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Colorado Space Grant ConsortiumColorado Space Grant ConsortiumMarch 19, 2001March 19, 2001

TodayToday

- Announcements- Announcements

- Discussion on Orbits- Discussion on Orbits

Announcements:Announcements:

- Status reports due last Friday- Status reports due last Friday

- Don’t forget about outreach requirementDon’t forget about outreach requirement

- I won’t be here on WednesdayI won’t be here on Wednesday

- Andrew Busbee and Steve Wichman will be Andrew Busbee and Steve Wichman will be speakingspeaking

Orbits:A Brief Historical Look

Arthur C. Clark

Discovered This Orbit

Ancient Orbit History:

“ORBIT” from Latin word “orbita” orbitus = circular orbis = orb

• 1800 B.C. Stonehenge

- Study of the vernal equinox

• 1500 B.C. Egyptians and Babylonians

- Written evidence of stellar observations- Time divided into 60 even units

Ancient Orbit History:

• 350 B.C. Aristotle

- Said earth is center of the universe- Dominated scientific thought for 1800 years

Aristarchus

- Said that is B.S.

Kinda Old Orbit History:

• 1543 A.D. Nicholas Copernicus

- Said Sun-centered rotations- Measurements crude but thinking shifts

• 1580 A.D. Tycho Brahe

- Accurate measurements of planets as a function of time- Even though telescope had not been invented

Orbit History :

• 1610 A.D. Galileo Galilei

- Good friends with Copernicus- Observations with TELESCOPE reinforced Copernicus- The wrath of the Catholic Church

Orbit History:

• 1600 A.D. Johannes Kepler

- Used Tycho’s careful observations to smash Aristotle theories- Presented 3 laws of planetary motion- Basis of understanding of spacecraft motion- However, “Why was not understood”- Calculus?

Orbit History:

Kepler’s 3 Laws of Planetary Motion1. All planets move in elliptical orbits,

sun at one focus

2. A line joining any planet to the sun,sweeps out equal areas in equal times

3. The square of the period of any planet about the sun is proportional to the cube of the of theplanet’s mean distance from the sun.

Orbit History:

• 1665 A.D. Isaac Newton

- At 23, plague while at Cambridge- Went to be one with nature- He studied gravity- Discovered “Newton’s Laws of Motion”

F = m a- 1666, he understood planetary motion- Did zip for 20 years until Edmund Halley

Newton’s Laws:

1st Law.....Body at rest stays at rest, a body in motion

stay in motion

2nd Law....F = m * a

3rd Law...For every action, there is an equal and opposite reaction

Newton’s Laws:

Newton Continued... - 1687, Principia Published

- Law of Universal Gravitation (Attraction)

FMmG

r 2 F

mV

r

2

Universal Gravitation, Applied:

- When in space why do you float? i.e. Weightlessness

mV

r

MmG

r

2

2

Types of Orbits:

Types of Orbits:

Types of Orbits:

Earth, the Moon, Mars, and the Stars Beyond

A Brief Discussion on Mission Design

Kepler:Kepler’s Laws...Orbits described by conic sections

Velocity of an orbit described by following equation

For a circle (a=r):

For a ellipse (a>0):

For a parabola (a=):

v2 r

a

GM

v r

v2 r

a

v2 r

Circular Orbit:

For a 250 km circular Earth Orbit

Orbital Velocity

Orbital Period

v r

v398600.4

(250 6378.14)

v7.75 km

sec 17,347 mph

P circumference

velocity

P 2r3

P 2 (250 6378.14)3

398600.4

P 5,370 sec 89.5 min

Circular Orbit:

For a 500 km circular Earth Orbit

Orbital Velocity

Orbital Period

Conclusions???

v r

v398600.4

(500 6378.14)

v7.61 km

sec 17,028 mph

P 2 r3

P 2(500 6378.14)3

398600.4P 5,676 sec 94.6 min

Changing Orbits:

How about 250 km to 500 km

How would you do it?

Changing Orbits:

Changing orbits usually involves an elliptical orbit

Perigee = closeApogee = far

Since orbit is elliptical a > 0, so

where

v2 r

a

a r1 r2

2

a (250 6378.14) (500 6378.14)

2a 6753 km

Changing Orbits:

Here’s what you need:1) Velocity of initial orbit

2) Velocity of final orbit

3) Velocity at perigee

4) Velocity at apogee

Then figure out your V’s

v f 7.61 km

sec

vi 7.75 km

sec

vper 2 r

a

vper 2* 398600.4

(250 6378.14)

398600.4

6753

vper 7.83 km

sec

vapo 7.54 km

sec v1 vper vi

v2 vapo v f

Changing Orbits:

Therefore:V1 is to start transfer

V2 is to circularize orbit

Time to do transfer is

v1 vper vi

v1 7.83 7.75

v1 .08 km

sec

v2 vapo v f

v2 7.54 7.61

v2 .07 km

sec

P 2 a3

*.5

P 2(6753)3

398600.4* .5

P 2,761 sec 46 min

v2v1

How well do you understand Hohmann Transfers?

123

• 1 to 2?

• 2 to 3?

• 3 to 1?

• 1 to 3?

Changing Orbits:

Also something called “Fast Transfer”

• It is more direct and quicker

• However it takes more fuel

• V1 and V2 are much bigger

From Earth Orbit to the Moon:

• Same as changing orbits but....- At apogee you don’t have empty space- Instead, you have a large and massive object

• Gravity from this object can act as a V against your spacecraft

• When going to the Moon the following could happen:1) Gravity will cause your spacecraft to crash into the surface2) Gravity will cause your spacecraft to zip off into space for a

long time

Getting to the Moon:

Gravity Assist

v2

v1

Apollo XIII:

Apollo XIII:

Apollo XIII:

Apollo XIII:

To the Moon for Money:

To the Moon for Money:

To the Moon for Money:

Earth to L1:

Earth to Mars:

Mars Orbit

Earth Orbit

Transfer Orbit

Initial Orbit

Final Orbit

v2v1

Earth to Beyond:

Say you are in a 250 km orbit...Orbital Velocity:

Velocity on parabolic (a=escape trajectory:

V needed:

V will not put you in a orbit, you will escape the Earth’s

gravity never to come back

sec 75.7 km

vi

v2 r

vesc 2 * 398600.4

(250 6378.14)

vesc 10.97 km

sec

vesc 3.22 km

sec

QuestionsQuestions