Planetary Mechanics:

Satellites A satellite is an object or a body that revolves

around another body due to the gravitational

attraction to the greater mass. Ex: The planets are

natural satellites of the Sun and moons are natural

satellites of the planets themselves.

Satellites Artificial satellites, conversely, are

human-made objects that orbit Earth or other bodies in the solar system.

Ex: CSA’s RADARSAT-1

and RADARSAT-2 and

the International

Space Station (ISS) are

examples of artificial

satellites.

Satellites Another well-known example of artificial

satellites is the network of 24 satellites

that make up the Global Positioning

System (GPS).

The data from 1 satellite

will show that the object is

somewhere along the

circumference of the circle.

Satellites Two satellites consulted simultaneously

will refine the location to one of two

intersection spots.

Satellites

With three satellites, the intersection of the

three circles will give the location of the

object to within 15 m of its actual position.

Satellites in Circular Orbits

When Newton developed the idea of universal

gravitation, he also theorized that the same

force that pulls objects to Earth also keeps the

Moon in its orbit. But of course the Moon does

not hit the Earth’s surface!

The Moon orbits Earth at a distance from

Earth’s centre - called the orbital radius.

The motion of the Moon depends upon the

centripetal force due to Earth’s gravity and the

Moon’s orbital velocity.

Moon Orbiting Earth

The Moon’s orbit,

similar to the orbits of

the planets around

the Sun, is actually

elliptical. The orbits

are closely

approximated as

circular orbits.

Analyzing Satellites in Circular Orbits For the motion of a satellite experiencing

uniform circular motion in a gravitational field:

From Newton’s Law of Universal Gravitation for an object in

Earth’s gravitational field: 𝑔 = 𝐺𝑚𝐸

𝑟2

From centripetal acceleration: 𝑎𝑐 =𝑣2

𝑟

Since the gravitational force provides the centripetal force for a satellite, m, in orbit: 𝐹𝑐 = 𝐹𝑔

𝑚𝑎𝑐 = 𝑚𝑔

𝑣2

𝑟=

𝐺𝑚𝐸

𝑟2

𝑣 =𝐺𝑚𝐸

𝑟

This eq’n gives the speed of an orbiting satellite/body within Earth’s gravitational field.

Analyzing Satellites in Circular Orbits To calculate the orbital speed around any large body

of mass m:

𝑣 =𝐺𝑚

𝑟

where v is the orbital speed of the satellite (m/s)

G is the universal gravitational constant (6.67 x 10-11 N·m2/kg2)

m is the central object’s mass about which the satellite orbits (kg)

r is the orbital radius (m)

ORBITAL SPEED = speed needed by a satellite to remain in orbit

Orbits: Example Problem 1 Determine the speeds of the 2nd and 3rd

planets from the Sun. The 2nd planet has an orbital radius of 1.08 x 1011 m while the 3rd has an orbital radius of 1.49 x 1011 m. The mass of the Sun is 1.99 x 1030 kg.

𝑣𝑣 =𝐺𝑚𝑆

𝑟𝑉= 3.51 x 104 m/s

𝑣𝐸 =𝐺𝑚𝑆

𝑟𝐸= 2.98 x 104 m/s

∴ 𝑨𝒔 𝑬𝒂𝒓𝒕𝒉 𝒊𝒔 𝒇𝒖𝒓𝒕𝒉𝒆𝒓 𝒇𝒓𝒐𝒎 𝒕𝒉𝒆 𝑺𝒖𝒏 𝒊𝒕 𝒕𝒓𝒂𝒗𝒆𝒍𝒔 𝒎𝒐𝒓𝒆 𝒔𝒍𝒐𝒘𝒍𝒚 𝒕𝒉𝒂𝒏 𝑽𝒆𝒏𝒖𝒔.

Orbits: Example Problem 2 The International Space Station (ISS) orbits Earth

at an altitude of about 350 km above Earth’s

surface. Determine:

A) The speed needed for the ISS to maintain its orbit.

B) The orbital period of the ISS in hours and minutes.

C) How many times in a 24 hour day would astronauts

aboard the ISS see the sun rise and set?

Given: mE = 5.98 x 1024 kg

rE = 6.378 x 106 m

hISS = 350 km = 3.5 x 105 m

Note: rISS = rE + hISS = 6.728 x 106 m

Orbits: Example Problem 2 Cont’d

A) 𝑣 =𝐺𝑚𝐸

𝑟𝐼𝑆𝑆= 7.699 𝑥 103 𝑚/𝑠 The ISS requires a

constant speed of 7.7 x 103 m/s to maintain its

orbit.

B) The distance travelled in 1 revolution is 2𝜋𝑟.

𝑇 = 2𝜋𝑟

𝑣= 5490. 4487𝑠

= 91.5075 𝑚𝑖𝑛

= 1.52512 ℎ

The ISS goes around the entire Earth in 1.5h!!!

c) Astronauts aboard the ISS would see the sun rise

and set apprx. 16 times a day! (every 45 min.)

