Lecture‐4 Conic section Conic sections are locus of points where distances from a fixed point and a fixed line are in constant ratio. Conic sections in 2‐D are curves which are locus of points whore position vector 
satisfies r

r    e.r 
where is a non‐negative constant called semi latus rectum and is a constant eccentricity e


e 0
vector and defined as lies in the plane of the conic section along the periapsis. In Fig. 1 the schematic of a conic section is shown. Fig. 1 From Fig. 1 we can write BF
 a constant = e BC
Moreover, FD  BF cos   BC  r cos   r
e
l
r
 1  e cos  
  r cos    r 

e
e
e


l
  r (1  e cos  )
r
The parameters and can also be written as l
e
l  a 1  e 2   rp 1  e  rp  0
e
where is the semi major axis,   a  
 is semi latus rectum, is eccentricity, and 

is the pericentre distance from the focus. The vectors r and e originate at the focus F. Case (1) If the point (focus) is chosen as origin of the Cartesian coordinate system with  x, y 

e
+ve x‐axis in the direction of then r    ex
Case (2) If the center of the conic section is the origin of the Cartesian coordinate system with +ve x‐axis in the direction of the eccentricity vector then x  ea,
y  0 provided e  1 Thus, shifting the origin to the center of the conic section i.e. replacing by x
x  ea
 r    e( x  ea)  a(1  e2 )  ex  e2 a  a  ex Case (3) On the other hand if the origin is translated from the focus Fr to the pericenter ”A“ ( x  rp , y  0)
then r    e( x  rp )  rp (1  e)  ex  erp.
 r  rp  ex
Note: radial distance is always measured from the focus ”F” The Cartesian equation of the conic section can be written as (with pericenter as the origin) r 2  (rp  x)2  y 2  (rp  ex) 2
 y 2  (1  e)  2rp x  1  e  x 2  This equation is valid for all orbits. It is universal in that no difficulty is encountered for a rp
transition from ellipse to parabola or to hyperbole by holding constant and allowing to "e"
e 1
e  1 increase continuously from to This is not the case when the centre of the orbit is the origin of coordinates. FIG‐16: Separate representation with origin as focus for ellipse FIG‐17: Separate representation with origin as focus for hyperbola FIG‐18: Separate representation with origin as focus for parabola (A) Focus directrix property : The conic sections are locus of points where distances from a fixed point and a fixed line are in constant ratio. Here, the fixed point is F and the fixed line is directrix. This implies BF
 e  a constant BD
e  0, the focus F and the centre C coincide and the conic section gets If the eccentricity reduced to a circle with r  .
Otherwise for e  0

 1  e cos
r
 r
   r cos. e e

r
  x
e
e
(B) Focal radii property If the origin is situated at center of the conic section then coordinates of F can be written as , 0 . The distances to a point on the conic section from the two foci are called focal radii. Figuer‐19 PF 2   x  ea   y 2
2
PF *2   x  ea   y 2
2
 PF *2   x  ea   y 2  4eax
2
 PF *2  PF 2  4eax
But PF  r  a  ex
 PF *2   a  ex   4eax
2
  a  ex 
2
which implies PF is  ve
 a  ex ellipse a  0
PF *  
   a  ex  hyperbola a  0, x  0
(B) Orbital Tangents (3rd property)‐ Focal‐radii to a point on an orbit make equal angles with the tangents to the curve at that point. [Prove yourself]. Calculation of semi‐major and semi‐minor axes “a” and “b” respectively: Figuer‐20 r
l
1  e cos 
For   900
l
l
1 0
and for  =0
rl 

1 e

ra 
1 e
 2a  rp  ra
rp 



1 e 1 e
1  1  e  1  e 
 1
 


2
 
1  e 1  e   1  e
2
2a 
1  e2
   a 1  e2 

 rp  a 1  e 
and ra  a 1  e 
b 2  r 2   a  a 1  e  

2
b 2  r 2  a 2e2

1  e cos  
b2 
2
a e
2 2
2
a 2 1  e 2 
2
1  e cos  
2
 a 2e2
Also  cos 
 Projection on axis = ae 1  e cos  .
  a 1  e 2  cos   ae  ae 2 cos 
  a cos   a e 2 cos   ae  ae 2 cos   e   cos 
Substituting is equation for b 2 b2 
a 2 1  e 2 
2
1  e 
2 2
 a 2e2
 a 2  a 2 e 2  a 2 1  e 2   b  a 1  e2
Now it can be proved that for the centre of the ellipse taken as origin x2 y 2

 1 a 2 b2
Rectilinear ellipse A rectilinear ellipse is defined by zero angular momentum of the particle in the orbit. In addition, e=1 i.e two focii move towards and the apoapsis and periapsis respectively. Using x2
y2

1
a 2 a 2 (1  e2 )
and taking limit as e  1
lim x 2 (1  e 2 )  y 2  a 2 (1  e 2 )
e 1
 y2  0  y  0
hence the ellipse degenerates into a line
Pair of straight lines If then e
x2
y2

1 a 2 a 2 1  e 2 
gets reduced to x2  a2  x  a This is equation of a pair of straight lines. Here, a hyperbola degenerates into a pair of straight lines Figuer‐21 Hyperbola If e  1 hyperbola. r

1  e cos 
when cos   
1
e
r 

   cos 1  1
e

defines asymptote and is known as true anomaly of the asymptote 
Now sin 2    cos 2    1
 sin 2    1 
 sin   
1 e2  1
 2 e2
e
e2  1
e
Vacant orbit is physically impossible because it requires repulsive gravitational field.   
b
 tan 
a
 b  a tan (180    )

a

a
sin 180    
cos 180    
a sin  
cos  
e2  1 e

e
1
 b   a e2  1
Figure‐22 1
 
1  1 
   cos   
 e
  2sin 1  
e
Equation of hyperbola x2 y 2

1
a 2 b2

ra 
1  e cos180


1 e
Since e  1  ra is negative