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Question Number 97271 by  M±th+et+s last updated on 07/Jun/20

hello every one  why do planets of the solar system  revolve around the sun in an eliptical  not circular orbit

$${hello}\:{every}\:{one} \\ $$$${why}\:{do}\:{planets}\:{of}\:{the}\:{solar}\:{system} \\ $$$${revolve}\:{around}\:{the}\:{sun}\:{in}\:{an}\:{eliptical} \\ $$$${not}\:{circular}\:{orbit} \\ $$

Commented by mr W last updated on 07/Jun/20

if we had a two−body system   consisting of the sun and a planet  and the only force acting between   them is the gravitational force and  both objects are seen as point masses,  it can be proved that the orbit of the  planet around the sun is an ellipse.  the shape of the ellipse depents on  the initial condition. in special  condition, the orbit can be a circle  which is also a special form of ellipse.  but the real solar system is much more  complicated than a two−body system,  there are several planets which  affect each other and both the sun  and the planets are not point masses,  therefore the real orbits of the planets  are not perfect ellipses, only near  ellipses. a circular orbit is too perfect,  which can not be kept stable in nature,  because any disturbance may let it  leave this perfect form. nevertheless  the orbit of Venus is almost a circle.

$${if}\:{we}\:{had}\:{a}\:{two}−{body}\:{system}\: \\ $$$${consisting}\:{of}\:{the}\:{sun}\:{and}\:{a}\:{planet} \\ $$$${and}\:{the}\:{only}\:{force}\:{acting}\:{between}\: \\ $$$${them}\:{is}\:{the}\:{gravitational}\:{force}\:{and} \\ $$$${both}\:{objects}\:{are}\:{seen}\:{as}\:{point}\:{masses}, \\ $$$${it}\:{can}\:{be}\:{proved}\:{that}\:{the}\:{orbit}\:{of}\:{the} \\ $$$${planet}\:{around}\:{the}\:{sun}\:{is}\:{an}\:{ellipse}. \\ $$$${the}\:{shape}\:{of}\:{the}\:{ellipse}\:{depents}\:{on} \\ $$$${the}\:{initial}\:{condition}.\:{in}\:{special} \\ $$$${condition},\:{the}\:{orbit}\:{can}\:{be}\:{a}\:{circle} \\ $$$${which}\:{is}\:{also}\:{a}\:{special}\:{form}\:{of}\:{ellipse}. \\ $$$${but}\:{the}\:{real}\:{solar}\:{system}\:{is}\:{much}\:{more} \\ $$$${complicated}\:{than}\:{a}\:{two}−{body}\:{system}, \\ $$$${there}\:{are}\:{several}\:{planets}\:{which} \\ $$$${affect}\:{each}\:{other}\:{and}\:{both}\:{the}\:{sun} \\ $$$${and}\:{the}\:{planets}\:{are}\:{not}\:{point}\:{masses}, \\ $$$${therefore}\:{the}\:{real}\:{orbits}\:{of}\:{the}\:{planets} \\ $$$${are}\:{not}\:{perfect}\:{ellipses},\:{only}\:{near} \\ $$$${ellipses}.\:{a}\:{circular}\:{orbit}\:{is}\:{too}\:{perfect}, \\ $$$${which}\:{can}\:{not}\:{be}\:{kept}\:{stable}\:{in}\:{nature}, \\ $$$${because}\:{any}\:{disturbance}\:{may}\:{let}\:{it} \\ $$$${leave}\:{this}\:{perfect}\:{form}.\:{nevertheless} \\ $$$${the}\:{orbit}\:{of}\:{Venus}\:{is}\:{almost}\:{a}\:{circle}. \\ $$