Orbits: Example Problem 3 What is the difference between a geosynchronous orbit

and a geostationary orbit?

A geosynchronous orbit is a satellite with an orbital speed that matches Earth’s period of rotation; it takes exactly 1 day to travel around the Earth. To an observer on Earth, the satellite will appear to travel through the same point in the sky every 24 h.

A geostationary orbit is a special type of geosynchronous orbit in which the satellite orbits directly over the equator. To an observer on Earth, the satellite would appear to remain fixed in the same point in the sky at all times.

Geosynchronous VS

Geostationary Satellite

More Info on Orbits To put you into a real spin….. Try pg. 303 #6,7,9,12

Check out:

Train Like an Astronaut: esamultimedia.esa.int/docs/.../en/PrimEduKit_ch4_en.pdf

http://www.businessinsider.com/watch-the-sun-rise-and-set-and-rise-again-from-the-international-space-station-2013-2

Physicsclassroom.com: Planetary and Satellite Motion

Early Astronomy Early Philosophers, Scientists, and

Mathematicians (Aristotle, Plato, Ptolemy)

believed in the geocentric

view of the universe;

Geo meaning Earth +

centric meaning centre

Scientists tried to explain

the motion of the stars

and planets

A Scientific Revolution Nicolas Copernicus (1476-1543) proposed the

heliocentric view of the solar system in which planets revolved in circles around the Sun; helios meaning Sun

He also deduced that planets

closer to the Sun have a higher

speed than those farther away;

supported by the orbital speed

equation 𝑣 =𝐺𝑚

𝑟

His work was supported and verified

by Galileo for which Galileo was

persecuted by the Catholic Church

Renaissance Astronomers Tycho Brahe (1564-1601) carried

out naked-eye observations using

large instruments (quadrants) to

accumulate the most complete and

accurate observations over 20 years

to support the heliocentric view

Tycho hired a brilliant young

mathematician, Johannes Kepler

(1571-1630), to assist in the

analysis of the data

Johannes Kepler

Kepler’s objective was to find an orbital shape for

the motions of the planets that best fit Tycho’s

data

Worked mainly with the orbit of Mars which had

the most complete records

The only shape that fit ALL of the data was the…

ellipse

He formulated three laws to explain the true

orbits of planets

Kepler’s First Law of Planetary Motion Law of Ellipses: Each planet moves around the

Sun in an elliptical orbit with the Sun at one focus of the ellipse.

Note: The orbits still very much resemble circles; distance from Earth to Sun varies by only apprx. 3% annually.

Kepler’s Second Law of Planetary Motion Law of Equal Areas: The straight line joining a

planet and the Sun sweeps out equal areas in space in equal intervals of time.

Kepler determined that Mars sped up as it approached the Sun and slowed down as it moved away

Law of Harmonies: The cube of the average radius r of a planet’s orbit is directly proportional to the square of the period T of the planet’s orbit.

𝑟3 ∝ 𝑇2

𝑟3 = 𝐶𝑠𝑇2

𝐶𝑠 = 𝑟3

𝑇2

where Cs = Kepler’s constant or the constant of proportionality for the Sun measured as 3.35 x 1018 m3/s2

Kepler’s Third Law of Planetary Motion

Kepler’s Laws: Example Problem 1 The average radius of orbit of Earth around

the Sun is 1.495 x 108 km. The period of

revolution is 365.26 days. Determine:

A) The constant Cs to four sig. digs.

B) An asteroid has a period of revolution around the

Sun of 8.1 x 107 s. What is the avg. radius of its

orbit?

Kepler’s Laws: Example Problem 1 Cont’d

A) rE = 1.495 x 108 km = 1.495 x 1011 m

TE = 365.26 days = 3.15585 x 107 s

𝐶𝑠 = 𝑟3

𝑇2 = 1.495 𝑥 1011 𝑚 3

3.15585 𝑥 107 𝑠 2 = 3.355 𝑥 1018 𝑚3/𝑠2

The Sun’s constant is 3.355 𝑥 1018 𝑚3/𝑠2 .

B) T = 8.1 x 107 s

𝑟 = 𝐶𝑠𝑇23

𝑟 = 3.355 𝑥 1018 𝑚3/𝑠2 8.1 𝑥 107𝑠 23

𝑟 = 2.8 𝑥 1011 m is the avg. radius of the asteroid’s orbit.

Kepler’s Laws: HW Problems

The equation for Kepler’s 3rd Law can be

obtained from a relationship between

Newton’s Law of Universal Gravitation and

the uniform circular motion of a planet around

the Sun. From 1st principles derive an

equation for the Sun’s constant that is

dependent only on the mass of the Sun.

What does orbital eccentricity mean?

And just for fun….How was the mass of the

Earth originally determined? How was Earth’s

radius calculated?