DERIVING the Orbit of Our Home Planet from Newton's Law of Gravity: Physics Mini Lesson

In one of the most remarkable discoveries of all time, Newton showed how to derive the orbit of our planet around the Sun. I'll walk you through how step by step. Get the notes for free here: courses.physic...
Get the PROBLEM SHEET, the SOLUTIONS, and the ANIMATIONS: www.physicswit...
Understanding the orbits from the effective potential: • The Trick that Makes U...
Learn the shortcut to deriving the orbits: • The Shortcut that Lets...
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About physics mini lessons:
In these intermediate-level physics lessons, I'll try to give you a self-contained introduction to some fascinating physics topics. If you're just getting started on your physics journey, you might not understand every single detail in every video---that's totally fine! What I'm really hoping is that you'll be inspired to go off and keep learning more on your own.
About me:
I’m Dr. Elliot Schneider. I love physics, and I want to help others learn (and learn to love) physics, too. Whether you’re a beginner just starting out with your physics studies, a more advanced student, or a lifelong learner, I hope you’ll find resources here that enable you to deepen your understanding of the laws of nature. For more cool physics stuff, visit me at www.physicswit....

[@dwaynep6174]

I’m a freshman physics major and this channel gives great insight into things I may be learning in the next few years. Love it!

[@PhysicswithElliot]

Thanks Dwayne, glad to hear it!

[@williamsutter2152]

I'm a maths and stats major, haven't done any physics courses, but for a C++ computing course I took I created C++ programs for modelling the trajectory of the Moon and Earth. I derived my equations of motion using the Euler-Lagrange equations, which I taught myself in high school. Back then the great resource of your videos didn't exist, the best available were MIT open courses and Susskind's lectures at Stanford.

[@maxst6647]

My dear sir, Right at 13:00 you leave out the crucial step in how to proceed from a DE in r'(\theta) to one in u'(\theta). You are therefore doing the same thing that articles and texts in mathematics do: "Clearly it follows..." or "We will leave it to the reader to show...". This is not pedagogically sound nor is it convincing. Without justifying the step at 13:00, you have proved exactly nothing. It's not that your conclusion is wrong, but rather that you have not proved it. Furthermore, you cannot treat dr/dt (or any other derivative) as a fraction and claim that the "dt's cancel"; this is just pure rubbish.

[@bastianherrera6408]

The clarity with which you explain these concepts is outstanding. Your videos are awesome, hope more people appreciates the quality of your content. Keep it up!

[@PhysicswithElliot]

Thanks Bastián!

[@vassilioskaxiras5724]

I never knew physics could be so "grounded". Keep up the good work!

[@PhysicswithElliot]

Grounded? I was going for "out of this world"

[@macrosense]

How would you know the mass of the sun, the earth, or the ratio between them?

[@chrisr9320]

Great video, just an addition/clarification to what you said about the perihelion precession of Mercury: the precession itself can be derived from Newtonian gravity if you take into account there are other planets that influence Mercury's orbit, plus other smaller perturbations. You will need General Relativity to get the right amount of arcseconds of precession, though (575"/century instead of the 531"/century predicted by Newtonian perturbation calculus, the famous missing 43")

[@PhysicswithElliot]

Thanks Chris! Yes, though what I was trying to say here is that even for a single planet orbiting a star the orbit will precess in a way that's not predicted by Newton's law of gravity

[@jmmahony]

Twice you said hyperbolic orbits were like a comet going by the sun. This is generally not true: comets are generally in elliptical orbits, but often in extremely eccentric orbits, where the part of the path we can measure (when it's close enough to the inner solar system to be observed from earth) may be indistinguishable from a parabola (the borderline case between a closed elliptical orbit and an open hyperbolic trajectory). But still these are thought to generally be on elliptical orbits, ie, still bound to the sun, in a closed orbit. Only very recently did we finally spot an object "'Oumuamua" that was clearly an extrasolar object on an open hyperbolic trajectory, veering by the sun on its way through the solar system. And then we found a second, but that's been it, so far.

[@PhysicswithElliot]

Very cool I hadn't heard of that object! Indeed comets can certainly follow elliptical orbits. I think I mentioned Halley's Comet in this video

[@carultch]

A hyperbolic "orbit" would be like a rogue comet that isn't bound to the sun, that finds its way into the solar system, only to escape and for us to never see it again.

[@TIO540S1]

The work you put into these is amazing. That you add notes and problems makes it even more so. I hope you can keep it up! Thank you!

[@PhysicswithElliot]

Thank you Rob!

