with high detail and justified confidence some of the topics I've added to the index. We all have to learn from each other.

I would like to talk about a bunch of things about our solar system but it seems I always get things wrong and I don't want to do that here.

Man, there is absolutely no problem in getting things wrong if you are able to learn and are open to criticism presenting your ideas with humility. You can talk about anything you want, at least I will not be judging you by this at all (I think no one will do that here). Physics in general is a hard topic, we have all made huge efforts and become desperate learning it, and we all  had strange ideas that we have embraced without any realization of how ignorant we really are. This is not a finished process for anyone, even for people that work in science. You always get things wrong. The important thing is to be able to detect the errors, the misconceptions, the biases, and ask yourself for more rigour each time. Even when you think you have archived high knowledge on a certain subject, science and nature are always there to tell you that you are just a presumptuous ignorant so don't be ashamed about that, I bet even Watsisname knows how it feels

Sorry if maybe I've been a little paternalistic with you, but It makes me sad to hear people that is afraid to talk and ask questions (I know is not your case) just because others have shown how the correct arguments and evidence are handled. This is not a competition of smartness, there is no losing here. Just learning about the errors and about other people. So keep thinking, talking and asking because that's the way (in my opinion). So as a recommendation (since you are very interested in these topics) just try to be clearer and question everything (even what you are going to say before saying it).

Guys, Is not about Watsisname nor me. You are all welcome to contribute. The great thing of SE's community is that even if we are all ignorants in one way or another, there's a lot of people that embraces the method, wants to learn, is able to make clever self-criticism, search for sources and wants to learn toghether. I mean, there are a lot of places where people want to learn toghether, but it can end up been a giant confusing nonesense with anti-educational potential. Here I feel there's enought people that is able to pinpoint the errors in reasoning of the others as to truly help in the overall learning.

Watsisname is a total reference here in terms of knowledge and I'm always glad to hear his opinion in every subject. But this thread is open to all of you. Even if we don't make perfect and correct contributions I know we as a community can correct that and make sense of the ideas while learning about our preconceived notions without reassuring them. Monolgues are great, I would totally read a physics book written by Watsisname, but we are in a forum and the idea is to bring questions (even if they sound stupid) so that some can learn where the focus of explanation should be placed and others can learn the subject. Also I know a lot of people here that can explain with high detail and justified confidence some of the topics I've added to the index. We all have to learn from each other.

I would like to talk about a bunch of things about our solar system but it seems I always get things wrong and I don't want to do that here.

Man, there is absolutely no problem in getting things wrong if you are able to learn and are open to criticism presenting your ideas with humility. You can talk about anything you want, at least I will not be judging you by this at all (I think no one will do that here). Physics in general is a hard topic, we have all made huge efforts and become desperate learning it, and we all  had strange ideas that we have embraced without any realization of how ignorant we really are. This is not a finished process for anyone, even for people that work in science. You always get things wrong. The important thing is to be able to detect the errors, the misconceptions, the biases, and ask yourself for more rigour each time. Even when you think you have archived high knowledge on a certain subject, science and nature are always there to tell you that you are just a presumptuous ignorant so don't be ashamed about that, I bet even Watsisname knows how it feels

Sorry if maybe I've been a little paternalistic with you, but It makes me sad to hear people that is afraid to talk and ask questions (I know is not your case) just because others have shown how the correct arguments and evidence are handled. This is not a competition of smartness, there is no losing here. Just learning about the errors and about other people. So keep thinking, talking and asking because that's the way (in my opinion). So as a recommendation (since you are very interested in these topics) just try to be clearer and question everything (even what you are going to say before saying it).

Thanks for that FFT! and sorry as well, I just thought you might have wanted the best teacher on the forum to help you to figure out about celestial mechanics.

How exactly was Neptune's existence theoreticaly predicted using only Newtons's Laws and Uranus perturbations?

Working on it now!

