哈工大天文学概论——银河

Gravitational Force Law
GM 1M 2 F 2 d
• • • • • F = force due to gravity. M1 = mass of the first object M2 = mass of the second object d = distance between their centers. G = “Gravitational Force Constant”
The Gravitational Force Constant
• The force constant, G, is a number which gives the size of the gravitational coupling between two massive objects. • G is very small, in metric units: • G=6.710–11 Newtons meter2 / kilogram2 • The Newton is the metric unit of force: • 4.41 Newtons = 1 pound • G has to be measured experimentally.
– Exact description of forces involved – How to quantify gravity ? – Galileo‟s Law of Falling Bodies – Newton, the Apple and the Moon !
Galileo‟s Mechanics Experiments
Towards an exact description
• Force of gravity between any two objects depends only upon:
– The masses of the two objects:
• More massive objects feel a stronger force.
• What is the force of the Earth on the apple
– F = GME MA/RE2
• What is the apple‟s acceleration (2nd Law):
– Newton‟s Laws then guarantee Galileo‟s observation of Falling Bodies – What is the dependence on distance ?
Comparing an apple to the moon
• Newton‟s fascinating thought process :
• Newton‟s clever idea : compare to apple‟s fall !
Comparing apple and Moon
• Apple on Earth falls in 1 second :
– dApple = 4.8 m
• Moon falls in 1 second :
The Idea of Universal Mutual Gravitation
• Newton, in his Principia, formulated the Law of Universal Mutual Gravitation:
– Gravity is an Attractive force:
What Newton knew about the Moon
• Distance to the Moon
– ~380,000 km = ~ 60 x Earth radius
• Sidereal Orbital Period
– 28.3 days
• Speed of Moon on its path : ~ 1000 m/sec
– Assumptions :
• Fall of an apple and path of moon governed by the same principle Gravitational attraction (only !) • Gravity independent of shape of Earth
– Careful numerical analysis leads to deeper understanding of gravity
What Newton knew about apples
• Falling apples on Earth :
– Constant acceleration near surface of Earth : a=9.8 meters/second2 – Radius of Earth : 6378 km = 6378,000 meters (Eratosthenes !)
d M1 d M2 d 2M2
F
GM 1M 2 d2
M2
G (2M 1 ) M 2 GM 1M 2 F 2 2 d d2
2M1
F
G (2 M 1 )(2 M 2 ) GM 1M 2 4 d2 d2
2M1
d M1 2d M1 d/2 M1 M2
F
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GM 1M 2 F d2
M2
GM1M 2 1 GM 1M 2 2 (2d ) 4 d2
Towards an exact description
• Newton‟s Idea :
– Force depends linearly on masses of objects
• Twice as much mass twice the force • Half as much mass half the force
– dMoon = 0.00136 meters
• Ratio of these distances = ratio of accelerations = ratio of forces
– dApple/dMoon = 4.8m/0.00136m ~ 3600
Comparing apple and Moon
• Law of Falling Bodies: – In the absence of air, heavy objects and light objects fall at the same, constant rate of acceleration.
Astronaut David R. Scott, Apollo 15 (Falcon) near Hadley Rille, August 1971
Example 1 : Weighing the Earth
• Measure the acceleration of gravity by dropping weights (Galileo):
• a = 9.8 meters/second2
• Measure the radius of the Earth using geometry (Eratosthenes):
• Prior to work with telescope, Galileo performed fundamental research on motion.
– Explored the rate of falling bodies by dropping different weights, or sliding them down inclined planes.
• RE=6378 kilometers = 6,378,000 meters
• Earth‟s Mass is:
2 aRE (9.8)(6,378,000)2 24 ME 5.98 10 kg 11 G (6.67 10 )
Example 2 : The Concept of Mutuality
Gravity and Orbits(2)
Stellar Mass
Newtonian Gravity
an explanation for planetary motions
• Newton‟s 3 Laws of Motion startingpoint • Necessary ingredient to calculate motions
Deflection
The curved path of the Moon
• How much does the moon have to fall in 1 second to „close the loop‟ ? • Simple geometry :
– 0.00136 meters (about 1.4 mm!)
Newton‟s exact formula
• The force of gravitational attraction between any two massive bodies is proportional to their masses and inversely proportional to the square of the distance between their centers.
M2
GM1M 2 GM1M 2 F 4 2 (d / 2) d2
Why is this such a powerful concept ?
• The Law of Gravity is Universal:
– Governs the fall of apples on the Earth – Governs the fall of the Moon around the Earth – Governs the fall of the Earth/Moon system around the Sun – Governs the fall of the Sun around the center of the Milky Way Galaxy. – Governs the fall of the Milky Way and Andromeda Galaxies in their mutual orbit...
• Works to bring massive objects closer together.
– Gravity is a Universal force:
• Works everywhere in the Universe.
– Gravity is a Mutual force:
• Works between pairs of massive objects.
A Curved Path
• But, of course the Moon really moves along a curved path:
– According to the first law, it is deflected from a straight-line path by the force of gravity. – This causes the moon to fall a little bit towards the Earth, deflecting its path into an arc.
– The distance between them:
• Objects closer together feel a stronger force.
– It does not depend at all on the shapes, colors, or compositions of the two objects.
• Ratio of forces :
– FApple/FMoon ~ 3600
• Ratio of distances from center of Earth :
– Dapple-Earth/DMoon-Earth ~ 60
• Price winning question :
– What is the relation between these two ratios ?