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


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

$h e l l o e v e r y o n e$ $w h y d o p l a n e t s o f t h e s o l a r s y s t e m$ $r e v o l v e a r o u n d t h e s u n i n a n e l i p t i c a l$ $n o t c i r c u l a r o r b i t$
Commented by mr W last updated on 07/Jun/20

$i f w e h a d a t w o - b o d y s y s t e m$ $c o n s i s t i n g o f t h e s u n a n d a p l a n e t$ $a n d t h e o n l y f o r c e a c t i n g b e t w e e n$ $t h e m i s t h e g r a v i t a t i o n a l f o r c e a n d$ $b o t h o b j e c t s a r e s e e n a s p o i n t m a s s e s,$ $i t c a n b e p r o v e d t h a t t h e o r b i t o f t h e$ $p l a n e t a r o u n d t h e s u n i s a n e l l i p s e .$ $t h e s h a p e o f t h e e l l i p s e d e p e n t s o n$ $t h e i n i t i a l c o n d i t i o n . i n s p e c i a l$ $c o n d i t i o n, t h e o r b i t c a n b e a c i r c l e$ $w h i c h i s a l s o a s p e c i a l f o r m o f e l l i p s e .$ $b u t t h e r e a l s o l a r s y s t e m i s m u c h m o r e$ $c o m p l i c a t e d t h a n a t w o - b o d y s y s t e m,$ $t h e r e a r e s e v e r a l p l a n e t s w h i c h$ $a f f e c t e a c h o t h e r a n d b o t h t h e s u n$ $a n d t h e p l a n e t s a r e n o t p o i n t m a s s e s,$ $t h e r e f o r e t h e r e a l o r b i t s o f t h e p l a n e t s$ $a r e n o t p e r f e c t e l l i p s e s, o n l y n e a r$ $e l l i p s e s . a c i r c u l a r o r b i t i s t o o p e r f e c t,$ $w h i c h c a n n o t b e k e p t s t a b l e i n n a t u r e,$ $b e c a u s e a n y d i s t u r b a n c e m a y l e t i t$ $l e a v e t h i s p e r f e c t f o r m . n e v e r t h e l e s s$ $t h e o r b i t o f V e n u s i s a l m o s t a c i r c l e .$
Commented by M±th+et+s last updated on 08/Jun/20