[@JulieanGalak]

11:37 - am I missing something here? Every calc book I've ever read emphasizes that "d/dt" (or whatever variable) is not really a fraction, it's an operator. So how can you "cancel" the "dt"s here? Am I missing something fundamental about the math involved?

[@gmncnr]

Welcome to physics

[@Darthvanger]

Just how genius Newton was to come up with all of these by himself in 17th century. Today ppl take the orbits for granted, as "obvious". Only a few realise how actually hard it is to solve this problem. Personally I was astonished at the difficulty level after solving simple stuff with no integrals in high school :)

[@sasamladenovic6120]

These videos are fantastic. Explanations are very clear and understandable, but without the loose of generality. I wish that I had youtube and this channel when I take a physics course in university! Keep going, great job!

[@PhysicswithElliot]

Thanks Sasa!

[@sairithvickg6663]

2:32 couldn't we just observe planets motion in Sun's frame, since in that case the modification we need to do would be converting k to k' and then we are set By the way great video

[@chrispw144]

Awesome video! You could say it was a "stellar" explanation :)

[@PhysicswithElliot]

Stellar? I have no "Earthly" idea what you're talking about Chris

[@normdesbiens726]

Great derivation, but can you spend a bit more time with the differential equatiron? Won't u(theta) = epsilon sin(theta) also solve the equation? Why do you pick cos? Also I'm not sure how you handle the + 1, since the differential of 1 with repsect ot theta is 0 so it would drop out with the first differentiation.

[@PhysicswithElliot]

It's the simple harmonic oscillator equation, but for u-1 instead of u. You could define a new variable z = u - 1, and then the equation is z'' = -z. Solve for z, and then get u back from u = z + 1. Yep, you could also write down a sine solution. Or equivalently, you could write the solution as cos(\theta - \theta_0) with another parameter \theta_0. Different choices for \theta_0 just amount to different choices of how you set up your coordinates. I set \theta_0 = 0, which means that the planet's point of closest approach to the star happens at \theta = 0.

[@TeamHarrisonMachine]

Thank you!!! I was just thinking about this kind of tutorial!! I learned that Uranus was discovered through mathematics before it was located visually and I wondered how. How did the math look? There’s others too. For example, Kepler realized the orbits were elliptical… how? How, by only looking trough a telescope were they able to figure out the precession of Mercury? I was thinking if I had enough money someday I would pay a physics instructor to make a tutorial about those things. This is a step in the right direction. Thanks for this!

[@gaeb-hd4lf]

Awesome channel bro, glad i found it!

[@javierrucal4813]

As a first year physics major, this is hugely inspiring. Your channel is fantastic! Please keep it up! Love from Guatemala.

[@PhysicswithElliot]

Thanks Javier!

[@dr20231may]

It would be better if the explanation done here is either done entirely on a paper and just record that paper work or use a neat font for computer explanation and just don't use digital pens for illustration , use pre-sets like LINE CURVE SHAPES SURFACES VOLUMES etc ,

[@sairithvickg6663]

I am amazed ! Thanks a Lot for this!!!

[@bygabop9368]

A straight line must also be a solution, i.e. initial angular speed equal to zero, I guess.

[@anthonyorf8827]

so how do I calculate dr/dt and dθ/dt when given the radius vector and the velocity vector

[@MaudWinston-t8n]

Jackson Betty Martin Jose Lee Carol

[@paulboro5278]

Awesome explanation.

[@varunahlawat9013]

Really really helpful! Please more lectures like this :)

[@andreaskleinert7756]

It was incredible deep and fun. Guys like Ptolemy spent their lifes thinking in this kind of problems. I want more of this types of videos!

[@PhysicswithElliot]

Thanks Andreas!

[@maxdemuynck9850]

Hi Elliot, love your videos, there just great:) was just wondering if maybe you could do a series on a second year E&M course?

[@PhysicswithElliot]

Thanks Max! I certainly hope to make more E&M videos. It's possible I'll make a full E&M course down the line. Right now I'm working on creating a course on Lagrangian mechanics.

[@observer137]

Why is there no solution where object falls into the sun?

[@monawykayla2156]

Very well explained. However, I didn't quite get the final conclusion. If the equation applies to circles, ellipses, parabolas and hyperbolas, why are the planets orbiting in ellipses? He mentioned the angular speed, but can someone explain it further? I think I didn't get the "that's why earth is orbiting in ellipses".

[@mikev4621]

The type of conic section followed by any solar satellite is dependant on the total energy the satellite has. They all 'sling-shot ' around the sun, but not all have enough energy to escape

[@jdalton4552]

These formulae could be more easily solved using the formulae for p, the semi-latis rectum of an ellipse: r=p/(1+e cos theta) and p=a(1-e squared). However the beauty of his calculus is worth the extra effort.

