Transcript ASTR2100 - Saint Mary's University | Astronomy & Physics

Chapter 4
Gravitation and the Waltz of the Planets
The important concepts of Chapter 4 pertain to
orbital motion of two (or more) bodies, central
forces, and the nature of orbits.
1. What we see in the sky results from the
rotation of the Earth on its axis, the orbital
motion of the Earth about the Sun, the orbital
motion of the Moon about Earth, and, to a small
extent, the gravitational effect of the Sun and the
Moon on the Earth’s axis of rotation.
2. The motions of the Earth produce a
fundamental frame of reference for stellar
observations.
Two great observers:
Hipparchus
Tycho Brahe
(2nd century BC)
(1546-1601 AD)
Angle measuring devices: mural quadrant
cross staff
Nicholas Copernicus (1473-1543)
revived the heliocentric model
for the solar system, where
planetary orbits are envisaged as
circular for simplicity. Even
circular orbits are sufficient for
understanding the difference
between sidereal (star) period of
a planet, Psid = time to orbit the
Sun, and its synodic period,
Psyn = time to complete a cycle
of phases as viewed from Earth.
The relationship between the
two is best demonstrated by considering the amount by
which two planets A and B advance in their orbits over
the course of one day.
Over the course of one day, planet A advances through
the angle
Planet B advances through
the angle
The difference in the
angles is the amount
by which planet A
has gained on planet
B, which is related
to its synodic period,
i.e.:
In other words,
Or:
If A is Earth, and B a superior planet (orbits outside
Earth’s orbit), then:
For Earth P = 365.256363 days (a little more than 365¼
days), i.e.:
Given two values, the third can be found !
The same technique can be
used to relate a planet’s
rotation rate and orbital
period to the length of its
day. The arrow is a fixed
feature on the planet.
For example, the planet Mars has a synodic period of
780 days, which means it returns to opposition from the
Sun every 2.14 years. But its true orbital period is 687
days, or 1.88 years, which means it returns to the same
point in its orbit about the Sun every 1.88 years.
Some further consequences:
Mercury: Prot = 58d.67, Psid = 88d.0, Pday = 176d.
Venus: Prot = 243d (retrograde rotation), Psid = 224d.7,
Pday = 117d.
Moon: Prot = 27d.3215, Psid = 365d.2564, Pday = 29d.5306.
Earth: Prot = 23h 56m, Psid = 365d.2564, Pday = 24h.
Which planet has the longest “day”?
Johannes Kepler (1571-1630)
How Kepler triangulated the orbit of Mars. He took
Tycho’s observations of Mars relative to the Sun
separated by the planet’s 687d orbital period (with Earth
at different parts of its orbit) and used them to
triangulate the location of Mars, which was at the same
point of its orbit.
Kepler’s study of the orbit of Mars resulted in his three
laws of planetary motion:
1. The orbits of the planets are ellipses with the Sun at
one focus. Actually they are conic sections.
2. The line from the Sun to a planet sweeps out equal
areas of orbit in equal time periods. Angular momentum
is conserved, i.e. mvr = constant.
3. The orbital period of a planet is related to the semimajor axis of its orbit by P2 = a3 (Harmonic Law).
centre distance, c = ae
2a
a = ½ string length 2a
b
c2 + b2 = a2
e = (1 ˗ b2/ a2) ½
ae
2b
Isaac Newton formulated
Kepler’s Laws into a
model of gravitation, in
which: a mass attracts
another mass with force
inversely proportional to
the square of the distance
between the two, i.e. F ~
1/d2. Forces produce
acceleration of an object
proportional to its mass,
i.e. F = m×a, and objects
stay at rest or in constant
motion in one direction
unless acted upon by a
force.
Objects in orbit
around Earth are
constantly falling
towards the Earth.
They are acted
upon by gravity,
but are in free-fall
towards
Earth.
They will not
“fall” to Earth if
their
transverse
speed is large
enough.
The importance of Kepler’s 3rd Law is that, as shown by
Newton, the constant of proportionality for a3 = P2
contains two constants, π (pi) and G (the gravitational
constant), plus the sum of the masses of the two coorbiting bodies. If one can determine orbital periods P
and semi-major axes a, then one can derive the masses of
the objects in the system: either planets or stars !