THE NEPTUNE PREDICTION
                    BY PROPULSION DISK

THE FIRST THEORY OF NEPTUNE'S EXISTENSE WAS MADE BY GALILEO GALILEI, ON DECEMBER 28, 1612, PROBABLY ONE OF THE MOST WELL KNOWN ASTRONOMERS, WAS ONCE LOOKING AT THE NIGHT SKY, SEARCHING FOR NEW PLANETS. THAT NIGHT, HE FOUND SOMETHING VERY INTERESTING, IT WAS WHAT SEEMED TO BE A STAR THAT DIDN'T TWINKLE, AND THEREFORE GALLIEO THOUGHT HE HAD FOUND A PLANET! HE LOOKED AT AGAIN A YEAR LATER TO FIGURE OUT THAT IT WAS STARTING TO GO RETROGRADE AND THEN PROVED THAT IT WAS INDEED A PLANET, HE WROTE IT IN HIS DRAWINGS OF THE SOLAR SYSTEM WITH NO NAME AND SHOWED IT TO THE PUBLIC, ASTRONOMERS KEPT LOOKING FOR NEPTUNE AND EVENTUALLY FOUND OUT THAT IT WAS JUST A STAR THAT WAS IN CONJUCTION WITH JUPITER WITCH IS WHAT MADE IT LOOK LIKE IT WAS MOVING TO GALLIEO, AND HIS SMALL TELESCOPE, AND WAS NEVER FOUGHT OF AGAIN INTILL 1821.

Here's the part you want to know FFT!
IN 1821, ASTRONOMER ALEXIS BOUVARD WAS OBSERVING URANUS, HE KEPT TRACK OF IT FOR MONTHS AND WROTE ON HIS TABLES THAT IT WAS CONSTANTLY GETTING PERTURBED BY WHAT COULD ONLY BE SOME UNKNOWN BODY, CAUSING HIM TO BELIEVE THE EXISTENCE OF NEPTUNE.

IN 1844, JOHN COUCH ADAMS WAS FACINATED BY NEPTUNE'S MYSTERIES, AND WANTED TO PROVE IT'S EXISTENCE, HE GATHERED A LOT OF DATA ON THE SUBJECT BY HIMSELF, BUT MAINLY TALKED TO OTHER FAMOUS ASTRONOMERS TO PROVE NEPTUNE'S EXISTENCE INTILL 1846.

And here's the part about Newton's laws.
THE FINAL TIME NEPTUNE WAS SEARCHED FOR WAS WHEN URBAIN DE VERRIER STARTED RESEARCHING IT, UNAWARE OF JOHN'S OBSERVATIONS, HE MADE CALCULATIONS OF WHERE NEPTUNE WOULD BE BY FINDING URANUS, AND WATCHING WHERE IT WAS BEING PULLED, AND HOW MUCH VIA NEWTON'S LAWS, THEN HE WENT TO MAKE A NOTE TO ASTRONOMER JOHANN GOTTFRIED GALLE THAT SAID TO USE THE REFRACTOR AT THE BERLIN OBSERVITORY AND TO USE HIS CALCULATIONS TO FIND NEPTUNE'S POSITION, WHEN GALLE GOT THE NOTE HE LOOKED FOR NEPTUNE IN THE NIGHT SKY AND FOUND IT! 1 DEGREE OF WHERE LE VERRIER FOUGHT IT WOULD BE.

And that's about it! Hope this answered your question FFT, and sorry if I missed some things, I was mainly trying to just get your question answered!

This is good writing, but please don't post it in all caps!

Also I wonder if you can expand on the details of Le Verrier's calculation.  Specifically, how with Newton's Laws was the location of Neptune pinned down with such accuracy?  I think this is what FFT (and I as well!) are most interested in.  It's not necessary to reproduce the entire calculation, but it would be nice to be able to understand how it works and potentially reproduce it for a general case of one planet perturbing another.

This is good writing, but please don't post it in all caps!

Yeah sorry about that. 

Also I wonder if you can expand on the details of Le Verrier's calculation.  Specifically, how with Newton's Laws was the location of Neptune pinned down with such accuracy?  I think this is what FFT (and I as well!) are most interested in.  It's not necessary to reproduce the entire calculation, but it would be nice to be able to understand how it works and potentially reproduce it for a general case of one planet perturbing another.

I can in a way, as I said, the calculations where made via Le Verrier's observations over the series of a few months to figure out Uranus' perturbation stats, and he used what he knew about gravity to figure out how big Neptune should be, but the problem was, Neptune could be very big and far away from Uranus, or small but close to Uranus, to be honest, he didn't even know that himself because these equations were imperfect, but sense he couldn't see it, then it was probably a size that could have enough gravity to pull on Uranus, but could not be seen to the unaided eye, so he actually had to go with his gut a little on it.

Can you show the equations, and how he used them to come up with a predicted position to within 1 degree of angle on the sky?