Today’s APOD

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The Sun Today

A100 Solar System

Kepler's Laws were a revolution in regards to understanding planetary motion, but there was no explanation why they worked

The explanation was provided by Isaac Newton when formulated his laws of motion and gravity

Newton recognized that the same physical laws applied both on Earth and in space.

And don’t forget the calculus! Isaac Newton

Remember the Definitions• Force: the push or pull on an object that

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: the amount of matter an object has

Newton's First LawAn object at rest will remain at rest, an

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

Outside Force

Newton's Second Law

When a force acts on a body, the resulting acceleration is equal to the force divided by the object's mass

Acceleration – a change in speed or direction

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

m

Fa maF or

Newton's Third Law

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• 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

Determining the Mass of the Sun

How do we determine the mass of the Sun? Put the Sun on a scale and determine its

weight??? Since gravity depends on the masses of both

objects, we can look at how strongly the Sun attracts the Earth

The Sun’s gravitational attraction keeps the Earth going around the Sun, rather than the Earth going straight off into space

By looking at how fast the Earth orbits the Sun at its distance from the Sun, we can get the mass of the Sun

Measuring Mass with Newton’s Laws - Assumptions to Simplify the

CalculationAssume a small

mass object orbits around a much more massive objectThe Earth around

the SunThe Moon around

the EarthCharon around Pluto

Assume the orbit of the small mass is a circle

Measuring the Mass of the Sun

The Sun’s gravity is the force that acts on the Earth to keep it moving in a circle

MSun is the mass of the Sun in kilograms MEarth is the mass of the Earth in kilograms r is the radius of the Earth’s orbit in kilometers

The acceleration of the Earth in orbit is given by:a = v2/r

where v is the Earth’s orbital speed

2rmGM

F EarthSung

rv

mamF EarthEarth

2

Measuring the Mass of the Sun

Set F = mEarthv2/r equal to F = GMSunmEarth/r2

and solve for MSun

MSun = (v2r)/G

The Earth’s orbital speed (v) can be expressed as the circumference of the Earth’s orbit divided by its orbital period:

v = 2r/P

Measuring the Mass of the Sun Combining these last two equations:

MSun = (42r3)/(GP2)

The radius and period of the Earth’s orbit are both known, G and π are constants, so the Sun’s mass can be calculated

This last equation in known as Kepler’s modified third law and is often used to calculate the mass of a large celestial object from the orbital period and radius of a much smaller mass

So what is the Mass of the Sun?

MSun = (42r3)/(GP2) rEarth = 1.5 x 1011 meters

PEarth = 3.16 x 107 seconds

G = 6.67 x 10-11 m3 kg-1 s-2

Plugging in the numbers gives the mass of the Sun

MSun = 2 x 1030 kg

What about the Earth?The same formula gives the mass of the

Earth

Using the orbit of the Moon:rMoon = 3.84 x 105 km

PMoon = 27.322 days = 2.36 x 106 seconds

Mass of Earth is 6 x 1024 kg

MEarth= (42r3)/(GP2)

How about Pluto?

The same formula gives the mass of Pluto, too

Using the orbit of Pluto’s moon Charon:rCharon = 1.96 x 104 km

PCharon = 6.38 days = 5.5 x 105 seconds

Mass of Pluto is 1.29 x 1022 kilograms

MPluto = (42r3)/(GP2)

Orbits tell us Mass We can measure the mass of any body that has an

object in orbit around it Planets, stars, asteroids

We just need to know how fast and how far away something is that goes around that object

But we can’t determine the mass of the Moon by watching it go around the Earth

To determine the mass of the Moon, we need a satellite orbiting the Moon

Surface Gravity

Surface gravity:Determines the weight of a mass at a

celestial object’s surfaceInfluences the shape of celestial objectsInfluences whether or not a celestial

object has an atmosphere

Surface gravity is the acceleration a mass undergoes at the surface of a celestial object (e.g., an asteroid, planet, or star)

Surface Gravity Calculations Surface gravity is determined from Newton’s Second Law and

the Law of Gravity:ma = GMm/R2

where M and R are the mass and radius of the celestial object, and m is the mass of the object whose acceleration a we wish to know

The surface gravity, denoted by g, is then:g = GM/R2

Notice dependence of g on M and R, but not m

gEarth = 9.8 m/s2

gMoon/gEarth= 0.18

gJupiter/gEarth = 3

Escape Velocity

To overcome a celestial object’s gravitational force and escape into space, a mass must obtain a critical speed called the escape velocity

Escape VelocityDetermines if a

spacecraft can move from one planet to another

Influences whether or not a celestial object has an atmosphere

Relates to the nature of black holes

Escape Velocity Calculation

The escape velocity, Vesc, is determined from Newton’s laws of motion and the Law of Gravity and is given by:

Vesc = (2GM/R)1/2

where M and R are the mass and radius of the celestial object from which the mass wishes to escape

Notice dependence of Vesc on M and R, but not m

Vesc,Earth = 11 km/s, Vesc,Moon = 2.4 km/s

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

TO DO LIST:

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