Central force motion

General central force motion
Gravitational (inverse square) force
Bound orbits
Scattering

General central force motion

Slide 1 of 2

Fig 1 — The central-force problem

Central force

\mathbf{F}(\mathbf{r}) = \hat{\mathbf{r}}f(r) = -\hat{\mathbf{r}} \dfrac{dV(r)}{dr}

(1)

Lagrangian

L = \tfrac{1}{2} \mu (\dot{r}^2 + r^2\dot{\varphi}^2) + V(r)

(2)

Reduced mass

\mu \equiv \dfrac{m_1 m_2}{m_1 + m_2}

(3)

Equivalently, $\mu^{-1} = m_1^{-1} + m_2^{-1}$ .
The results on this page are for the equivalent one-body problem (as shown in Fig. 1) of the central force motion of two bodies $m_1$ and $m_2$ about their center of mass. Thus $m$ is replaced by $\mu$ .

Orbital EoM

\displaystyle\varphi(r) = \frac{\pm l}{\sqrt{2\mu}}\int^{r'}{\dfrac{dr}{r^2(E-V_{\text{eff}})}} + \varphi_0

(4)

Effective potential

V_{\text{eff}}(r) = V(r) + \dfrac{l^2}{2 \mu r^2}

(5)

Angular momentum

l = |\mathbf{r} \times \mathbf{p}| = \mu r^2 \dot{\varphi} = \text{constant}

(6)

Energy

E = \tfrac{1}{2} \mu (\dot{r}^2+r^2\dot{\varphi}^2) + V(r) = \text{constant}

(7)

Differential EoM

\displaystyle\frac{d^2}{d\theta^2} \left(\frac{1}{r}\right) + \frac{1}{r}=-\frac{\mu r^2}{l^2}F(r)

(8)

Differential form of

Eq. (4)

Gravitational (inverse-square) force

Gravitational potential

V(r) = - \dfrac{\gamma \mu}{r}

(9)

\gamma \equiv G(m_1 + m_2)

(10)

Solution of orbital EoM

\dfrac{1}{r} = C(1-\varepsilon \cos{\varphi})

(11)

Solution of
Eq. (4)
.
This is the equation of a conic section with one focus at the origin, where $\varphi_0=0$ by convention.
Keep in mind that here $E$ and $l$ are the energy and angular momentum in the center-of-mass frame.

C \equiv \dfrac{\gamma \mu^2}{l^2}

(12)

Eccentricity

\varepsilon = \left(1 + \dfrac{2El^2}{\mu^3 \gamma^2} \right)^{1/2}

(13)

Eccentricity	Energy	Orbit
$\varepsilon>1$	$E>0$	Hyperbola
$\varepsilon=1$	$E=0$	Parabola
$0<\varepsilon<1$	$V_{\text{min}}<E<0$	Ellipse
$\varepsilon=0$	$E=V_{\text{min}}$	Circle

Bound orbits

Bound, elliptic orbits ( $E < 0$ ) in a gravitational potential.

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Fig 1 — Bound (elliptic) orbit

Semimajor axis

a = \dfrac{1}{C(1-\varepsilon^2)} = -\dfrac{\mu \gamma}{2E} = \dfrac{d+d'}{2}

(14)

Semiminor axis

b = a \sqrt{1-\varepsilon^2} = \sqrt{a^2-f^2}

(15)

Focus

f = \varepsilon a

(16)

Kepler's laws

1st Law

r^{-1}=C(1-\varepsilon \cos{\varphi})

(17)

Planets move in elliptical orbits about the Sun with the Sun at one focus.

2nd Law

\dfrac{dA}{dt} = \dfrac{l}{2\mu} = \text{constant}

(18)

The area per unit time swept out by a radius vector from the Sun to a planet is constant.

3rd Law

\tau^2 = \dfrac{4 \pi^2 a^3}{\gamma} \cong \dfrac{4 \pi^2 a^3}{Gm_{\text{Sun}}}

(19)

The square of a planet's period is proportional to the cube of the major axis of the planet's orbit

Scattering

Unbounded, hyperbolic orbits ( $E > 0$ ) in a gravitational potential.

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Fig 1 — Scattering from an attractive central potential

Energy

E = \tfrac{1}{2} \mu v_0^2

(20)

Energy in the center-of-mass frame.
$v_0$ is the relative speed of particles approaching from infinity.

Angular momentum

l = \mu v_0 b

(21)

Angular momentum in the center-of-mass frame.
$b$ is called the impact parameter and is the distance of closest approach if the trajectory were undeflected.

Eccentricity

\varepsilon = \bigg[1 + \bigg(\dfrac{v_0^2 b}{\gamma}\bigg)^2 \bigg]^{1/2}

(22)

Distance of closest approach

r_{\text{min}} = b \left( \dfrac{\varepsilon - 1}{\varepsilon + 1} \right)^{1/2}

(23)

Deflection angle

\theta = 2 \arctan{\dfrac{\gamma}{v_0^2 b}}

(24)

General scattering orbits

Deflection angle

\theta = |\pi - 2\varphi_m|

(25)

Angle to $r_{\text{min}}$

\displaystyle \varphi_m = \dfrac{l}{\sqrt{2\mu}}\int_{\infty}^{r_{\text{min}}}{\dfrac{dr}{r^2(E-V_{\text{eff}})}} + \pi

(26)

Distance of closest approach

E-V(r_{\text{min}}) = \dfrac{l^2}{2 \mu r_{\text{min}}^2}

(27)

$r_{\text{min}}$ is the root of the equation.

Cross section

Differential elastic scattering cross section

d\sigma_{\text{el}}=\bigl(\tfrac{d\sigma}{d\Omega}\bigr)_{\text{el}}d\Omega=2\pi b\,db \Rightarrow \bigl(\tfrac{d\sigma}{d\Omega}\bigr)_{\text{el}}=\tfrac{b}{\sin{\theta}}\bigr|\tfrac{db}{d\theta}\bigl|

(28)

Solid angle

d\Omega=\tfrac{dA}{R^2}=\sin\theta\,d\theta d\varphi

(29)

Total elastic scattering cross section

\sigma_T=\int d\sigma_{\text{el}}

(30)

Resources

Wikipedia