Orbital Mechanics I

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Newtonian mechanics

Parametric vectors

Displacement $\vec{r}(t)=x(t)\hat{i}y(t)\hat{j}+z(t)\hat{k}$

distance: $r(t)=|\vec{r}(t)|=\sqrt{\vec{r}\cdot \vec{r}}$

Newtons laws

First law

• Isaac newton(1642-1727)
• an objects’ velocity remains constant unless a net outside force acts upon it
• $\vec{v}(t)=\overrightarrow{v_0}=const$

second law

• $\vec{F}=m\vec{a}(t)$
• $\vec{F}=\frac{d\vec{p}(t)}{dt}$
• $d\vec{v}/dt=\vec{f}/m$
• force changes velocity
• used a lot in computational math

third law

• forces come in pairs, equal in magnitude, and opposite in direction

Newtonian gravity

• a force, grav, exsits between any two objects having mass m and M, prop to the product of their masses mM and inversely proportinal tothe square of the separtation distance r of their centers
• for coordinates centered on M:
• $\vec{F}=-G\frac{Mm}{|\vec{r}{|}^2}\hat{r}$

Displacement vector and polar coordinates

• cartesian coordinates are often written a (x,y,z) in a coordinate system centered on mass M
• Axis orientations are chosen so that the planet orbits in the x-y plane
• Displacement $\vec{r}(t)=x(t)\hat{i}+y(t)\hat{j}$

velocity vector and polar coordinates

• unit vectors in polar coordinates vary with $\theta (t)$
• $\frac{d\hat{r}(t)}{dt}=\frac{d\hat{r}(t)}{d\theta }\frac{d\theta (t)}{dt}=\frac{d\theta (t)}{dt}\hat{\theta }(t)$
• .
• .
• .
• $\vec{v}(t)=v_r\hat{r}+v_t\hat{\theta }$
• two velocity components in polar coords

Kepler laws: angular momentum

• .

keplers 2nd law = consv, angular momentum

• $d\vec{L}/dt=0$
• $\vec{L}=\vec{R}\times \vec{p}=\vec{r}\times m\vec{v}=const$
• $\Rightarrow |\vec{v}|=L=mr{v}_1$

Keplers Laws

Keplers First Law

• $\frac{d\vec{v}}{dt}=-\frac{GM}{r_2}\hat{r}$
• $\frac{L}{GMm}\frac{d\vec{v}}{dt}=\frac{d\hat{\theta }}{dt}$
• $\frac{L}{GMm}\vec{v}=\hat{\theta }+e\hat{j}$
• take dot product of both sides with unit vector $\hat{\theta },using$
• $\hat{j}\cdot \hat{\theta }=\cos \theta$
• $\vec{v}\cdot \hat{\theta }=v_t=\frac{L}{mr}$

Kepler III

• we know that $\frac{dA}{dt}=\frac{l}{2m}=const$
• area of a ellipse $a=\pi ab$ of orb period p.
• $\therefore \frac{A}{P}=\frac{\pi ab}{P}=\frac{L}{2m}$
• eclipse geo : $b^2=a^2(1-e^2)$
• also, $\frac{L^2}{m^2}GMa(1-e^2)$
• $P^2=\frac{4{\pi }^2}{GM}{a}^3$