Physic Labs
Step 1 of 4: A central force conserves angular momentum
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Intuition

Gravity always points straight at the Sun, so it cannot 'bend' the orbit out of its plane nor speed up or slow down the rotational part of the motion — like swinging a ball on a string: the string pulls toward the center but cannot by itself 'twist' the orbit.

F⃗=−GMmr2r^ ⇒ τ⃗=r⃗×F⃗=0 ⇒ L⃗=const\vec{F} = -\frac{GMm}{r^2}\hat{r} \ \Rightarrow \ \vec{\tau} = \vec{r} \times \vec{F} = 0 \ \Rightarrow \ \vec{L} = \text{const}
Analysis

The torque τ⃗=r⃗×F⃗\vec{\tau} = \vec{r} \times \vec{F} vanishes because F⃗\vec{F} is parallel to r⃗\vec{r}. Hence L⃗=m r⃗×r⃗˙\vec{L} = m\,\vec{r} \times \dot{\vec{r}} is a constant vector — in magnitude and direction. The fixed direction of L⃗\vec{L} confines the orbit to a plane perpendicular to L⃗\vec{L}, so polar coordinates (r,θ)(r, \theta) suffice.