Commented by  M±th+et+s last updated on 08/Jun/20

thank you sir

$${thank}\:{you}\:{sir} \\ $$

Answered by Sourav mridha last updated on 07/Jun/20

okk let′s see the trajectory of  body moving in central force   field...but Here i am not using  GTR..only the point of view of  old classical concept of Newtonian  Lagrangian Mechanics.  let′s pick up the example of   our solar system....where sun is  at the focus and earth moves   round sun in account of Gravitational force   feild.   so our Lagrangian constracted  in plane polar coordinate like that       L=(1/2)[(r^• )^2 +r^2 (𝛉^• )^2 ]−v(r)  now using Euler−Lagrange eq^n   (from Variational calculas)  (∂L/∂r)−(d/dt)((∂L/∂r^• ))=0 and (∂L/∂𝛉)−(d/dt)((∂L/∂𝛉^• ))=0  from 1st eq^n  we get     r^(••) −r(𝛉^• )^2 =−(∂/∂r)[v(r)]=F(r)=(k/r^2 )                 .................(i)  F(r)=gravitetional force ,k=constant  k=Gm_s m_e .   and 2nd eq^n  gives us  r^2 𝛉^• =constant,multi:𝛍 both sides  we get L=𝛍r^2 𝛉=constant                   ...........(ii)  where 𝛍=((m_s m_e )/(m_s +m_e ))=reduce mass  for central force−  torque 𝛕 =r^→ ×f(r)=0 or (dL/dt)=constant  where L=angular momentam.    now let u=(1/r) now diff wrt to   𝛉 and using eq^n   (ii) we get  (du/d𝛉)=((−1)/r^2 ).((dr/dt)/(d𝛉/dt))=−(1/r^2 )(r^• /𝛉^• )=−(𝛍/L)r^•   now (d^2 u/d𝛉^2 )=−(𝛍^2 /(L^2 u^2 )).r^(••)   .  so we get r^(••) =−((L^2 u^2 )/𝛍^2 ).(d^2 u/d𝛉^2 ) and                         r(𝛉^• )^2 =((u^3 L^2 )/𝛍^2 )...  now putting this two results at  eq^n   (i) we get eq^(n ) of motion of   earth around sun is                       (d^2 u/d𝛉^2 )+u=−((𝛍^2 k)/L^2 )(constant)  by solving this easy 2nd ODE    we get u(𝛉)=−((𝛍^2 k)/L^2 )+Acos𝛉+Bsin𝛉         or            =−((𝛍^2 k)/L^2 )+𝛄cos𝚽   we considerate at first u=(1/r)  so now r(𝚽)=(1/(−((𝛍^2 k)/L^2 )+γcos(𝚽)))                               =((−(L^2 /(𝛍^2 k)))/(1−((L^2 𝛄)/(μ^2 k))cos(𝚽)))  compare this result with   eq^n  of a ellipse in plane polar  coordinate r(𝛉)=(l/(1+_− ecos(𝛉)))  in our case−−  sami letus rectum of the ellipticak  path of our planet is  l=a(1−e^2 )=−(L^2 /(𝛍^2 k))  and escentricity e=((L^2 𝛄)/(𝛍^2 k))  this is why planets arw moving  in a elliptical path...  1st proposed by Kepler in his  1st las of planetory motion..

