Dynamical Astronomy Flashcards
one-body systems
systems with two masses, where
one is significantly larger than the other.
only the lighter body moves significantly
Kepler’s 1st Law
The orbit of each planet is an ellipse with the Sun at one focus
(centre of mass of the system at the focus but sun is much more massive than planets)
Kepler’s 2nd Law
For any planet, the line connecting the planet to the Sun
sweeps out an equal area in an equal time
Kepler’s 3rd Law
The cubes of the semi-major axes of the planetary orbits are proportional to the squares of the planet’s orbital periods
a^3/T^2=constant
Newton’s 1st law
Every body continues in its state of rest or uniform motion in a
straight line until acted on by an external force
Newton’s 2nd law
The rate of change of momentum of a body is proportional to
the applied force, and is in the direction of the force
Newton’s 3rd Law
To every action, there is an equal and opposite reaction
a planet moving in a circular orbit must…
be feeling a force and must be accelerating
proving kepler 3
using centripetal acceleration and gravity
natural units for the solar system
au and sidereal years
geostationary
if a satellite orbits Earth in the equatorial plane with a period of 1 sidereal day
calculating orbital radius
using Newton’s second law and universal law of gravitation
Conservation of angular momentum in elliptical orbits
It is conserved in orbits because the force of gravity acts along the line joining
the masses, so cannot exert a torque on them.
how is angular momentum changed?
twisting forces (torque)
why is L=mva for circular motion
sin(pi/2)=1
eccentricity e=0
circle
eccentricity e= close to 1
long,thin orbit
eccentricity e=1
parabola (not a closed orbit)
eccentricity e>1
hyperbola
relating the parameter in an ellipse
AC+BC=constant
perihelion
point in orbit when closest to the sun
rp=a-ae
aphelion
point in orbit when furthest from the sun
ra=a+ae
name of perihelion and aphelion for orbits around earth
perigee and apogee
name of perihelion and aphelion for orbits around stars
periastron and apastron
name of perihelion and aphelion for orbits in general
periapsis and apoapsis
how to equate angular momentum at A and P
using Kepler’s second law
L=mva(a+ae)=mvp(a-ae)
which rearranges to
vp/va=1+e/1-e
how to find speed of circular orbit
equation gravitational force with centripetal force
case 1 - kick in the direction of motion
speed of mass increases
new orbit is elliptical
radius of old orbit becomes periastron of new orbit
case 2 - kick in the opposite direction to motion
speed of mass decreases
new orbit is elliptical
radius of old orbit becomes apastron of the new orbit
case 3 - radial kicks perpendicular to direction of motion
no torque applied so no change in angular momentum
velocity does not change
radius of old orbit becomes semilatus rectum of new orbit
radial kicks always…
increase kinetic energy
conserved quantities for two stars orbiting common centre of mass
linear momentum m1v1=m2v2
velocities have opposite directions
angular momentum
Ltotal=m1r1v1sinθ1+m2r2v2sinθ2
conservation of energy - total energy of a planet or satellite
kinetic energy and gravitational potential energy
gravitational potential energy
the minus the work done to separate two masses to an infinite distance apart
derivation of gravitational potential energy
first consider work done for small change in distance
integrate with limits of r and infinity
total energy for one-body orbits
Etotal=1/2mv^2-Gm1m2/r
escape velocity
if E>0 then the mass escapes the gravitational attraction
the speed at which it just escapes is the escape velocity
polar coordinates radial vector
r defined from the focus
polar coordinates true anomaly
theta, defined from the periapsis side of the major axis to r
most fuel effecient method of changing orbit
Hohmann transfer orbit
stage 1 Hohmann transfer orbit
boost speed to inject into transfer orbit (velocity required at perigee)
Hohmann transfer orbit intermission
wait until it reaches exit point, the apogee point of the orbit in half a period
stage 2 Hohmann transfer orbit
need to change velocity again to enter GEO
Are Hohmann transfers effective?
minimises fuel required (just two small burns)
not designed to minimise trip time
aerobraking
make elliptical orbit more circular
Mar’s thin atmosphere is a source of friction which reduces periapsis speed. This reduces semi-major axis and eccentricity
how does a gravity assist work?
spacecraft approaches stationary planet at a speed > escape velocity.
Trajectory of the spacecraft is deflected into a hyperbolic orbit
accelerates as it is attracted to planet the decelerates as it flies away
gravity assist - where does the extra momentum and energy come from?
actually slows down the planet (change infinitesimal)
deriving for the two body problem
centre of mass remains stationary
use normal gravitational force and sub in m1r1=m2r2 for r2 and define a mass
two-body problem
two masses always sit on same line passing through centre of mass
trace out same ellipse, one just larger
eccentricities same
changing equations for 2-body systems
M=m1+m2
a=a1+a2
v=v1+v2