$t h a n k y o u s i r$
	Answered by Sourav mridha last updated on 07/Jun/20

	$o k k {l e t}^{'} s s e e t h e t r a j e c t o r y o f$ $b o d y m o v i n g i n c e n t r a l f o r c e$ $f i e l d . . . b u t H ere i a m n o t u s i n g$ $G T R . . only t h e p o i n t o f v i e w o f$ $o l d c l a s s i c a l c o n c e p t o f N e w t o n i a n$ $L a g r a n g i a n M e c h a n i c s .$ ${l e t}^{'} s p i c k u p t h e e x a m p l e o f$ $o u r s o l a r s y s t e m . . . . w h e r e s u n i s$ $a t t h e f o c u s a n d e a r t h m o v e s$ $r o u n d s u n i n a c c o u n t o f G r a v i t a t i o n a l f o r c e$ $f e i l d .$ $s o o u r L a g r a n g i a n c o n s t r a c t e d$ $i n p l a n e p o l a r c o o r d i n a t e l i k e t h a t$ $L = \frac{1}{2} [{(\overset{∙}{r})}^{2} + r^{2} {(\overset{∙}{θ})}^{2}] - v (r)$ $n o w u s i n g E u l e r - L a g r a n g e {e q}^{n}$ $(f r o m V a r i a t i o n a l c a l c u l a s)$ $\frac{\partial L}{\partial r} - \frac{d}{d t} (\frac{\partial L}{\partial \overset{∙}{r}}) = 0 a n d \frac{\partial L}{\partial θ} - \frac{d}{d t} (\frac{\partial L}{\partial \overset{∙}{θ}}) = 0$ $f r o m 1 s t {e q}^{n} w e g et$ $\overset{∙ ∙}{r} - r {(\overset{∙}{θ})}^{2} = - \frac{\partial}{\partial r} [v (r)] = F (r) = \frac{k}{r^{2}}$ $. . . . . . . . . . . . . . . . . (i)$ $F (r) = g r a v i t e t i o n a l f o r c e, k = c o n s t a n t$ $k = {G m}_{s} m_{e} .$ $a n d 2 n d {e q}^{n} g i v e s u s$ $r^{2} \overset{∙}{θ} = c o n s t a n t, m u l t i : μ b o t h s i d e s$ $w e g e t L = μ r^{2} θ = c o n s t a n t$ $. . . . . . . . . . . (i i)$ $w h e r e μ = \frac{m_{s} m_{e}}{m_{s} + m_{e}} = r e d u c e m a s s$ $f o r c e n t r a l f o r c e -$ $t o r q u e τ = \vec{r} \times f (r) = 0 o r \frac{d L}{d t} = c o n s t a n t$ $w h e r e L = a n g u l a r m o m e n t a m .$ $n o w l e t u = \frac{1}{r} n o w d i f f w r t t o$ $θ a n d u s i n g {e q}^{n} (i i) w e g e t$ $\frac{d u}{d θ} = \frac{- 1}{r^{2}} . \frac{\frac{d r}{d t}}{\frac{d θ}{d t}} = - \frac{1}{r^{2}} \frac{\overset{∙}{r}}{\overset{∙}{θ}} = - \frac{μ}{L} \overset{∙}{r}$ $n o w \frac{d^{2} u}{d θ^{2}} = - \frac{μ^{2}}{L^{2} u^{2}} . \overset{∙ ∙}{r} .$ $s o w e g e t \overset{∙ ∙}{r} = - \frac{L^{2} u^{2}}{μ^{2}} . \frac{d^{2} u}{d θ^{2}} a n d$ $r {(\overset{∙}{θ})}^{2} = \frac{u^{3} L^{2}}{μ^{2}} . . .$ $n o w p u t t i n g t h i s t w o r e s u l t s a t$ ${e q}^{n} (i) w e g e t {e q}^{n} o f m o t i o n o f$ $e a r t h a r o u n d s u n i s$ $\frac{d^{2} u}{d θ^{2}} + u = - \frac{μ^{2} k}{L^{2}} (c o n s t a n t)$ $b y s o l v i n g t h i s e a s y 2 n d O D E$ $w e g e t u (θ) = - \frac{μ^{2} k}{L^{2}} + A c o s θ + B s i n θ$ $or = - \frac{μ^{2} k}{L^{2}} + γ c o s Φ$ $w e c o n s i d e r a t e a t f i r s t u = \frac{1}{r}$ $s o n o w r (Φ) = \frac{1}{- \frac{μ^{2} k}{L^{2}} + γ c o s (Φ)}$ $= \frac{- \frac{L^{2}}{μ^{2} k}}{1 - \frac{L^{2} γ}{μ^{2} k} c o s (Φ)}$ $c o m p a r e t h i s r e s u l t w i t h$ ${e q}^{n} o f a e l l i p s e i n p l a n e p o l a r$ $c o o r d i n a t e r (θ) = \frac{l}{1 \underline{+} e c o s (θ)}$ $i n o u r c a s e - -$ $s a m i l e t u s r e c t u m o f t h e e l l i p t i c a k$ $p a t h o f o u r p l a n e t i s$ $l = a (1 - e^{2}) = - \frac{L^{2}}{μ^{2} k}$ $a n d e s c e n t r i c i t y e = \frac{L^{2} γ}{μ^{2} k}$ $t h i s i s w h y p l a n e t s a r w m o v i n g$ $i n a e l l i p t i c a l p a t h . . .$ $1 s t p r o p o s e d b y K e p l e r i n h i s$ $1 s t l a s o f p l a n e t o r y m o t i o n . .$
	Commented by smridha last updated on 08/Jun/20

	$o h h s o r r y i t s h o u l d b e \frac{d L}{d t} = 0$ $s o L = c o n s t a n t .$
	Commented by M±th+et+s last updated on 08/Jun/20

	$g r e a t w o r k t h a n k s$
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