For example: Jupiter’s mass from the Galilean satellites.
Astronomers try to keep the calculations simple, so they
usually omit π and G. Thus, the Newtonian version of
Kepler’s 3rd Law is usually written as:
where the sum of the masses of the two co-orbiting
objects, “M1” and “M2”, is calculated in terms of the
Sun’s mass, the orbital semi-major axis (~radius) “a” is
calculated in terms of the Earth’s distance from the Sun,
the Astronomical Unit, and the orbital period “P” is
expressed in Earth years.
The point to be emphasized is that a measurement of two
of the parameters permits one to calculate a value for the
third parameter. Astronomers use the relationship to
measure the masses of planets and stars.
Vis-Viva Equation.
There is a useful relationship for orbital speed that can be
obtained from the energy equation:
Solving the equation for the velocity v gives:
which is the vis-viva equation, where a is the semi-major
axis of the orbit, r is any point in the orbit, and v is the
speed in the orbit at r. Escape velocity is attained when
a → ∞, i.e.:
Circular orbits apply when r = a everywhere, i.e.:
Earth’s orbital velocity.
And escape velocity from Earth orbit is:
The maximum encounter velocity for a solar system
object is the sum of the previous two values, i.e.:
which is close to the encounter velocity of meteoroids
associated with Halley’s Comet.
Sending a satellite to the Sun.
Here, the situation is pictured at
right, where an orbit from
Earth to the Sun will have a
semi-major axis of ½a. By
Kepler’s 3rd Law the orbital
period is calculated as:
But aphelion to perihelion
constitutes exactly half an orbit,
so the time to reach the Sun is:
Astronomical Terminology
Rotation. The act of spinning on an axis.
Revolution. The act of orbiting another object.
Geocentric. = Earth-centred.
Heliocentric. = Sun-centred.
Opposition. When a planet is opposite (180° from) the
Sun.
Conjunction. When a planet is in the same direction as.
Typically refers to conjunction with the Sun.
Inferior planet. A planet orbiting inside Earth’s orbit.
Superior planet. A planet orbiting outside Earth’s orbit.
Prograde motion. When a planet’s RA increases nightly.
Retrograde motion. When a planet’s RA decreases
nightly.
Astronomical Unit = A.U. The average distance between
Earth and the Sun.
Inertia. An object’s resistance to its state of motion.
Inertial reference frame. = non-accelerated frame.
Astronomical Terminology 2
Eccentricity. The amount of non-circularity of an orbit,
from round (e = 0.0) to very flattened (e = 0.9).
Semi-major Axis. Half the length of the long axis of an
ellipse, equivalent to the “radius” of an orbit.
Orbital Period. The time taken for one object to orbit
another object.
Synodic Period. The time taken for an object to cycle
through its phases as viewed from Earth.
Inferior planet. A planet orbiting inside Earth’s orbit.
Superior planet. A planet orbiting outside Earth’s orbit.
Prograde motion. When a planet’s RA increases nightly.
Retrograde motion. When a planet’s RA decreases
nightly.
Gravity. The force exerted by an object on any other
object in the universe.
“Zero gravity.” A fictional term referring to the
apparent weightlessness of an object in free fall.
Sample Questions
1. Imagine a planet moving in a perfectly
circular orbit around the Sun. Because the
orbit is circular, the planet is moving at a
constant speed. Is the planet experiencing
acceleration? Explain your answer.
Answer: Yes, it is. The planet experiences
acceleration since it is constantly falling
towards the Sun.
2. Suppose that astronomers discovered a
comet approaching the Sun in a hyperbolic
orbit. What would that say about the origin
of the planet?
Answer. Objects in hyperbolic orbits are
not bound to the object they are orbiting.
Astronomers would therefore conclude that
the comet is not bound to the solar system
and must therefore have originated from
“outside” the solar system.
3. Why is the term “zero gravity”
meaningless? Is there a place in the
universe where no gravitational forces
exist?
Answer. All objects are subject to the
attractive force of every other object, in
proportion to the inverse square of the
separation r from the other object. For one
object
to
experience
no
outside
gravitational forces, i.e. zero gravity, it
would have to be an infinite distance away
from every other object, which is not
possible. So the term “zero gravity” cannot
apply anywhere in the known universe.