开普勒行星运动定律

Three laws distilled from astronomy

The area rate is half the cross product

The area law is a perpendicularity statement

Speed varies as the orbit widens and narrows

A particle has position $\mathbf{r}(t)$ and velocity $\mathbf{v}(t)$. What is the rate at which the position vector sweeps area?

Why does a central force $\mathbf{a}(t) = -\lambda(t)\,\mathbf{r}(t)$ make the areal velocity constant?

A comet follows a stretched ellipse with the sun at one focus. Where is the comet moving fastest?

A particle traverses the unit circle as $\mathbf{r}(t) = (\cos t, \sin t, 0)$. How much area does the position vector sweep from $t = 0$ to $t = \pi$, in terms of $\pi$? Give the coefficient of $\pi$, as a plain number or fraction like 1/2.

A planet orbits with semi-major axis $4$ in units where $T^2 = a^3$. Give its period, as a plain number.