PH6L19 1

Central forces

20 )cos(Lkm

Au −−= φφ

)cos(102 φφ −+−= A

Lkm

r

⎟⎟⎠

⎞⎜⎜⎝

⎛−−−= )cos(11

0

2

2 φφkmAL

Lkm

r(2)

Similar to equation

φε cos1+=rl

(3)

Equation of conic section with origin at the focus

PH6L19 2

Central forces1 cosl e

rϕ= +

PH6L19 3

2 2

1 1/ 1 ( / )cos

rmk L AL mk ϕ

=− + −⎟

⎟⎠

⎞⎜⎜⎝

⎛−−−= )cos(11

0

2

2 φφkmAL

Lkm

r

ϕcos1

12

2

⎟⎟⎠

⎞⎜⎜⎝

⎛−+

⋅−=

mkALmk

Lr0

11 cos

ere ϕ+

=+

2 2

01

1AL Le and rmk mk e

= − = −+

2

11

1Lrmk e

= −−

r0 is the minimum value of the radius (perihelion for the earth’s orbit and perigee for an orbit around Earth)

Let r1 be the largest radius (aphelion or apogee)

r0 , r1 are the turning points for the orbit where Veff = E

PH6L19 4

Central forces2

22

12 2

k LE mrr mr

= + + At turning point 0r =

22 1 1 02L k Em r r⎛ ⎞ ⎛ ⎞+ − =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ quadratic equation

( )( )( )

1 / 22 2

2

4 / 212 / 2

k k L m E

r L m

⎡ ⎤− ± − −⎣ ⎦=

1/ 22

2 2 20

1 2km mk mEr L L L

⎡ ⎤⎛ ⎞= − + +⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

1/ 22

2 2 21

1 2mk mk mEr L L L

⎡ ⎤⎛ ⎞= − − +⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

PH6L19 5

Central forces

V eff

O-K

/r

2

2

2mrL

r

r0 r1E

PH6L19 6

01

1 coser r

e ϕ+

=+

Central forces

( )

( )

2

2 20

20

2

2 21

21

1 1 1

1

1 1 1

1

m k m k A Ler m kL L

m k Ar L

m k m k A Ler m kL L

m k Ar L

⎛ ⎞= − + = − −⎜ ⎟

⎝ ⎠

= − +

⎛ ⎞= − − = − +⎜ ⎟

⎝ ⎠

= − −

1/ 22

2 2 20

1 2mk mk mEr L L L

⎡ ⎤⎛ ⎞= − + +⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

1/ 22

2 2 21

1 2mk mk mEr L L L

⎡ ⎤⎛ ⎞= − − +⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

From the solution for the equation for the orbit

1 / 22 2 2

4 2 2

2 21m k mE ELA eL L mk

⎡ ⎤= + = +⎢ ⎥⎣ ⎦

Comparing the two equations

PH6L19 7

Central forcesNature of the Orbits

The nature of the orbit is entirely determined by the value of the eccentricity e . The eccentricity depends on total energy E

2

2

21 ELemk

= +

Value of energy Value of eccentricity

Nature of the orbit

E > 0 e > 1 hyperbola

E = 0 e = 1 parabola

(Veff)min < E < 0 0 < e < 1 ellipse

E = (Veff)min e = 0 Circle

PH6L19 8

r1

r2

F1 F2

EllipseC

(0,0)

r1 + r2 = 2a

Major axis=2a

Min

or a

xis=

2b

kE

= =Major axis

(c,0)

Dire

trix

2axc

=

2 2

2 2 1x ya b

+ =

The ellipse can also be defined as the locus of the points whose distance from the focus is proportional (e times) to the horizontal distance from a vertical line known as directrix

PH6L19 9

2

21 bea

= − 2 2 2(1 )b a e= −

2 2 2 2 2a b a e c− = =

2 2 cb a c c ae ea

= − = =2(1 )

1 cosa er

e ϕ−

=+

The maximum and minimum distances from the focus are called the apoapsis and periapsis, and are given by

)1(

)1(

earr

earr

periapsis

apoapsis

−==

+==

−

+

Area = π a b

PH6L19 10

Any point on a parabola is the same distance from the directrix as it is from the focus.

Vertex (0,0) Focus (a,0)

x=-a

Semi Latus recturm (2a)

PH6L19 11

Equation of a parabola

axy 42 = latus rectumThe quantity is known as the 4a .

If the vertex is at instead of (0, 0), the equation of the parabola is

)(4)( 02

0 xxayy −=−

If the parabola instead opens upwards, its equation is

)(4)(4

02

0

2

yyaxxorayx

−=−

=

y

x

PH6L19 12

Parametric equation for a parabola

atyandatx 22 ==

Polar equation

In polar coordinates, the equation of a parabola with parameter and center (0, 0) is given by a

21 cos

arφ

=− φ

PH6L19 13

The equation of a hyperbola with centre located at (x0,y0) is

1)()(2

20

2

20 =

−−

−b

yya

xx

• A hyperbola is the locus of all the points where the differencebetween the distances to each foci is constant.

r2

r1

arr 212 =−

PH6L19 1414

Like noncircular ellipses, hyperbolae have two distinct foci and two associated conic section directrices, each directrix being perpendicular to the line joining the two foci

inverse square law orbits

In polar coordinates centered at a focus, 

φcos1)1( 2

eear

−−=

00

1 1 [1 cos( )]er r

ϕ ϕ= + ±or, equivalently,