[@palfers1]

Your value for the eccentricity is >1, which (according to you) is hyperbolic if E>0. I assume that one can prove that E

[@AffaAu]

Brilliant. Have been looking for such mathematical explanation!

[@steffenleo5997]

Great Videos..... 👍👍..... Thanks a lot..... Have a nice weekend

[@PhysicswithElliot]

Thanks Steffen!

[@astridlovespie]

Amazing video and great explanation! I am so glad I came across your channel while studying this topic. I have a question regarding the video though, how do we know for sure that the integration constant from 14:33 is indeed the eccentricity of the orbit? Thank you for the video and keep going!

[@peterhall6656]

Feynman did an even more basic version of the path derivation in Volume 1 of his Lectures on Physics pages 9-6 to 9-9. That derivation did not involve all the calculus employed here.

[@zack_120]

So cool! Literally you pictured the pathes of planets by actual numbers. In some cultures you would be painted as a supernatrural figure, fortune teller or magic.

[@agrajyadav2951]

best physcis channel!

[@andreaq6529]

Thank you, I was trying to derive the ellipse equation by kinematics only and totally forgot about using energy.

[@kmahadevan6110]

Thank You! Today I learnt how a topic can presented in its simplest but rigorous form.

[@r2k314]

I always think when I see this kind of derivation, how do you know that you haven't irretrievably lost a variable or constant or some other relationship. I wish there was a course on the limits of substitutions, differentiation, etc. Is there a reference which deals with all the legitimate tricks in one place?

[@informalchipmunk5775]

Wow! Just found this now, I love how you explain everything clearly whilst also forcing me to do some work as I watch

[@joaco4ever11]

is there a way you can solve the equations with respect to time ?

[@itoshiigrauben]

Actually, by combining some geometry with the fact that the area-sweeping speed is constant , one is able to solve the time respective to the position of the planet t(x,y). However, the equations of x(t) and y(t) for the inverse-square law orbit are non-elementary. You can look up Keplers' Equations for elliptical orbits.

[@raulcantu6998]

Is there a position vs time formula, asside from aproximations, and if so, could you give the solution :o

[@Daniel-c3l1k]

For the first time I have come across what I wanted to learn most: how Newtonian laws apply

[@jordanfreidel1751]

Why at 5:15 do you draw the perpendicular component to the velocity? I don't see why.

[@josemanuel9305]

Cross product by definition needs two perpendicular vectors.

[@truthprevail2742]

I really thank you very much, i was looking forward this concept mathematically for so long🙏

[@press2701]

A very clear, clean and thoughtful lecture, sir. Well done.

[@alejandroalmarza8447]

You are excellent profesor …keep this great physic videos coming

[@anishsharma6702]

this videos are like gold , waiting to be explored , by a curious mind,

[@ravikantpatil3398]

Excellent very exciting .. thanks Eliot

[@sarthaksamal182]

Great explanation man! Where can I get the animation that you showed in the end?

[@sarthaksamal182]

Actually I want to present a ppt on the runge-lenz vector so it would really be a help if I get the similar kind of simulation, that you showed, somewhere

[@PhysicswithElliot]

Thanks Sarthak! You can find all the links here: www.physicswithelliot.com/orbits-mini

[@JasonAStillman]

That was awesome, really enjoyed that, thanks E.

[@realcygnus]

Nifty

[@jordanfreidel1751]

Why can’t I use sin theta in the final equation? Or does it not make a difference, or am I just blind and short circuiting when doing the second derivative?

[@robert.ehrlich8942]

There is something missing at this stage in the video. After saying that cos theta or sin theta are solution of the equation without the "1", he keeps only the cos variant without further explanation. For such linear equations, any linear combination of solutions is also a solution, so the general solution would be A cos theta + B sin theta. Not surprising, a second order differential equation usually depends on 2 integration constants. This can be rewritten as C cos (theta - phi) with the two new constants C = sqrt(A²+B²) and phi = atan(B/A). In the following he assumes that phi = 0, which can always be obtained by rotating the x,y axes by phi, so eliminating one of the two constants. This is why all the following trajectories are symetrical relatively to the x axis, which is not implied by the initial description of the problem.

[@jordanfreidel1751]

@@robert.ehrlich8942 then that would mean that using sin theta would be an acceptable answer correct?

