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Lecture 1

ASTA01H3 Lecture Notes - Lecture 1: Angular Momentum, Tidal Acceleration, Escape Velocity

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Parandis Tajbakhsh

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Types of Orbits
Newton’s work made him realize that orbits come in a number of different shapes.
Orbits come in 2 different types: closed and open.
To understand the types of orbits, we need to understand conic sections.
Conic sections are those curves which are produced due to the intersection of a plane
with a cone.
There are three types of conic sections: ellipses, parabola and hyperbola.
Orbits in Gravitationally-Bound Systems
Two bodies of different mass, balance at their centre of mass, which is located closer to
the more massive object.
Two gravitationally-bound object orbit their common centre of mass or barycentre, in
elliptical orbits, with the common centre of mass at the common focus.
The common centre of mass of the Earth-moon system is 4671 kms from the centre of the
Earth, located inside the Earth (radius of Earth is 6378 km).
Newton’s version of Kepler’s third law relates the orbital period of any two orbiting objects
to the distance between their centres.
P2=(4π2/[G(M1 + M2)])a3
Compare this to P2yr = a3AU: For the case of Planets and Sun, the mass of the planet is
almost negligible.
Types of Orbits
A closed orbit is one on which the orbiting object returns to its starting point. Circular and
elliptical orbits are examples of closed orbits.
For any planet, one can define an escape velocity which depends on the mass of the
planet and the distance to the planet.
ve = (2GM/R)(1/2)
M is the mass of the planet, R is the radius of the planet and G is a constant called the
gravitational constant.
If an object achieves a velocity equal to the escape velocity, it will leave the gravitational
field of the planet.
If the velocity is equal to the escape velocity, the orbit is a parabola.
If the velocity is larger than the escape velocity, the orbit is a hyperbola.
The velocity needed to stay in a circular orbit is called circular velocity.
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