Acceleration vector Kepler's laws of planetary motion
from heliocentric point of view consider vector planet
r
=
r
r
^
{\displaystyle \mathbf {r} =r{\hat {\mathbf {r} }}}
r
{\displaystyle r}
distance planet ,
r
^
{\displaystyle {\hat {\mathbf {r} }}}
unit vector pointing towards planet.
d
r
^
d
t
=
r
^
˙
=
θ
˙
θ
^
,
d
θ
^
d
t
=
θ
^
˙
=
−
θ
˙
r
^
{\displaystyle {\frac {d{\hat {\mathbf {r} }}}{dt}}={\dot {\hat {\mathbf {r} }}}={\dot {\theta }}{\hat {\boldsymbol {\theta }}},\qquad {\frac {d{\hat {\boldsymbol {\theta }}}}{dt}}={\dot {\hat {\boldsymbol {\theta }}}}=-{\dot {\theta }}{\hat {\mathbf {r} }}}
where
θ
^
{\displaystyle {\hat {\boldsymbol {\theta }}}}
unit vector direction 90 degrees counterclockwise of
r
^
{\displaystyle {\hat {\mathbf {r} }}}
, ,
θ
{\displaystyle \theta }
polar angle, , dot on top of variable signifies differentiation respect time.
differentiate position vector twice obtain velocity vector , acceleration vector:
r
˙
=
r
˙
r
^
+
r
r
^
˙
=
r
˙
r
^
+
r
θ
˙
θ
^
,
{\displaystyle {\dot {\mathbf {r} }}={\dot {r}}{\hat {\mathbf {r} }}+r{\dot {\hat {\mathbf {r} }}}={\dot {r}}{\hat {\mathbf {r} }}+r{\dot {\theta }}{\hat {\boldsymbol {\theta }}},}
r
¨
=
(
r
¨
r
^
+
r
˙
r
^
˙
)
+
(
r
˙
θ
˙
θ
^
+
r
θ
¨
θ
^
+
r
θ
˙
θ
^
˙
)
=
(
r
¨
−
r
θ
˙
2
)
r
^
+
(
r
θ
¨
+
2
r
˙
θ
˙
)
θ
^
.
{\displaystyle {\ddot {\mathbf {r} }}=({\ddot {r}}{\hat {\mathbf {r} }}+{\dot {r}}{\dot {\hat {\mathbf {r} }}})+({\dot {r}}{\dot {\theta }}{\hat {\boldsymbol {\theta }}}+r{\ddot {\theta }}{\hat {\boldsymbol {\theta }}}+r{\dot {\theta }}{\dot {\hat {\boldsymbol {\theta }}}})=({\ddot {r}}-r{\dot {\theta }}^{2}){\hat {\mathbf {r} }}+(r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}){\hat {\boldsymbol {\theta }}}.}
so
r
¨
=
a
r
r
^
+
a
θ
θ
^
{\displaystyle {\ddot {\mathbf {r} }}=a_{r}{\hat {\boldsymbol {r}}}+a_{\theta }{\hat {\boldsymbol {\theta }}}}
where radial acceleration is
a
r
=
r
¨
−
r
θ
˙
2
{\displaystyle a_{r}={\ddot {r}}-r{\dot {\theta }}^{2}}
and transversal acceleration is
a
θ
=
r
θ
¨
+
2
r
˙
θ
˙
.
{\displaystyle a_{\theta }=r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}.}
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