PX282 - I3 - newtons' laws

newton (1687) used kepler's laws to infer the law of universal gravitation
simplified derivation using circular orbits

\begin{matrix} P^{2} = k r^{3} \\ P = 2 π \frac{r}{v} \\ ⟹ 4 π^{2} \frac{r^{2}}{v^{2}} = k r^{3} \\ ∴ v^{2} = \frac{4 π^{2}}{k r} \end{matrix}

using newton's second law:

F = m a = \frac{m v^{2}}{r} = \frac{4 π^{2} m}{k r^{2}}

using PX155 - A3 - newton's third law:

F = \frac{4 π^{2} m}{k r^{2}} = \frac{4 π^{2} M}{k^{'} r^{2}}

where, $m$ is the mass of the planet, and $M$ is the mass of the sun

note: the sun moving is inconsistent with kepler's laws, where the sun is fixed on a focus of the ellipse

k M = k^{'} M = k^{″}

F = \frac{4 π^{2} M m}{k^{″} r^{2}}

defining $G = 4 π^{2} / k^{″}$ , the gravitational constant:

∴ F = \frac{G M m}{r^{2}}

newton was able to derive all three of kepler's laws using this law of universal gravitation (see [chapter 2, Carroll & Ostile, 2007])
the constant in kepler's third law, assuming $M ≫ m :$

k = \frac{k^{″}}{M} = \frac{4 π^{2}}{G M} ⟹ P^{2} = \frac{4 π^{2}}{G M} a^{3}

this is called the one body problem
kepler's third law widely used to obtain the masses of astronomical objects
eg:
- for the mass of the sun: $P = 1$ yr, $a = 1$ AU $⟹ M_{⊙} = 1.99 \times 10^{30}$ kg
- the mass of jupiter from its moons
- blackholes in galaxies