Introduction to Astronomy AST0111-3 (Astronomía) Semester 2014B Prof. Thomas H. Puzia
Newton s Laws Big Ball Fail
Universal Law of Gravitation Every mass attracts every other mass through a force called gravity. The strength of the gravitational force between two bodies is directly proportional to the mass of the two bodies. (doubling the mass of one doubles the force). The strength of gravity between two objects decreases as the square of the distance between them. Gravitational force is an inverse square law Newton s Laws (doubling the distance weakens the force by 4). G = gravitational constant = 6.67 x 10-11 m 3 /(kg s 2 )
Gravity Example: Calculate the force of the Sun above the Earth Where: m=6x10 24 kg M=2x10 30 kg r =1.5x10 11 m G = 6.67x10-11 m 3 /kg. s 2 F = GMm / r 2 = mass of the Earth = mass of the Sun = Earth-Sun distance = Universal constant of Gravitation The force of the Sun on the Earth is F = 3.6x10 22 m kg/s 2 Using F=ma we can calculate the accelerations: a Earth = F/m = 6x10-3 m/s 2 a Sun = F/M = 1.8x10-8 m/s 2 (compare to what we feel from Earth = 9.8 m/s 2 )
Gravity Newton: What is the force of gravity? Force that keeps us glued to Earth Force that keeps the Moon in its orbit Force that keeps Earth bound to the Sun Force that keeps the Sun revolving around the Milky Way etc.
Gravity
GMp g Rp
Gravity You apparent weight on surface of other worlds: Earth Moon Mars Saturn Jupiter Sun WD = 80 kg = 13 kg = 30 kg = 85 kg = 190 kg = 2160 kg = 104.000.000 kg NS = 11.000.000.000.000 kg
Newton s Laws Angular Momentum In general, L = r x p = r x mv is constant, or at least conserved in a rotating system when no external forces are acting or when the only forces are directed toward the point of origin (central forces) The second law of Kepler is an example of this conservation principle. Newton was able to derive by balancing angular momentum + gravity P 2 /a 3 = constant = 4π 2 /GM Example: a dancer is spinning and open her arms; the rotation of a pulsar, which was a normal star whose nucleus that collapsed. A very important concept in Astronomy. Formation of the solar system, planets, stars, galaxies, black holes.
Importance of Newton s Laws Understanding the movements and forces led several inventors to produce machines that profitably used those forces. This led to the industrial revolution some 100 years after Newton's work. We finally understood the movements in the Solar System. Newton s Laws
Key Concepts: How did we historically come to understand the basic laws of physics and our place in our solar system? How do we describe motion? Functional use of Kepler s and Newton s laws How is mass different from weight? What is gravity? What is angular momentum? Why is it important?
According to the law of universal gravitation, what would happen to Earth if the Sun were somehow replaced by a black hole of the same mass A. Earth would immediately get sucked into the black hole. B. Earth would slowly spiral into the black hole over the next few years. C. Earth s orbit would not change. D. Earth s orbit would become more eccentric.
Orbits
Gravitational Energy Total energy = Kinetic E + Potential E = constant (in closed systems) E= Fdx Kinetic Energy = 1/2 m1v 2 Potential Energy = -G m1m2/r Gravitational Binding energy is the mechanical energy required to completely separate some system. A bound system typically has a lower mechanical (kinetic) energy than its gravitational binding energy.
Example of orbit Earth and Moon Apogee 406655k 358087km Perigee 2 Abr 11 17 Apr 11
Satellites The orbit of a satellite depends on the launch velocity. Orbits can be bound (parabolic) or unbound (hyperbolic). Low-Earth orbit: Vcirc=8 km/s, height = 200 km, period = 90 min. Geosynchronous Orbit: height = 36000 km, period = 1 sidereal day = 23h56m Escape velocity: for Earth, Vesc = 11 km/s = 40000 km/h. Note: Ve=1.4 Vcirc for Sun, Vesc = 42 km/s = 150000 km/h. this velocity was reached by Pioneer 10/11 and Voyager 1/2.
Interplanetary Satellites Suppose we want to travel to Mars, and arrive with VMars. Orbit of least energy between Earth & Mars: Mars a=1.26 AU (semi-major axis), P=1.4 yr, travel length 0.7 yr (0.3?). 1.5au Earth 1au Use VE=30 km/s, need Vsat=33 km/s. Interplanetary travel è change of orbit = change of ship s energy. aphelion Launch Window: each 780 d
Earth-bound Satellites Over 6000 objects in space, only 900 of which are operational Positions of ~1000 satellites above the Earth in real time. www.science.nasa.gov/realtime/jtrack/3d/jtrack3d.html
Interplanetary Satellites Assisted orbits: e.g., Flybys of the planets by Voyager. V = 42 km/s = 150.000 km/h
The escape velocity of an object depends on... A. The mass of the object we are trying to escape in (e.g., a rocket) B. The mass of the body we are trying to escape from (e.g., a planet) C. How far the object is trying to travel (e.g., the Moon or another planet) D. The amount and type of energy imparted to the object E. More than one of the above.
Orbits of Binary Systems m=m Center of system (center of mass) Recall: Light from moving objects will be affected by motion
m<m Orbits of Binary Systems
Orbits of Binary Systems m<<m e.g. star with planet Only need to see motion in star spectrum to know that planet is there
Gravitational slingshot, Gravity assist maneuver, or Swing-by
Key Concepts: What types of orbits are there? How can orbits be used and understood (orbital energy, Vesc, etc.)?