Chapter 4 : Understanding Orbits

Question 1

At what rate is everything falling towards Earth?

Answer 1

every 5 meters in the horizontal direction, an object would drop 8 meters in the vertical direction, or 7.9 km/s

Question 2

How does a satellite not fall back to Earth?

Answer 2

It does fall back to Earth! Anything in orbit is always in free fall towards the earth, however the horizontal (or tangential) speed keeps the object from dropping

Question 3

List the Motion Analysis Process

Answer 3

1. define coordinate system
2. equation of motion
3. simplifying assumptions
4. initial conditions
5. testing the model and error analysis

Question 4

What is Newton’s first law?

Answer 4

body at rest tends to stay at rest to until an outside force acts upon it.
body in motion tends to stay in motion until an outside force acts upon it.

Question 5

What is Newton’s second law?

Answer 5

time rate of change for momentum. 
dρ/dt=(d(mv))/dt=m (dv/dt)+v (dm/dt)
a= dv/dt, 
F=ma if mass is constant
F=ma + (dm/dt)v is m and v are changing

Question 6

What is Newton’s third law?

Answer 6

for every force acted upon a body, there is an equal force acted in the opposite direction

Question 7

Explain the force of gravity

Answer 7

Fg=mg=(GMm)/(r^2)
ma = (Gm)/(r^2)
where G=6.67E-11 Nm^2/kg^2
μ,earth = GM,earth = 3.986E14 m^3/s^2

Question 8

Describe the law of conservation of energy and the equations of energy.

Answer 8

for a orbiting space craft
KE=1/2 mv^2
PE= -mμ/R
E=1/2 mv^2 - mμ/R

Question 9

Describe the geocentric-equatorial coordinate system

Answer 9

1. origin is at Earth’s center
2. fundamental plane = equator
3. principle direction = vernal equinox (the line from Earth to the sun on the first day of spring)
4. third axis found using right hand rule

Question 10

What is included in equation of motion (the total external forces)

Answer 10

Forces of gravity, drag, thrust, 3rd body, and other, all equal mass*acceleration

Question 11

List the assumptions made with the equation of motion.

Answer 11

1. F,drag = 0 (high enough from atmosphere)
2. F,thrust = 0 (space craft wont maneuver)
3. F,3rd body = 0 (close to earth so ignore the moon and the sun… earth is primary force)
4. F,other = 0 (compared to Earth’s gravity, everything else is negligible.
5. M,earth&raquo_space; m,spacecraft
6. Earth has uniform density (treat as point mass)
7. spacecraft mass = constant
8. there is sufficient inertia (newtons laws apply)

Question 12

Orbital Geometry, R = ?

Answer 12

• spacecrafts position vector measured from Earth’s center

- radius from the focus of the ellipse to the orbiting object

Question 13

Orbital Geometry, V = ?

Answer 13

spacecraft’s velocity vector

Question 14

Orbital Geometry, F and F’ = ?

Answer 14

the primary (occupied) and vacant (unoccupied) foci of the ellipse (Earth’s center is at the primary foci.

Question 15

Orbital Geometry, R,p = ?

Answer 15

radius of perigee (Earth) or periapsis (other)

radius of the closest approach

Question 16

Orbital Geometry, R,a = ?

Answer 16

radius of apogee(Earth) or apoapsis (other)

radius of the farthest approach

Question 17

Orbital Geometry, 2a = ?

Answer 17

major axis

2a=R,p + R,a

Question 18

Orbital Geometry, 2b = ?

Answer 18

minor axis

Question 19

Orbital Geometry, 2c = ?

Answer 19

distance between the foci

R,a - R,p

Question 20

Orbital Geometry, a = ?

Answer 20

semi-major axis
a > 0 circle
a > 0 ellipse
a = infinity parabola
a < 0 hyperbola

Question 21

Orbital Geometry, b = ?

Answer 21

semi-minor axis

Question 22

Orbital Geometry, ν (nu) = ?

Answer 22

true anomaly - polar angle measured from perigee to the spacecraft’s position vector R, in the direction of the spacecraft’s motion. 0 < v <360

Question 23

Orbital Geometry, φ = ?

Answer 23

flight path angle - measured from the loacl horizon to the velocity vector V. The local horizontal is a line perpendicular to the position vector R at spacecraft
φ > 0 outbound (perigee to apogee)
φ < 0 inbound (apogee to perigee)
φ = 0 at perigee and apogee

Question 24

Orbital Geometry, e = ?

Answer 24

eccentricty - ration of the distance between the foc (2c) to the length of the ellipse (2a)
e = 2c/2a
defines the shape or type of conic section
e=0 , circle
e approaching 1 = long narrow ellipse, 01 , hyperbola

Question 25

What is specific mechanical energy?

Answer 25

ε, mechanical energy that does not depend on mass

Question 26

Describe the specific mechanical energy at perigee and apogee.

Answer 26

@ perigee - PE is min (shortest distance), KE is max (fastest velocity)
@ apogee - PE is max (farthest distance), KE is min (slowest velocity)

Question 27

What do the signs of ε stand for? ( +, -, 0)

Answer 27

ε = (-) for circular, a = (+)
ε = (-) for elliptical, a = (+)
ε = 0 for parabola, a = infinity
ε = (+) for hyperbola, a = (-)

Question 28

What two things are conserved in the 2-body diagram?

Answer 28

specific mechanical energy and

angular momentum