Download Orbits and Kepler`s Laws Lesson

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 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
𝐺𝑚
Earth’s gravitational field:
𝑔 = 2𝐸
From centripetal acceleration:
𝑎𝑐 =
𝑟
𝑣2
𝑟
Since the gravitational force provides the centripetal force for a
satellite, m, in orbit:
𝐹𝑐 = 𝐹𝑔
𝑚𝑎𝑐 = 𝑚𝑔
𝑣2
𝑟
=
𝑣=
𝐺𝑚𝐸
𝑟2
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
𝑣𝐸 =
𝐺𝑚𝑆
𝑟𝐸
= 3.51 x 104 m/s
= 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 .
2𝜋𝑟
𝑇=
= 5490. 4487𝑠
𝑣
= 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.
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_ch
4_en.pdf
 http://www.businessinsider.com/watch-the-sunrise-and-set-and-rise-again-from-theinternational-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
Kepler’s Third Law of Planetary Motion

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
𝑟 3 = 𝐶𝑠 𝑇 2
𝑟3
𝐶𝑠 = 2
𝑇
where Cs = Kepler’s constant or the constant of
proportionality for the Sun measured as
3.35 x 1018 m3/s2
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 =
B)
𝑟3
𝑇2
=
1.495 𝑥 1011 𝑚
3.15585 𝑥 107 𝑠
3
18
3/𝑠2
=
3.355
𝑥
10
𝑚
2
The Sun’s constant is 3.355 𝑥 1018 𝑚3/𝑠2 .
T = 8.1 x 107 s
𝑟=
3
3
𝐶𝑠 𝑇 2
𝑟=
3.355 𝑥 1018 𝑚3/𝑠2 8.1 𝑥 107𝑠 2
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?