[@Paolo-sy7wf]

Hi, your videos are really amazing for those who want to go deeper in the theories and know where the laws come from. But i had an issue with the solution of the differential equation r(theta). As you explained (for example) in the pendulum video the solution for that DE is: a sin(x)+b cos(x) and i don't get how did you ended up choosing b cos(x). Did you consider the initial conditions to determine that? I tried also to figure it out with another method (taking square root of dr/dt and separating variables after the division of the 2 equations) but i can't solve the integral which should result in theta+theta0 = arccos((b^2 - ar)/(sqrt(a^2 - b^2))), with a and b being some rearrangements of G, M, m, L (this method is presented in a book where it misses the solving part).

[@PhysicswithElliot]

Thanks Paolo! Picking the cosine solution just means that I set up the coordinates so that theta = 0 when the planet is at its point of closet approach to the star. You could also add a sine term-it would just amount to rotating the ellipse around.

[@Ninjamanguy732]

Thanks for the great video! Can i ask why you aren’t factoring the Earth’s spin into the angular momentum and kinetic energy calculations? Thanks!

[@Jackie-wn5hx]

I'm definitely not Elliot, but I was curious as well. He used point mechanics to derive the orbits and celestial mechanics. Were you asking whether the rotational motion and kinetic energy of the earth will have a significant impact on the shape of its orbits around the sun? I've seen gravitational problems in general relativity that have to factor in the effects of frame dragging and spinning black holes and neutron stars. For most orbital problems in Newtonian mechanics and relativistic physics, point masses are used.

[@briannguyen6994]

I don't understand the step at 11:50 when you divide by (dtheta/dt)^2 to get the green equation. more specifically, where did the (l/mr)^2 term came from as a multiplication. Could you care to explain in a little more detail for me please? :D

[@PhysicswithElliot]

I'm dividing (dr/dt)^2 by (d\theta/dt)^2 there to get (dr/d\theta)^2. From the angular momentum equation d\theta/dt = L/(mr^2). I skipped a step where I multiplied that (L/(mr^2))^2 in the denominator on the right-hand-side over to the numerator on the left-hand-side of the equation.

[@alfredkokou2013]

Great video. But how does the value of epsilon get lower than 1 given that you derived it as epsilon = sqrt ( 1 + x) where x is always positive ( x = 2EL^2 / mk^2) ?

[@PhysicswithElliot]

E is negative for circular and elliptical orbits, so no x doesn't need to be positive!

[@mandystarr404]

Does anyone know if this equation could be written as a vector expression with i^ and j^

[@AwfulnewsFM]

Yes, you can use the simple transform equation r_vector = r*cos(theta)*i^ + r*sin(theta)*j^ Just put in the expansion for r

[@mandystarr404]

@@AwfulnewsFM thanks 👏🏼

[@sergiolucas38]

Great video, man, so much quality :)

[@arkopro30]

Why we failed to find celestial objects moving in parabolic or hyperbolic curves ?

[@carultch]

Because those aren't the orbits that stand the test of time. By definition, those trajectories are escape paths of bodies that don't stay in the solar system. A parabolic "orbit" is a special case of orbit theory, where an object has exactly enough energy to escape the gravitational field. If you had a spacecraft that you put in motion exactly at escape velocity, and then continue to coast, it would follow a parabolic "orbit" as it escapes its home world. A hyperbolic "orbit" is what you get when you have more than enough energy to escape.

[@MarceloRobertoJimenez]

Congratulations for the excelent content of your videos. Can I suggest you a video? Use the data Newton had available to derive the orbits of the planets. Best regards.

[@PhysicswithElliot]

Thanks Marcelo! That's an interesting idea!

[@過客-c5u]

Very clear and concise explanation.

[@billthomas7644]

A bit of a sleight of hand here. Makes the assumption that the m in Newton's gravitational force equation has anything to do with the "m" used in the momentum and energy equations. Using the same symbol doesn't justify it.

[@JulieanGalak]

Huh? Mass is mass, it's not just the same symbol, it's the same quantity....

[@billthomas7644]

@@JulieanGalak Nope. You have to prove that inertial mass and gravitational mass are the same. Look up the principle of equivalence. Hundreds of experiments have been carried out attempting to either prove or disprove this.

[@JulieanGalak]

@@billthomas7644 - interesting, I'll have to look at that.

[@JeffreyHoeper]

@@billthomas7644 Great point. Part of the genius of Einstein was that he made it a fundamental law of nature that inertial mass and gravitational mass are the same. Thus, there is no difference in nature (and physics) between a phone booth that is accelerating in space and a phone booth that is suspended in a gravitational field. That leads to the complex transformation equations of non-Euclidian curved space. And that leaves me very, very confused!

