Chapter 11

1. Chapter 11 Gravity, Planetary Orbits, and the Hydrogen Atom

2. Newton’s Law of Universal Gravitation • Every particle in the Universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them • G is the universal gravitational constant and equals 6.673 x 10-11 Nm2 / kg2

3. Law of Gravitation, cont • This is an example of an inverse square law • The magnitude of the force varies as the inverse square of the separation of the particles • The law can also be expressed in vector form

4. Notation • is the force exerted by particle 1 on particle 2 • The negative sign in the vector form of the equation indicates that particle 2 is attracted toward particle 1 • is the force exerted by particle 2 on particle 1

5. More About Forces • The forces form a Newton’s Third Law action-reaction pair • Gravitation is a field force that always exists between two particles, regardless of the medium between them • The force decreases rapidly as distance increases • A consequence of the inverse square law

6. G vs. g • Always distinguish between G and g • G is the universal gravitational constant • It is the same everywhere • g is the acceleration due to gravity • g = 9.80 m/s2 at the surface of the Earth • g will vary by location

7. Gravitational Force Due to a Distribution of Mass • The gravitational force exerted by a finite-sized, spherically symmetric mass distribution on a particle outside the distribution is the same as if the entire mass of the distribution were concentrated at the center • For the Earth, this means

8. Measuring G • G was first measured by Henry Cavendish in 1798 • The apparatus shown here allowed the attractive force between two spheres to cause the rod to rotate • The mirror amplifies the motion • It was repeated for various masses

9. Gravitational Field • Use the mental representation of a field • A source mass creates a gravitational field throughout the space around it • A test mass located in the field experiences a gravitational force • The gravitational field is defined as

10. Gravitational Field of the Earth • Consider an object of mass m near the earth’s surface • The gravitational field at some point has the value of the free fall acceleration • At the surface of the earth, r = RE and g = 9.80 m/s2

11. Representations of the Gravitational Field • The gravitational field vectors in the vicinity of a uniform spherical mass • fig. a – the vectors vary in magnitude and direction • The gravitational field vectors in a small region near the earth’s surface • fig. b – the vectors are uniform

12. Structural Models • In a structural model, we propose theoretical structures in an attempt to understand the behavior of a system with which we cannot interact directly • The system may be either much larger or much smaller than our macroscopic world • One early structural model was the Earth’s place in the Universe • The geocentric model and the heliocentric models are both structural models

13. Features of a Structural Model • A description of the physical components of the system • A description of where the components are located relative to one another and how they interact • A description of the time evolution of the system • A description of the agreement between predictions of the model and actual observations • Possibly predictions of new effects, as well

14. Kepler’s Laws, Introduction • Johannes Kepler was a German astronomer • He was Tycho Brahe’s assistant • Brahe was the last of the “naked eye” astronomers • Kepler analyzed Brahe’s data and formulated three laws of planetary motion

15. Kepler’s Laws • Kepler’s First Law • Each planet in the Solar System moves in an elliptical orbit with the Sun at one focus • Kepler’s Second Law • The radius vector drawn from the Sun to a planet sweeps out equal areas in equal time intervals • Kepler’s Third Law • The square of the orbital period of any planet is proportional to the cube of the semimajor axis of the elliptical orbit

16. Notes About Ellipses • F1 and F2 are each a focus of the ellipse • They are located a distance c from the center • The longest distance through the center is the major axis • a is the semimajor axis

17. Notes About Ellipses, cont • The shortest distance through the center is the minor axis • b is the semiminor axis • The eccentricity of the ellipse is defined as e = c /a • For a circle, e = 0 • The range of values of the eccentricity for ellipses is 0 < e < 1

18. Notes About Ellipses, Planet Orbits • The Sun is at one focus • Nothing is located at the other focus • Aphelion is the point farthest away from the Sun • The distance for aphelion is a + c • For an orbit around the Earth, this point is called the apogee • Perihelion is the point nearest the Sun • The distance for perihelion is a – c • For an orbit around the Earth, this point is called the perigee

19. Kepler’s First Law • A circular orbit is a special case of the general elliptical orbits • Is a direct result of the inverse square nature of the gravitational force • Elliptical (and circular) orbits are allowed for bound objects • A bound object repeatedly orbits the center • An unbound object would pass by and not return • These objects could have paths that are parabolas and hyperbolas

20. Orbit Examples • Pluto has the highest eccentricity of any planet (a) • ePluto = 0.25 • Halley’s comet has an orbit with high eccentricity (b) • eHalley’s comet = 0.97

21. Kepler’s Second Law • Is a consequence of conservation of angular momentum • The force produces no torque, so angular momentum is conserved

22. Kepler’s Second Law, cont. • Geometrically, in a time dt, the radius vector r sweeps out the area dA, which is half the area of the parallelogram • Its displacement is given by

23. Kepler’s Second Law, final • Mathematically, we can say • The radius vector from the Sun to any planet sweeps out equal areas in equal times • The law applies to any central force, whether inverse-square or not

24. Can be predicted from the inverse square law Start by assuming a circular orbit The gravitational force supplies a centripetal force Ks is a constant Kepler’s Third Law

