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PHGN324: Celestial mechanicsFred Sarazin ([email protected])Physics Department, Colorado School of Mines
(Celestial) MechanicsBack to PHGN100 and a touch more…
PHGN324: Celestial mechanicsFred Sarazin ([email protected])Physics Department, Colorado School of Mines
Newton’s three laws of motion(adapted from wikipedia)
• First law: in an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a net force.
• Second law: in an inertial reference frame, the vector sum of force �⃑� on an object is equal to the time derivative of the momentum of the object:
• Third law: when one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.
𝑑�⃑�𝑑𝑡 = 𝑚�⃑� =)�⃑�
�
�(assuming constant m)
PHGN324: Celestial mechanicsFred Sarazin ([email protected])Physics Department, Colorado School of Mines
Work and energy
• The work done by a net force �⃑� along a trajectory 𝑑𝑠is given by: W = ∫ �⃑�. 𝑑𝑠01
• The work is a measurement of the energy needed to move the object from its initial state (position) i to its final state (position) f. This corresponds to:
• A change of potential energy: 𝑈0 − 𝑈1 = ∆𝑈 = −𝑊
• A change of kinetic energy: 𝐾0 − 𝐾1 = 𝑊
• Hence, the work represent a conversion between potential and kinetic energy.
• Conservation of energy of the system: 𝐸 = 𝐾 + 𝑈
PHGN324: Celestial mechanicsFred Sarazin ([email protected])Physics Department, Colorado School of Mines
Newton’s law of universal gravitation
• Newton’s law of universal gravitation states that a particle attracts every other particle in the Universe using a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
• Potential energy (deduced from 𝐹 = −9:9;
for example):
m1
m2
𝑟=>? 𝐹=> = 𝐺𝑚=𝑚>
𝑟=>>𝑟=>? = −𝐺
𝑚>𝑚=
𝑟>=>𝑟>=? = −𝐹>=
𝑟>=? (Newton’s 3rd law)
𝑈 = −𝐺𝑚=𝑚>𝑟
Gravitational constant: G=6.67x10-11 m3.kg-1.s-2
(adapted from wikipedia)
PHGN324: Celestial mechanicsFred Sarazin ([email protected])Physics Department, Colorado School of Mines
Exercise
• Calculate the escape velocity near the Earth’s surface with G = 6.67x10-11 m3.kg-1.s-2, MEarth = M⊕ = 5.97 x 1024 kg and rEarth = r⊕ = 6371 km.
PHGN324: Celestial mechanicsFred Sarazin ([email protected])Physics Department, Colorado School of Mines
Exercise
What is the weight of my daughter’s pet bunny (Luna) on Earth and on Jupiter? Her mass is 1.2 kg.
DATA: M⊕ (Earth) = 5.972x1024 kg, MJ (Jupiter) = 1.898x1027 kgR ⊕ (Earth) = 6371 km, RJ (Jupiter) = 71492 kmG = 6.67x10-11 m3.kg-1.s-2
PHGN324: Celestial mechanicsFred Sarazin ([email protected])Physics Department, Colorado School of Mines
The virial theorem (applied to gravity)
• In mechanics, the virial theorem provides a general equation that relates the average over time of the total kinetic energy 𝑇 of a stable (static) system consisting of N particles bound by potential forces with that of the total potential energy 𝑈 .
• Applied to gravity, the virial theorem simplifies considerably:
• Hence, the total energy of the system 𝐸 can also be written as:
• We will apply the virial theorem a few times over the course of this class…
(adapted from wikipedia)
𝑇 = −12) 𝐹D. 𝑟D
E
DF=
𝑇 = −12 𝑈
𝐸 = 𝑇 + 𝑈 =12 𝑈