History of Astronomy - Part II

• After the Copernican Revolution, astronomers strived for more observations to help better explain the universe around them

• During this time (1600-1750) many major advances in science and astronomy occurred– Kepler's Laws of Planetary Motion– Newton's Laws of Motion and Gravity

• Warning! - Math and Equations Ahead!

Tycho Brahe - An Observer

• Tycho Brahe was a prominent scholar and aristocrat in Denmark in the mid-late 1500's

• He made a huge number of observations of the stars and planets, all with the naked eye– Even without a telescope, he was

very accurate in his measurements

• Also recorded the appearance of comets and supernovae– The Tycho supernova remnant is

still visible today

Tycho (1546-1601)

Johannes Kepler - A Theorist

• Shortly before his death, Tycho began working with another scientist named Kepler

• Kepler was put to the task of creating a model to fit all of Tycho's planetary data

• Kepler spent the remainder of his life formulating a set of laws that explained the motion of the planets

Kepler (1571 - 1630)

Kepler's First Law• Kepler first noted that the orbital

path of a planet around the Sun is an ellipse, not a perfect circle

• The Sun lies at one of the foci of the ellipse

• The eccentricity of an ellipse is a measure of how 'squished' from a circle the shape is

• Most planets in the Solar System are very close to a perfect circle– Eccentricity, e ~ 0 for a circle

Focus Focus

Kepler's 1st Law: The orbital paths of the planets are elliptical

with the Sun at one focus.

Kepler's First Law

=closest to the Sun=farthest from the Sun

Kepler's Second Law

• Kepler also noticed that the planets sweep out equal areas in their orbit over equal times

• Notice that this means the planet must speed up and slow down at different points

• If it takes the same amount of time to go through A as it does C, at what point is it moving faster?– C, when it is closest to the

Sun

Kepler's 2nd Law: An imaginary line connecting the Sun to any planet

sweeps out equal areas of the ellipse over equal intervals of time.

Kepler's Third Law• Finally, Kepler noticed that

the period of planet's orbit squared is proportional to the cube of its semi major axis

• This law allowed the orbits of all the planets to be calculated

• It also allowed for the prediction of the location of other possible planets

32 aP Kepler's 3rd Law Simplified

NOTE: In order to use the equation as shown, you must be talking about a planet in the Solar System, P must be in years, and

a must be in A.U. !!!

Kepler's Third Law - Examples

• Suppose you found a new planet in the Solar System with a semi major axis of 3.8 A.U.

• A planet with a semi major axis of 3.8 A.U. would have an orbital period of 7.41 years

32 aP

872.548.3 32 P

41.7872.54872.54 21

P years

Kepler's Third Law - Examples

• Suppose you want to know the semi major axis of a comet with a period of 25 years

• A planet with an orbital period of 25 years would have a semi major axis of 8.55 A.U.

23 Pa

6252523 a

55.8625625 331

a A.U.

Isaac Newton• Kepler's Laws were a revolution in

regards to understanding planetary motion, but there was no explanation why they worked

• That explanation would have to wait until Isaac Newton formulated his laws of motion and the concept of gravity

• Newton's discoveries were important because they applied to actions on Earth and in space

• Besides motion and gravity, Newton also developed calculus Newton (1642-1727)

Some terms• Force: the push or pull on an object that in some way

affects its motion• Weight: the force which pulls you toward the center of the

Earth (or any other body)• Inertia: the tendency of an object to keep moving at the

same speed and in the same direction• Mass: basically, the amount of matter an object has

• The difference between speed and velocity– These two words have become identical in common language, but in

physics, they mean two different things– Speed is just magnitude of something moving (25 km/hr)– Velocity is both the magnitude and direction of motion (35 km/hr to

the NE)

Newton's First Law• Newton's first law states: An object at rest will remain at

rest, an object in uniform motion will stay in motion - UNLESS acted upon by an outside force

• This is why you should always wear a seat belt!

Outside Force

Newton's Second Law

• Acceleration is created whenever there is a change in velocity– Remember, this can mean a change in magnitude AND/OR

direction

• Newton's Second Law states: When a force acts on a body, the resulting acceleration is equal to the force divided by the object's mass

• Notice how this equation works:– The bigger the force, the larger the acceleration– The smaller the mass, the larger the acceleration

m

Fa maF or

Newton's Third Law• Newton's Third Law states:

For every action, there is an equal and opposite reaction

• Simply put, if body A exerts a force on body B, body B will react with a force that is equal in magnitude but opposite direction

• This will be important in astronomy in terms of gravity– The Sun pulls on the Earth

and the Earth pulls on the Sun

Newton and the Apple - Gravity

• After formulating his three laws of motion, Newton realized that there must be some force governing the motion of the planets around the Sun

• Amazingly, Newton was able to connect the motion of the planets to motions here on Earth through gravity

• Gravity is the attractive force two objects place upon one another

The Gravitational Force

• G is the gravitational constant – G = 6.67 x 10-11 N m2/kg2

• m1 and m2 are the masses of the two bodies in question

• r is the distance between the two bodies

221

r

mGmFg

Gravity - Examples• Weight is the force you feel due to the gravitational force

between your body and the Earth– We can calculate this force since we know all the variables

26

242

211

221

)10378.6(

)1097.5)(72)(1067.6(

m

kgkgkgmN

r

mGmFg

NFg 7051 Newton is approximately 0.22 pounds

lbsN

lbsNFg 155

1

22.0705

Gravity - Examples• What if we do the same calculation for a person standing on

the Moon?– All we have to do is replace the Earth's mass and radius with the

Moon's

26

222

211

221

)10738.1(

)1035.7)(72)(1067.6(

m

kgkgkgmN

r

mGmFg

NFg 1171 Newton is approximately 0.22 pounds

lbsN

lbsNFg 26

1

22.0117

Gravity - Examples• If gravity works on any two bodies in the universe, why don't

we all cling to each other?– Replace the from previous examples with two people and the distance

with 5 meters

2

2

211

221

)5(

)65)(72)(1067.6(

m

kgkgkgmN

r

mGmFg

NNFg81025.10000000125.0

1 Newton is approximately 0.22 pounds

lbsN

lbsNFg

98 1075.21

22.01025.1

Revisions to Kepler's 1st Law

• Newton's law of gravity required some slight modifications to Kepler's laws

• Instead of a planet rotating around the center of the Sun, it actually rotates around the center of mass of the two bodies

• Each body makes a small elliptical orbit, but the Sun's orbit is much much smaller than the Earth's because it is so much more massive

Revisions to Kepler's 3rd Law

• Gravity also requires a slight modification to Kepler's 3rd Law

• The sum of the masses of the two bodies is now included in the equation

• For this equation to work, the masses must be in units of solar mass (usually written as M)

• Why did this equation work before?

21

32

MM

aP

Remember - for this equation to work:

P must be in years!

a must be in A.U.

M1 and M2 must be in solar masses