$$\boldsymbol{{okk}}\:\boldsymbol{{let}}'\boldsymbol{{s}}\:\boldsymbol{{see}}\:\boldsymbol{{the}}\:\boldsymbol{{trajectory}}\:\boldsymbol{{of}} \\ $$$$\boldsymbol{{body}}\:\boldsymbol{{moving}}\:\boldsymbol{{in}}\:\boldsymbol{{central}}\:\boldsymbol{{force}}\: \\ $$$$\boldsymbol{{field}}...\boldsymbol{{but}}\:\boldsymbol{{H}}\mathrm{ere}\:\boldsymbol{{i}}\:\boldsymbol{{am}}\:\boldsymbol{{not}}\:\boldsymbol{{using}} \\ $$$$\boldsymbol{{GTR}}..\mathrm{only}\:\boldsymbol{{the}}\:\boldsymbol{{point}}\:\boldsymbol{{of}}\:\boldsymbol{{view}}\:\boldsymbol{{of}} \\ $$$$\boldsymbol{{old}}\:\boldsymbol{{classical}}\:\boldsymbol{{concept}}\:\boldsymbol{{of}}\:\boldsymbol{{Newtonian}} \\ $$$$\boldsymbol{{Lagrangian}}\:\boldsymbol{{M}}\mathrm{e}\boldsymbol{{chanics}}. \\ $$$$\boldsymbol{{let}}'\boldsymbol{{s}}\:\boldsymbol{{pick}}\:\boldsymbol{{up}}\:\boldsymbol{{the}}\:\boldsymbol{{example}}\:\boldsymbol{{of}}\: \\ $$$$\boldsymbol{{our}}\:\boldsymbol{{solar}}\:\boldsymbol{{system}}....\boldsymbol{{where}}\:\boldsymbol{{sun}}\:\boldsymbol{{is}} \\ $$$$\boldsymbol{{at}}\:\boldsymbol{{the}}\:\boldsymbol{{focus}}\:\boldsymbol{{and}}\:\boldsymbol{{earth}}\:\boldsymbol{{moves}}\: \\ $$$$\boldsymbol{{round}}\:\boldsymbol{{sun}}\:\boldsymbol{{in}}\:\boldsymbol{{account}}\:\boldsymbol{{of}}\:\boldsymbol{{Gravitational}}\:\boldsymbol{{force}}\: \\ $$$$\boldsymbol{{feild}}. \\ $$$$\:\boldsymbol{{so}}\:\boldsymbol{{our}}\:\boldsymbol{{Lagrangian}}\:\boldsymbol{{constracted}} \\ $$$$\boldsymbol{{in}}\:\boldsymbol{{plane}}\:\boldsymbol{{polar}}\:\boldsymbol{{coordinate}}\:\boldsymbol{{like}}\:\boldsymbol{{that}} \\ $$$$\:\:\:\:\:\mathscr{L}=\frac{\mathrm{1}}{\mathrm{2}}\left[\left(\overset{\bullet} {\boldsymbol{{r}}}\right)^{\mathrm{2}} +\boldsymbol{{r}}^{\mathrm{2}} \left(\overset{\bullet} {\boldsymbol{\theta}}\right)^{\mathrm{2}} \right]−\boldsymbol{{v}}\left(\boldsymbol{{r}}\right) \\ $$$$\boldsymbol{{now}}\:\boldsymbol{{using}}\:\boldsymbol{{Euler}}−\boldsymbol{{Lagrange}}\:\boldsymbol{{eq}}^{\boldsymbol{{n}}} \\ $$$$\left(\boldsymbol{{from}}\:\boldsymbol{{V}}\mathrm{a}\boldsymbol{{riational}}\:\boldsymbol{{calculas}}\right) \\ $$$$\frac{\partial\mathscr{L}}{\partial\boldsymbol{{r}}}−\frac{\boldsymbol{{d}}}{\boldsymbol{{dt}}}\left(\frac{\partial\mathscr{L}}{\partial\overset{\bullet} {\boldsymbol{{r}}}}\right)=\mathrm{0}\:\boldsymbol{{and}}\:\frac{\partial\mathscr{L}}{\partial\boldsymbol{\theta}}−\frac{\boldsymbol{{d}}}{\boldsymbol{{dt}}}\left(\frac{\partial\mathscr{L}}{\partial\overset{\bullet} {\boldsymbol{\theta}}}\right)=\mathrm{0} \\ $$$$\boldsymbol{{from}}\:\mathrm{1}\boldsymbol{{st}}\:\boldsymbol{{eq}}^{\boldsymbol{{n}}} \:\boldsymbol{{we}}\:\boldsymbol{{g}}\mathrm{et} \\ $$$$\:\:\:\overset{\bullet\bullet} {\boldsymbol{{r}}}−\boldsymbol{{r}}\left(\overset{\bullet} {\boldsymbol{\theta}}\right)^{\mathrm{2}} =−\frac{\partial}{\partial\boldsymbol{{r}}}\left[\boldsymbol{{v}}\left(\boldsymbol{{r}}\right)\right]=\boldsymbol{{F}}\left(\mathrm{r}\right)=\frac{\boldsymbol{{k}}}{\boldsymbol{{r}}^{\mathrm{2}} } \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:.................\left(\boldsymbol{{i}}\right) \\ $$$$\boldsymbol{{F}}\left(\mathrm{r}\right)=\boldsymbol{{gravitetional}}\:\boldsymbol{{force}}\:,\boldsymbol{{k}}=\boldsymbol{{constant}} \\ $$$$\boldsymbol{{k}}=\boldsymbol{{Gm}}_{\boldsymbol{{s}}} \boldsymbol{{m}}_{\boldsymbol{{e}}} . \\ $$$$\:\boldsymbol{{and}}\:\mathrm{2}\boldsymbol{{nd}}\:\boldsymbol{{eq}}^{\boldsymbol{{n}}} \:\boldsymbol{{gives}}\:\boldsymbol{{us}} \\ $$$$\boldsymbol{{r}}^{\mathrm{2}} \overset{\bullet} {\boldsymbol{\theta}}=\boldsymbol{{constant}},\boldsymbol{{multi}}:\boldsymbol{\mu}\:\boldsymbol{{both}}\:\boldsymbol{{sides}} \\ $$$$\boldsymbol{{we}}\:\boldsymbol{{get}}\:\boldsymbol{{L}}=\boldsymbol{\mu{r}}^{\mathrm{2}} \boldsymbol{\theta}=\boldsymbol{{constant}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...........\left(\boldsymbol{{ii}}\right) \\ $$$$\boldsymbol{{where}}\:\boldsymbol{\mu}=\frac{\boldsymbol{{m}}_{\boldsymbol{{s}}} \boldsymbol{{m}}_{\boldsymbol{{e}}} }{\boldsymbol{{m}}_{\boldsymbol{{s}}} +\boldsymbol{{m}}_{\boldsymbol{{e}}} }=\boldsymbol{{reduce}}\:\boldsymbol{{mass}} \\ $$$$\boldsymbol{{for}}\:\boldsymbol{{central}}\:\boldsymbol{{force}}− \\ $$$$\boldsymbol{{torque}}\:\boldsymbol{\tau}\:=\overset{\rightarrow} {\boldsymbol{{r}}}×\boldsymbol{{f}}\left(\boldsymbol{{r}}\right)=\mathrm{0}\:\boldsymbol{{or}}\:\frac{\boldsymbol{{dL}}}{\boldsymbol{{dt}}}=\boldsymbol{{constant}} \\ $$$$\boldsymbol{{where}}\:\boldsymbol{{L}}=\boldsymbol{{angular}}\:\boldsymbol{{momentam}}. \\ $$$$\:\:\boldsymbol{{now}}\:\boldsymbol{{let}}\:\boldsymbol{{u}}=\frac{\mathrm{1}}{\boldsymbol{{r}}}\:\boldsymbol{{now}}\:\boldsymbol{{diff}}\:\boldsymbol{{wrt}}\:\boldsymbol{{to}}\: \\ $$$$\boldsymbol{\theta}\:\boldsymbol{{and}}\:\boldsymbol{{using}}\:\boldsymbol{{eq}}^{\boldsymbol{{n}}} \:\:\left(\boldsymbol{{ii}}\right)\:\boldsymbol{{we}}\:\boldsymbol{{get}} \\ $$$$\frac{\boldsymbol{{du}}}{\boldsymbol{{d}\theta}}=\frac{−\mathrm{1}}{\boldsymbol{{r}}^{\mathrm{2}} }.