[@tgylfason]

Brilliantly lucid lectures, thank you.

[@pedrojuan5550]

Very well put. Thank you.

[@chritophergaafele8922]

Can you explain the mecury perihelion shift solution to einstein's field equations

[@PhysicswithElliot]

Possibly! I was thinking about whether I could cover it in an accessible way

[@briangoad8016]

Can you use this equation to tell where a planet will be in the night sky? Does time T represent calendar time?

[@carultch]

Yes. This is exactly what they use to predict the positions of planets. The t doesn't necessarily represent calendar time, but rather just time relative to an arbitrary point we chose to call t=0. Usually lowercase t represents a specific point in time, while in this context, capital T represents the period of orbit.

[@Structure-Dubai]

hi sir, what is the physical meaning of a position vector? is it some kind of position coordinates or direction of position or what?

[@PhysicswithElliot]

It's an arrow pointing to the location of a particle from whatever origin you've chosen

[@chritophergaafele8922]

I like orbital mechanics

[@glennrubel4326]

So how do you get an elliptical orbit from an eccentricity that is always greater than one?

[@itoshiigrauben]

Total mechanical energy E is a negative quantity for elliptical orbits

[@kaypee9187]

Sir, A very good video. One question though, would one have a Hyperbola in an attractive central force such as gravity. Even comets have highly eccentric orbits. The voyager spacecraft would be a parabola of course its angular momentum might be changed by the rocket thrust at the appropriate moment. Also the derivation of the locus was an exercise in mathematics!. Congratulations!

[@PhysicswithElliot]

Thanks Kaypee! The trajectory is an ellipse when then eccentricity is less then 1 and a hyperbola when it's greater than 1.

[@Salien1999]

Or in physical terms, if the object has reached escape velocity, it will follow a hyperbolic path. Otherwise, it will follow an elliptical path (at least according to classical mechanics)

[@agrajyadav2951]

0:50 well not really, pretty sophisticated models, even though not as good as newton's, but still pretty good existed before that, for example, surya siddhanta en.wikipedia.org/wiki/Surya_Siddhanta

[@asifshahriyar6382]

How amazing!

[@yosemiguelsotomayorcarnead9909]

you are amazing man

[@owen7185]

Another awesome lecture

[@mcalkis5771]

So the final equation still requires us to know a planet's total energy and angular momentum? How is that done?

[@PhysicswithElliot]

Those just amount to the initial conditions, like when you throw a ball you need to tell me what it was doing at t = 0 to be able to write down the trajectory after that.

[@mcalkis5771]

@@PhysicswithElliot So how would we know the angular momentum of the earth if we were just standing on its surface? How did Kepler and Newton do it?

[@Tobi21089]

@@mcalkis5771 I guess by watching the skye and looking how long it takes for one orbit and maybe they did calculate R by triangulation of the planet but that's just a guess for their velocities around the sun

[@carultch]

@@mcalkis5771 At the time of Kepler and Newton, we couldn't know what the momentum of the Earth is, in kg-m/s. We could only get the relative masses of the planets from extrapolating the observational data, and comparing it to the theory behind orbital mechanics. For instance, we could know the Earth is 81 times the mass of the moon, and that the sun is 333000 times the mass of the Earth, but we couldn't know how many kilograms each one of them were. The same is also true with knowing how many meters or kilometers away the sun is. We could know the relative scale of the solar system, but not the actual distance It wasn't until Cavendish, whose work was in the century after Newton's life, that we could know the actual mass of the Earth and celestial bodies. Cavendish "weighed" lead spheres in each others' gravitational fields, in order to determine the universal gravitational constant, and isolate the G from the GM product of astronomical bodies. From that information, we could solve for M to know their masses, as an application of this knowledge. Similarly, it also wasn't until Captain James Cook, and Lewis Swift, until we could know the actual scale of the solar system, and know that the sun is 150 million km away. They observed the transit of Venus from a known distance apart on opposite sides of the world, to determine this.

[@jcffilms256]

GOAT

[@frankreashore]

Really nice!

[@davidrandell2224]

Galilean relative motion has the earth approaching the released object. D=1/2 at^2.The earth is expanding at 16 feet per second per second constant acceleration: gravity. Or 1/770,000th its size. Where are the lower case “ a” and “ b” in Newton’s first Proposition? The Final Theory: Rethinking Our Scientific Legacy “, Mark McCutcheon for gravity facts.

[@carultch]

The earth is not expanding toward objects that fall toward it. We would notice that by now.

[@asifshahriyar6382]

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