Download Kepler`s Law Applied to the Planets 10 868 870 9.54 29.5 Saturn 13

Kepler’s Law Applied to the Planets
Planet
T (yr)
R (AU)
T2
R3
Speed
(km/s)
Mercur
y
0.24
0.39
0.06
0.06
61
Venus
0.62
0.72
0.39
0.37
38
Earth
1
1
1
1
30
Mars
1.88
1.52
3.53
3.51
24
Jupiter
11.9
5.2
142
141
13
Saturn
29.5
9.54
870
868
10
Critical Speeds to Remember
Speed of Solar Wind: 400 km/s
Orbital Speed of Earth around Sun: 30 km/s
Speed of Satellite around the Earth in Low Earth
Orbit: 8 km/s
Critical Speeds to Remember
Speed of Solar Wind: 400 km/s
Orbital Speed of Earth around Sun: 30 km/s
Planetary speeds go as 1/r1/2 e.g. Mars 24 km/s
Speed of Satellite around the Earth in Low Earth
Orbit: 8 km/s
How to Get to Mars?
Place the
planets ready
for a mission
from Earth to
Mars
Does the
Configuration
work?
No! The
planets are
moving at high
speed around
the Sun
Mars
Earth
Possible Solution
(1) Start Earth
Behind Mars
Mars
Earth
Speeding Up and Slowing Down
(a) To move outwards
into the solar system,
spacecraft has to
increase speed
relative to the
planet
(b) Faster you
travel from Earth
means more
braking required
at arrival at the
planet
Mars at
arrival
Earth at
arrival
Mars at
launch
Earth at
launch
Increase Orbital
speed
e.g. Earth 30 km/s, spacecraft + 7 km/s; Mars is at 24 km/s; Braking 13 km/s
Low Energy/ Low Cost Run
One-Way Trip
(c) Lowest energy
configuration, 6-9 month
trip with changes in speed
of only 3-5 km/s from
Earth speed
(d) Use atmospheric
drag and air-bags to drop
to the planet at safe
speeds
Mars at
arrival
Earth at
arrival
Mars at
launch
Earth at
launch
Delta V = 3 km/s
Return Trips
Mars at
arrival
(1) 3 Months Later
Requires
7km/s
acceleration
plus
deceleration at
Mars
Earth at
arrival
Mars at
launch
Earth at
launch
Stay On Mars
(2) 1 Month Stay
on Mars
Mars at
departure
Mars at
arrival
Earth at
departure
Earth at
arrival
Mars at
launch
Earth at
launch
Orbital Transfer
(3) To get back to Earth,
Mars at
arrival
Must reduce speed
(4) But Earth
is in front of
Mars, and
both are
moving
around the
Sun
Requires
planetary
alignment
before
return
attempted
Earth at
arrival
Delta V = -3 km/s
Mars at
launch
Earth at
launch
Departure from Mars
Wait on Mars for
1.5 Yrs
Earth at
arrival
Earth gets
behind
Mars
Earth at
departure
Total Mission
Time: ~ 2.6 yrs
Earth on
Return
Mars on
Return
Mars at departure
Return planetary trips is very difficult
Fast Mars Mission
Travel faster to
Mars – requires
speeds of 20 km/s
Requires much
greater
power/fuel
Advantage:
Earth never
overtakes Mars
Quick Return
Home Possible
Mission < 0.5 yr
Earth on
Return
Earth at
departure
Earth at
arrival
Mars on
Return
Mars at
departure
Mars at
arrival
Mars at
launch
Earth at
launch
Downside- high speeds
are needed at each step
Conservation of Momentum
Momentum: Mass times velocity
Conserved quantity : sum of parts all the same
Mcraft × Vcraft + Mfuel × Vfuel = 0 (initially at rest)
Mcraft × Vcraft = − Mfuel × Vfuel at all times
Speed limit set by Fuel Characteristics
Vcraft = − Mfuel × Vfuel
Mcraft
Minus sign because fuel
goes in the opposite
direction of spacecraft
Optional: Rocket equation takes into account for finite burn time of
fuel:
M fuel = M craft exp( v craft / v fuel )
Faster you want go, you have to carry increasingly more fuel
Chemical Propellants
H2 + O
Î
H2 O
Advanced Chemical
3-4 km/s
5 km/s
Any Chemical system Î Mass of Fuel Greatly Exceeds
Mass of Payload
Requirement is hotter/faster propellants