Dynamical Astronomy

Question 1

one-body systems

Answer 1

systems with two masses, where
one is significantly larger than the other.

only the lighter body moves significantly

Question 2

Kepler’s 1st Law

Answer 2

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)

Question 3

Kepler’s 2nd Law

Answer 3

For any planet, the line connecting the planet to the Sun
sweeps out an equal area in an equal time

Question 4

Kepler’s 3rd Law

Answer 4

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

Question 5

Newton’s 1st law

Answer 5

Every body continues in its state of rest or uniform motion in a
straight line until acted on by an external force

Question 6

Newton’s 2nd law

Answer 6

The rate of change of momentum of a body is proportional to
the applied force, and is in the direction of the force

Question 7

Newton’s 3rd Law

Answer 7

To every action, there is an equal and opposite reaction

Question 8

a planet moving in a circular orbit must…

Answer 8

be feeling a force and must be accelerating

Question 9

proving kepler 3

Answer 9

using centripetal acceleration and gravity

Question 10

natural units for the solar system

Answer 10

au and sidereal years

Question 11

geostationary

Answer 11

if a satellite orbits Earth in the equatorial plane with a period of 1 sidereal day

Question 12

calculating orbital radius

Answer 12

using Newton’s second law and universal law of gravitation

Question 13

Conservation of angular momentum in elliptical orbits

Answer 13

It is conserved in orbits because the force of gravity acts along the line joining
the masses, so cannot exert a torque on them.

Question 14

how is angular momentum changed?

Answer 14

twisting forces (torque)

Question 15

why is L=mva for circular motion

Answer 15

sin(pi/2)=1

Question 16

eccentricity e=0

Answer 16

circle

Question 17

eccentricity e= close to 1

Answer 17

long,thin orbit

Question 18

eccentricity e=1

Answer 18

parabola (not a closed orbit)

Question 19

eccentricity e>1

Answer 19

hyperbola

Question 20

relating the parameter in an ellipse

Answer 20

AC+BC=constant

Question 21

perihelion

Answer 21

point in orbit when closest to the sun
rp=a-ae

Question 22

aphelion

Answer 22

point in orbit when furthest from the sun
ra=a+ae

Question 23

name of perihelion and aphelion for orbits around earth

Answer 23

perigee and apogee

Question 24

name of perihelion and aphelion for orbits around stars

Answer 24

periastron and apastron

Question 25

name of perihelion and aphelion for orbits in general

Answer 25

periapsis and apoapsis

Question 26

how to equate angular momentum at A and P

Answer 26

using Kepler's second law L=mva(a+ae)=mvp(a-ae) which rearranges to vp/va=1+e/1-e

Question 27

how to find speed of circular orbit

Answer 27

equation gravitational force with centripetal force

Question 28

case 1 - kick in the direction of motion

Answer 28

speed of mass increases new orbit is elliptical radius of old orbit becomes periastron of new orbit

Question 29

case 2 - kick in the opposite direction to motion

Answer 29

speed of mass decreases new orbit is elliptical radius of old orbit becomes apastron of the new orbit

Question 30

case 3 - radial kicks perpendicular to direction of motion

Answer 30

no torque applied so no change in angular momentum velocity does not change radius of old orbit becomes semilatus rectum of new orbit

Question 31

radial kicks always...

Answer 31

increase kinetic energy

Question 32

conserved quantities for two stars orbiting common centre of mass

Answer 32

linear momentum m1v1=m2v2 velocities have opposite directions angular momentum Ltotal=m1r1v1sinθ1+m2r2v2sinθ2

Question 33

conservation of energy - total energy of a planet or satellite

Answer 33

kinetic energy and gravitational potential energy

Question 34

gravitational potential energy

Answer 34

the minus the work done to separate two masses to an infinite distance apart

Question 35

derivation of gravitational potential energy

Answer 35

first consider work done for small change in distance integrate with limits of r and infinity

Question 36

total energy for one-body orbits

Answer 36

Etotal=1/2mv^2-Gm1m2/r

Question 37

escape velocity

Answer 37

if E>0 then the mass escapes the gravitational attraction the speed at which it just escapes is the escape velocity

Question 38

polar coordinates radial vector

Answer 38

r defined from the focus

Question 39

polar coordinates true anomaly

Answer 39

theta, defined from the periapsis side of the major axis to r

Question 40

most fuel effecient method of changing orbit

Answer 40

Hohmann transfer orbit

Question 41

stage 1 Hohmann transfer orbit

Answer 41

boost speed to inject into transfer orbit (velocity required at perigee)

Question 42

Hohmann transfer orbit intermission

Answer 42

wait until it reaches exit point, the apogee point of the orbit in half a period

Question 43

stage 2 Hohmann transfer orbit

Answer 43

need to change velocity again to enter GEO

Question 44

Are Hohmann transfers effective?

Answer 44

minimises fuel required (just two small burns) not designed to minimise trip time

Question 45

aerobraking

Answer 45

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

Question 46

how does a gravity assist work?

Answer 46

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

Question 47

gravity assist - where does the extra momentum and energy come from?

Answer 47

actually slows down the planet (change infinitesimal)

Question 48

deriving for the two body problem

Answer 48

centre of mass remains stationary use normal gravitational force and sub in m1r1=m2r2 for r2 and define a mass

Question 49

two-body problem

Answer 49

two masses always sit on same line passing through centre of mass trace out same ellipse, one just larger eccentricities same

Question 50

changing equations for 2-body systems

Answer 50

M=m1+m2 a=a1+a2 v=v1+v2