Aero 319

Question 1

Five assumptions of a circular orbit

Answer 1

1. Small satellite orbits a massive body on a circular orbit
2. m«M
3. Acceleration of massive body is negligible
4. Massive body is spherically symmetrical
5. The only force acting is Newtonian gravity.

Question 2

Equation for gravitational force, F

Answer 2

GMm/r^2, where r is the distance between the bodies, and G is the universal gravitational constant.

Question 3

Equation for centripetal acceleration, a

Answer 3

(v^2)/r

Question 4

Period of motion, T

Answer 4

2πr/v = 2π*√(r^3/GM), where r is replaced by a, the semi-major axis, for elliptical orbits.

Question 5

What is µ?

Answer 5

GM

Question 6

Kepler’s three laws

Answer 6

1. The planets orbit the sun on elliptical orbits - the sun is at one of the foci of each planet’s ellipse.
2. A line drawn between a planet’s centre and the sun’s centre sweeps out an area at a constant rate as the planets orbit.
3. The ratio of (orbital period)^2/(mean distance from sun)^3 is constant.

Question 7

Angular velocity, omega

Answer 7

2π/T

Question 8

What do the stars of a binary system orbit?

Answer 8

A common centre of mass.

Question 9

Mass centre equation for binary system circular orbit

Answer 9

M1R1 = M2R2, where R1 + R2 = D, the distance between the stars (measured from their centres).

Question 10

In Keplerian orbits, L/r = 1 + eCostheta. What do L and e represent?

Answer 10

L is a length parameter that defines the size of the orbit, and e is the eccentricity.

Question 11

e = 0

Answer 11

Circular orbit

Question 12

0 < e < 1

Answer 12

Elliptical orbit

Question 13

e = 1

Answer 13

Parabola - escape trajectory; orbit no longer closed.

Question 14

e > 1

Answer 14

Hyperbola

Question 15

Equation for L

Answer 15

(h^2)/µ, where h is the specific angular momentum.

Question 16

When does the minimum value of r occur, and what’s it called.

Answer 16

It occurs when theta = 0. It’s called the periapsis.

Question 17

Gravitational potential per unit mass at r, V(r)

Answer 17

-µ/r

Question 18

Equation for escape velocity, v

Answer 18

√(2GM/r)

Question 19

Tsiolkovsky’s equation

Answer 19

∆V = Velnµ, where ∆V is the achievable velocity increment, Ve is the effective exhaust velocity, and µ is the mass ratio.

Question 20

Mass ratio, µ

Answer 20

M0/(M0-mf), where M0 is the total initial mass and mf is the mass of propellant.

Question 21

Total initial mass, M0

Answer 21

mf + ms + mp

Question 22

Effective exhaust velocity, Ve

Answer 22

Isp*g0, where Isp is the specific impulse.

Question 23

Structural coefficient

Answer 23

ms/(ms + mf)

Question 24

Burnout velocity, Vbo

Answer 24

VeLnµ - gtbo, where tbo is the burnout time.

Question 25

Height above launch as a function of burn time, H(tb)

Answer 25

((Vetbµ)/(µ-1))(1-(1/µ)(1 + Lnµ)) - 0.5g*tb^2

Question 26

Assuming constant burn rate, burn time, tb

Answer 26

((µ-1)Ve)/(µphig0), where phi is the thrust to weight ratio.

Question 27

Total energy, E

Answer 27

((M0Ve^2)/µ)[(1/phi)(1-(1/µ)(1+Lnµ)) - 0.5((µ-1)/(µphi))^2 + 0.5(Lnµ - ((µ-1)/(µphi)))^2]

Question 28

Mach, M

Answer 28

v/a

Question 29

Gamma

Answer 29

cp/cv

Question 30

Conservation of energy

Answer 30

h1 + 0.5v1^2 = h2 + 0.5v2^2, where h is enthalpy.

Question 31

Change in enthalpy, ∆h (ideal gas only)

Answer 31

cp∆T

Question 32

Isentropic flow

Answer 32

Flow is adiabatic and reversible (constant entropy)

Question 33

P/rho^gamma

Answer 33

Constant

Question 34

p1/p2

Answer 34

(rho1/rho2)^gamma

Question 35

T1/T2

Answer 35

(rho1/rho2)^gamma-1 = (p1/p2)^(gamma-1/gamma)

Question 36

Important to note about shockwaves

Answer 36

Flow is irreversible, so isentropic relationships can’t be used.

Question 37

Exit velocity formula

Answer 37

Ve/a0 = √(2/(gamma-1))(1-(pe/p0)^(gamma-1)/gamma

Question 38

Maximum possible exit plane velocity, Vemax

Answer 38

a0*√(2/(gamma-1)) = √(2cpT0)

Question 39

Choked mass flow rate, mdot

Answer 39

rho0a0At*(2/(gamma+1))^(gamma + 1)/2(gamma-1))

Question 40

F/mdot*a0

Answer 40

Ve/a0 + a0/(gammaVe)(pe/p0 - pb/p0)(pe/p0)^-1/gamma

Question 41

Maximum F/mdot*a0

Answer 41

√(2/gamma-1)(1 - (pb/p0)^(gamma-1)/gamma)

Question 42

Rocket motor thrust coefficient, Cf

Answer 42

F/p0At = (mdot/p0At)*(Ve + (a0^2/gammaVe)(pe/p0 - pb/p0)(pe/p0)^-1/gamma

Question 43

mdot/p0At

Answer 43

(gamma/a0^2)(2/(gamma+1))^(gamma+1)/2(gamma-1)

Question 44

Characteristic velocity, C*

Answer 44

p0At/mdot

Question 45

Specific impulse, Isp

Answer 45

F/(mdot*g0) = Ve/g0

Question 46

Hohmann transfer

Answer 46

Used to transfer a spacecraft from one circular orbit up to a higher circular orbit.

Question 47

How does a Hohmann transfer work?

Answer 47

An elliptical transfer using two tangential burns, which are assumed to be impulsive and produce a ∆v that’s tangent to the original orbit.

Question 48

Eccentricity formula

Answer 48

(Rmax - Rmin)/(Rmax + Rmin)

Question 49

Semi-major axis formula

Answer 49

0.5(Rmax + Rmin)

Question 50

Specific orbital energy formula, E

Answer 50

0.5((h/L)^2)(e^2 - 1)

Question 51

Circular orbit orbital velocity, v

Answer 51

√(µ/r)