Aero 319 Flashcards
Five assumptions of a circular orbit
- Small satellite orbits a massive body on a circular orbit
- m«M
- Acceleration of massive body is negligible
- Massive body is spherically symmetrical
- The only force acting is Newtonian gravity.
Equation for gravitational force, F
GMm/r^2, where r is the distance between the bodies, and G is the universal gravitational constant.
Equation for centripetal acceleration, a
(v^2)/r
Period of motion, T
2πr/v = 2π*√(r^3/GM), where r is replaced by a, the semi-major axis, for elliptical orbits.
What is µ?
GM
Kepler’s three laws
- The planets orbit the sun on elliptical orbits - the sun is at one of the foci of each planet’s ellipse.
- 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.
- The ratio of (orbital period)^2/(mean distance from sun)^3 is constant.
Angular velocity, omega
2π/T
What do the stars of a binary system orbit?
A common centre of mass.
Mass centre equation for binary system circular orbit
M1R1 = M2R2, where R1 + R2 = D, the distance between the stars (measured from their centres).
In Keplerian orbits, L/r = 1 + eCostheta. What do L and e represent?
L is a length parameter that defines the size of the orbit, and e is the eccentricity.
e = 0
Circular orbit
0 < e < 1
Elliptical orbit
e = 1
Parabola - escape trajectory; orbit no longer closed.
e > 1
Hyperbola
Equation for L
(h^2)/µ, where h is the specific angular momentum.
When does the minimum value of r occur, and what’s it called.
It occurs when theta = 0. It’s called the periapsis.
Gravitational potential per unit mass at r, V(r)
-µ/r
Equation for escape velocity, v
√(2GM/r)
Tsiolkovsky’s equation
∆V = Velnµ, where ∆V is the achievable velocity increment, Ve is the effective exhaust velocity, and µ is the mass ratio.
Mass ratio, µ
M0/(M0-mf), where M0 is the total initial mass and mf is the mass of propellant.
Total initial mass, M0
mf + ms + mp
Effective exhaust velocity, Ve
Isp*g0, where Isp is the specific impulse.
Structural coefficient
ms/(ms + mf)
Burnout velocity, Vbo
VeLnµ - gtbo, where tbo is the burnout time.
Height above launch as a function of burn time, H(tb)
((Vetbµ)/(µ-1))(1-(1/µ)(1 + Lnµ)) - 0.5g*tb^2
Assuming constant burn rate, burn time, tb
((µ-1)Ve)/(µphig0), where phi is the thrust to weight ratio.
Total energy, E
((M0Ve^2)/µ)[(1/phi)(1-(1/µ)(1+Lnµ)) - 0.5((µ-1)/(µphi))^2 + 0.5(Lnµ - ((µ-1)/(µphi)))^2]
Mach, M
v/a
Gamma
cp/cv
Conservation of energy
h1 + 0.5v1^2 = h2 + 0.5v2^2, where h is enthalpy.
Change in enthalpy, ∆h (ideal gas only)
cp∆T
Isentropic flow
Flow is adiabatic and reversible (constant entropy)
P/rho^gamma
Constant
p1/p2
(rho1/rho2)^gamma
T1/T2
(rho1/rho2)^gamma-1 = (p1/p2)^(gamma-1/gamma)
Important to note about shockwaves
Flow is irreversible, so isentropic relationships can’t be used.
Exit velocity formula
Ve/a0 = √(2/(gamma-1))(1-(pe/p0)^(gamma-1)/gamma
Maximum possible exit plane velocity, Vemax
a0*√(2/(gamma-1)) = √(2cpT0)
Choked mass flow rate, mdot
rho0a0At*(2/(gamma+1))^(gamma + 1)/2(gamma-1))
F/mdot*a0
Ve/a0 + a0/(gammaVe)(pe/p0 - pb/p0)(pe/p0)^-1/gamma
Maximum F/mdot*a0
√(2/gamma-1)(1 - (pb/p0)^(gamma-1)/gamma)
Rocket motor thrust coefficient, Cf
F/p0At = (mdot/p0At)*(Ve + (a0^2/gammaVe)(pe/p0 - pb/p0)(pe/p0)^-1/gamma
mdot/p0At
(gamma/a0^2)(2/(gamma+1))^(gamma+1)/2(gamma-1)
Characteristic velocity, C*
p0At/mdot
Specific impulse, Isp
F/(mdot*g0) = Ve/g0
Hohmann transfer
Used to transfer a spacecraft from one circular orbit up to a higher circular orbit.
How does a Hohmann transfer work?
An elliptical transfer using two tangential burns, which are assumed to be impulsive and produce a ∆v that’s tangent to the original orbit.
Eccentricity formula
(Rmax - Rmin)/(Rmax + Rmin)
Semi-major axis formula
0.5(Rmax + Rmin)
Specific orbital energy formula, E
0.5((h/L)^2)(e^2 - 1)
Circular orbit orbital velocity, v
√(µ/r)