\frac{\frac{\boldsymbol{{dr}}}{\boldsymbol{{dt}}}}{\frac{\boldsymbol{{d}\theta}}{\boldsymbol{{dt}}}}=−\frac{\mathrm{1}}{\boldsymbol{{r}}^{\mathrm{2}} }\frac{\overset{\bullet} {\boldsymbol{{r}}}}{\overset{\bullet} {\boldsymbol{\theta}}}=−\frac{\boldsymbol{\mu}}{\boldsymbol{{L}}}\overset{\bullet} {\boldsymbol{{r}}} \\ $$$$\boldsymbol{{now}}\:\frac{\boldsymbol{{d}}^{\mathrm{2}} \boldsymbol{{u}}}{\boldsymbol{{d}\theta}^{\mathrm{2}} }=−\frac{\boldsymbol{\mu}^{\mathrm{2}} }{\boldsymbol{{L}}^{\mathrm{2}} \boldsymbol{{u}}^{\mathrm{2}} }.\overset{\bullet\bullet} {\boldsymbol{{r}}}\:\:. \\ $$$$\boldsymbol{{so}}\:\boldsymbol{{we}}\:\boldsymbol{{get}}\:\overset{\bullet\bullet} {\boldsymbol{{r}}}=−\frac{\boldsymbol{{L}}^{\mathrm{2}} \boldsymbol{{u}}^{\mathrm{2}} }{\boldsymbol{\mu}^{\mathrm{2}} }.\frac{\boldsymbol{{d}}^{\mathrm{2}} \boldsymbol{{u}}}{\boldsymbol{{d}\theta}^{\mathrm{2}} }\:\boldsymbol{{and}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{r}}\left(\overset{\bullet} {\boldsymbol{\theta}}\right)^{\mathrm{2}} =\frac{\boldsymbol{{u}}^{\mathrm{3}} \boldsymbol{{L}}^{\mathrm{2}} }{\boldsymbol{\mu}^{\mathrm{2}} }... \\ $$$$\boldsymbol{{now}}\:\boldsymbol{{putting}}\:\boldsymbol{{this}}\:\boldsymbol{{two}}\:\boldsymbol{{results}}\:\boldsymbol{{at}} \\ $$$$\boldsymbol{{eq}}^{\boldsymbol{{n}}} \:\:\left(\boldsymbol{{i}}\right)\:\boldsymbol{{we}}\:\boldsymbol{{get}}\:\boldsymbol{{eq}}^{\boldsymbol{{n}}\:} \boldsymbol{{of}}\:\boldsymbol{{motion}}\:\boldsymbol{{of}}\: \\ $$$$\boldsymbol{{earth}}\:\boldsymbol{{around}}\:\boldsymbol{{sun}}\:\boldsymbol{{is}} \\ $$$$\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\boldsymbol{{d}}^{\mathrm{2}} \boldsymbol{{u}}}{\boldsymbol{{d}\theta}^{\mathrm{2}} }+\boldsymbol{{u}}=−\frac{\boldsymbol{\mu}^{\mathrm{2}} \boldsymbol{{k}}}{\boldsymbol{{L}}^{\mathrm{2}} }\left(\boldsymbol{{constant}}\right) \\ $$$$\boldsymbol{{by}}\:\boldsymbol{{solving}}\:\boldsymbol{{this}}\:\boldsymbol{{easy}}\:\mathrm{2}\boldsymbol{{nd}}\:\boldsymbol{{ODE}} \\ $$$$\:\:\boldsymbol{{we}}\:\boldsymbol{{get}}\:\boldsymbol{{u}}\left(\boldsymbol{\theta}\right)=−\frac{\boldsymbol{\mu}^{\mathrm{2}} \boldsymbol{{k}}}{\boldsymbol{{L}}^{\mathrm{2}} }+\boldsymbol{{Acos}\theta}+\boldsymbol{{Bsin}\theta} \\ $$$$\:\:\:\:\:\:\:\mathrm{or}\:\:\:\:\:\:\:\:\:\:\:\:=−\frac{\boldsymbol{\mu}^{\mathrm{2}} \boldsymbol{{k}}}{\boldsymbol{{L}}^{\mathrm{2}} }+\boldsymbol{\gamma{cos}\Phi} \\ $$$$\:\boldsymbol{{we}}\:\boldsymbol{{considerate}}\:\boldsymbol{{at}}\:\boldsymbol{{first}}\:\boldsymbol{{u}}=\frac{\mathrm{1}}{\boldsymbol{{r}}} \\ $$$$\boldsymbol{{so}}\:\boldsymbol{{now}}\:\boldsymbol{{r}}\left(\boldsymbol{\Phi}\right)=\frac{\mathrm{1}}{−\frac{\boldsymbol{\mu}^{\mathrm{2}} \boldsymbol{{k}}}{\boldsymbol{{L}}^{\mathrm{2}} }+\gamma\boldsymbol{{cos}}\left(\boldsymbol{\Phi}\right)} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{−\frac{\boldsymbol{{L}}^{\mathrm{2}} }{\boldsymbol{\mu}^{\mathrm{2}} \boldsymbol{{k}}}}{\mathrm{1}−\frac{\boldsymbol{{L}}^{\mathrm{2}} \boldsymbol{\gamma}}{\mu^{\mathrm{2}} \boldsymbol{{k}}}\boldsymbol{{cos}}\left(\boldsymbol{\Phi}\right)} \\ $$$$\boldsymbol{{compare}}\:\boldsymbol{{this}}\:\boldsymbol{{result}}\:\boldsymbol{{with}}\: \\ $$$$\boldsymbol{{eq}}^{\boldsymbol{{n}}} \:\boldsymbol{{of}}\:\boldsymbol{{a}}\:\boldsymbol{{ellipse}}\:\boldsymbol{{in}}\:\boldsymbol{{plane}}\:\boldsymbol{{polar}} \\ $$$$\boldsymbol{{coordinate}}\:\boldsymbol{{r}}\left(\boldsymbol{\theta}\right)=\frac{\boldsymbol{{l}}}{\mathrm{1}\underset{−} {+}\boldsymbol{{ecos}}\left(\boldsymbol{\theta}\right)} \\ $$$$\boldsymbol{{in}}\:\boldsymbol{{our}}\:\boldsymbol{{case}}−− \\ $$$$\boldsymbol{{sami}}\:\boldsymbol{{letus}}\:\boldsymbol{{rectum}}\:\boldsymbol{{of}}\:\boldsymbol{{the}}\:\boldsymbol{{ellipticak}} \\ $$$$\boldsymbol{{path}}\:\boldsymbol{{of}}\:\boldsymbol{{our}}\:\boldsymbol{{planet}}\:\boldsymbol{{is}} \\ $$$$\boldsymbol{{l}}=\boldsymbol{{a}}\left(\mathrm{1}−\boldsymbol{{e}}^{\mathrm{2}} \right)=−\frac{\boldsymbol{{L}}^{\mathrm{2}} }{\boldsymbol{\mu}^{\mathrm{2}} \boldsymbol{{k}}} \\ $$$$\boldsymbol{{and}}\:\boldsymbol{{escentricity}}\:\boldsymbol{{e}}=\frac{\boldsymbol{{L}}^{\mathrm{2}} \boldsymbol{\gamma}}{\boldsymbol{\mu}^{\mathrm{2}} \boldsymbol{{k}}} \\ $$$$\boldsymbol{{this}}\:\boldsymbol{{is}}\:\boldsymbol{{why}}\:\boldsymbol{{planets}}\:\boldsymbol{{arw}}\:\boldsymbol{{moving}} \\ $$$$\boldsymbol{{in}}\:\boldsymbol{{a}}\:\boldsymbol{{elliptical}}\:\boldsymbol{{path}}... \\ $$$$\mathrm{1}\boldsymbol{{st}}\:\boldsymbol{{proposed}}\:\boldsymbol{{by}}\:\boldsymbol{{K}}\mathrm{e}\boldsymbol{{pler}}\:\boldsymbol{{in}}\:\boldsymbol{{his}} \\ $$$$\mathrm{1}\boldsymbol{{st}}\:\boldsymbol{{las}}\:\boldsymbol{{of}}\:\boldsymbol{{planetory}}\:\boldsymbol{{motion}}.. \\ $$$$ \\ $$

Commented by smridha last updated on 08/Jun/20

ohh sorry it should be (dL/dt)=0  so L=constant.

$$\boldsymbol{{ohh}}\:\boldsymbol{{sorry}}\:\boldsymbol{{it}}\:\boldsymbol{{should}}\:\boldsymbol{{be}}\:\frac{\boldsymbol{{dL}}}{{d}\boldsymbol{{t}}}=\mathrm{0} \\ $$$$\boldsymbol{{so}}\:\boldsymbol{{L}}=\boldsymbol{{constant}}. \\ $$

Commented by  M±th+et+s last updated on 08/Jun/20

great work thanks

$${great}\:{work}\:{thanks} \\ $$

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