25. Kepler’s Third Law, cont • This can be extended to an elliptical orbit • Replace r with a • Remember a is the semimajor axis • Ks is independent of the mass of the planet, and so is valid for any planet

26. Kepler’s Third Law, final • If an object is orbiting another object, the value of K will depend on the object being orbited • For example, for the Moon around the Earth, KSun is replaced with KEarth

27. Energy in Satellite Motion • Consider an object of mass m moving with a speed v in the vicinity of a massive object M • M >> m • We can assume M is at rest • The total energy of the two object system is E = K + Ug

28. Energy, cont. • Since Ug goes to zero as r goes to infinity, the total energy becomes

29. Energy, Circular Orbits • For a bound system, E < 0 • Total energy becomes • This shows the total energy must be negative for circular orbits • This also shows the kinetic energy of an object in a circular orbit is one-half the magnitude of the potential energy of the system

30. Energy, Elliptical Orbits • The total mechanical energy is also negative in the case of elliptical orbits • The total energy is • r is replaced with a, the semimajor axis

31. Escape Speed from Earth • An object of mass m is projected upward from the Earth’s surface with an initial speed, vi • Use energy considerations to find the minimum value of the initial speed needed to allow the object to move infinitely far away from the Earth

32. Escape Speed From Earth, cont • This minimum speed is called the escape speed • Note, vesc is independent of the mass of the object • The result is independent of the direction of the velocity and ignores air resistance

33. Escape Speed, General • The Earth’s result can be extended to any planet • The table at right gives some escape speeds from various objects

34. Escape Speed, Implications • This explains why some planets have atmospheres and others do not • Lighter molecules have higher average speeds and are more likely to reach escape speeds • This also explains the composition of the atmosphere

35. Black Holes • A black hole is the remains of a star that has collapsed under its own gravitational force • The escape speed for a black hole is very large due to the concentration of a large mass into a sphere of very small radius • If the escape speed exceeds the speed of light, radiation cannot escape and it appears black

36. Black Holes, cont • The critical radius at which the escape speed equals c is called the Schwarzschild radius, RS • The imaginary surface of a sphere with this radius is called the event horizon • This is the limit of how close you can approach the black hole and still escape

37. Black Holes and Accretion Disks • Although light from a black hole cannot escape, light from events taking place near the black hole should be visible • If a binary star system has a black hole and a normal star, the material from the normal star can be pulled into the black hole

38. Black Holes and Accretion Disks, cont • This material forms an accretion disk around the black hole • Friction among the particles in the disk transforms mechanical energy into internal energy

39. Black Holes and Accretion Disks, final • The orbital height of the material above the event horizon decreases and the temperature rises • The high-temperature material emits radiation, extending well into the x-ray region • These x-rays are characteristics of black holes

40. Black Holes at Centers of Galaxies • There is evidence that supermassive black holes exist at the centers of galaxies • Theory predicts jets of materials should be evident along the rotational axis of the black hole • An HST image of the galaxy M87. The jet of material in the right frame is thought to be evidence of a supermassive black hole at the galaxy’s center.

41. Gravity Waves • Gravity waves are ripples in space-time caused by changes in a gravitational system • The ripples may be caused by a black hole forming from a collapsing star or other black holes • The Laser Interferometer Gravitational Wave Observatory (LIGO) is being built to try to detect gravitational waves

42. Importance of the Hydrogen Atom • A structural model can also be used to describe a very small-scale system, the atom • The hydrogen atom is the only atomic system that can be solved exactly • Much of what was learned about the hydrogen atom, with its single electron, can be extended to such single-electron ions as He+ and Li2+

43. Light From an Atom • The electromagnetic waves emitted from the atom can be used to investigate its structure and properties • Our eyes are sensitive to visible light • We can use the simplification model of a wave to describe these emissions

44. Wave Characteristics • The wavelength, l, is the distance between two consecutive crests • A crest is where a maximum displacement occurs • The frequency, ƒ, is the number of waves in a second • The speed of the wave is c = ƒ l

45. Atomic Spectra • A discrete line spectrum is observed when a low-pressure gas is subjected to an electric discharge • Observation and analysis of these spectral lines is called emission spectroscopy • The simplest line spectrum is that for atomic hydrogen

46. Uniqueness of Atomic Spectra • Other atoms exhibit completely different line spectra • Because no two elements have the same line spectrum, the phenomena represents a practical and sensitive technique for identifying the elements present in unknown samples

47. Emission Spectra Examples

48. Absorption Spectroscopy • An absorption spectrum is obtained by passing white light from a continuous source through a gas or a dilute solution of the element being analyzed • The absorption spectrum consists of a series of dark lines superimposed on the continuous spectrum of the light source

49. Absorption Spectrum, Example • A practical example is the continuous spectrum emitted by the sun • The radiation must pass through the cooler gases of the solar atmosphere and through the Earth’s atmosphere

50. Balmer Series • In 1885, Johann Balmer found an empirical equation that correctly predicted the four visible emission lines of hydrogen • His red,  = 656.3 nm • H is green,  = 486.1 nm • His blue,  = 434.1 nm • H is violet,  = 410.2 nm