 PHYS Astronomy Kepler’s 3rd Law

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PHYS Astronomy Kepler’s 3rd Law The square of any planet's period of orbital revolution, P, is proportional to the cube of its mean distance, r, from the sun. From Kepler’s 2nd Law Speed around orbit: Circumference (2r)/ time P=period, time of 1 orbit

What is the mass of the Sun?
PHYS Astronomy What is the mass of the Sun? G = 6.67 x Nm2/ kg2 r = x 1011m - AU P = x 107 s So: Msun = 2 x 1030 kg

Geosynchronous/Geostationary Orbits
PHYS Astronomy Geosynchronous/Geostationary Orbits A geosynchronous orbit has a period the same as the rotational speed of the Earth - e.g., it orbits in the same amount of time that the Earth rotates - 1 sidereal day. A geostationary orbit is a geosynchronous orbit at the equator - it always stays above the same place on the Earth - communications satellites, satellite TV, etc… What is the altitude of a geostationary orbit calculated from Newton’s formulation of Kepler’s 3rd Law: G = 6.67 x Nm2/ kg2 P = 86,164 s (sidereal day) MEarth = 5.97 X 1024 kg So: r = 42,164 km above the center of the Earth and the altitude is 35,768 km.

Another Way of Calculating Geosynchronous Orbit
PHYS Astronomy Another Way of Calculating Geosynchronous Orbit For an orbiting body, the inward and outward forces must equal each other (Newtons 3rd Law) - the centrifugal force from orbital motion has to equal the centripetal force from gravity:  is angular velocity - at geosynchronous orbit,  of satellite is equal to the angular velocity of the Earth  = 2/86164 (length of sidereal day) M = 5.97 X 1024 kg G= 6.67 x Nm2/ kg2 Plug in the numbers a you get r = 42,164 km - same as when we used Kepler’s 3rd Law

PHYS Astronomy Escape Velocity If a projectile is fired straight up with a large enough velocity, it will escape the Earth’s gravity. It will travel slower and slower due to the Earth’s gravity, but never to zero. Escape velocity - velocity at which gravity can not stop outward motion. Note that the gravitational attraction of Earth never ceases, it just gets infinitesimally small. Escape velocity is calculated by using conservation of energy - a body achieves escape velocity when the all of its initial gravitational potential energy is converted to kinetic energy.

Gravitational Force - vector r is the unit vector in outward direction
PHYS Astronomy Potential energy Gravitational Force - vector r is the unit vector in outward direction Gravitational potential energy at distance r from reference point Kinetic energy Escape velocity Starting from the surface of the Earth: r = X 106 m, M = 5.97 X 1024 kg, G= 6.67 x Nm2/ kg2 v= 11,174 m/s

Types of Orbits Parabolic Hyberbolic Elliptical Circular
PHYS Astronomy Types of Orbits Parabolic Hyberbolic Elliptical Circular

PHYS Astronomy Center of Mass Note: the previous calculations assumed that the mass of the orbiting body was much smaller that the central body - center of orbit at center of central body Newton showed that two objects attracted to each other by gravity actually orbit about their center of mass - the point at which the objects would balance if they were connected. Center of Mass - Binary Star This idea is used to find planets orbiting other stars - massive planets cause star to move against background stars

Einstein 1905 - “The Year of Physics”
Submitted doctoral thesis "A New Determination of Molecular Dimensions” Published five pioneering papers in "Annalen der Physik" - revolutionized physics: "On A Heuristic Point of View Concerning the Production and Transformation of Light" - electromagnetic radiation must consist of quantums or photons - explained the photoelectric effect - became the foundation of quantum theory - what he received the Nobel Prize for in 1921 "On the Electrodynamics of Moving Bodies" - special relativity - new interpretation of the conception of space and time - observer can never detect their uniform motion except relative to other objects - coordinate systems - speed of light constant - independent of motion relative to light source "Does the Inertia of a Body Depend upon its Energy Content?" - the equivalence of mass and energy - E = mc2 PHYS Astronomy

PHYS Astronomy General Relativity 1916 published "The Foundation of the General Theory of Relativity” - generalized special theory of relativity - observer cannot distinguish between inertial forces due to acceleration and uniform gravitational forces - gravity is curvature of space-time - curvature dependent on mass - acceleration of mass dependent on space-time curvature Numerous implications on astronomy and astrophysics - orbital motion - black holes - big bang - formation and structure of galaxies

Proof of General Relativity
PHYS Astronomy Proof of General Relativity Theory predicted the deflection of light in a gravitational field Einstein convinced that light deflection by the gravitational field of the sun could be observed during a total solar eclipse - photograph section of sky where eclipse would occur - during eclipse, photograph same section and measure difference in positions - predicted deflection of 1.75 arcseconds for starlight grazing Sun’s surface Several failed observations of total solar eclipses before proof in 1919 - observed eclipse in island of Principe in the Gulf of Guinea in western Africa and Sobral, Brazil - found shift in stars outward from Sun - 1.61±0.30 arcseconds at Principe - 1.98±0.12 arcseconds at Sobral

PHYS Astronomy

PHYS Astronomy Einstein’s eclipse .

Proof of General Relativity Precession of long axis of Mercury’s orbit
PHYS Astronomy Proof of General Relativity Precession of long axis of Mercury’s orbit Newton’s formulation predicted precession of arcseconds per century - actually arcseconds more - about 29 km past position predicted by Newton per orbit - accumulative - 12,000 km per century - easily measured Einstein predicted arcseconds per century difference Effect since observed on Venus, Earth, and asteroid Icarus

GPS Satellite clock drift - relativistic effect
PHS Astronomy GPS Satellite clock drift - relativistic effect satellites move at 3874 m/s - relativistic time dilation means time runs slower on GPS satellite than on Earth ~ 7.2 microseconds per day satellite at km height exposed to a much weaker field of gravitation than the observer on Earth gravitational time dilation means clock on board of a satellite is running faster than one on Earth this effect about six times stronger than time dilation frequency standard onboard each satellite given rate offset prior to launch makes clock run slightly slower than the desired frequency on Earth at MHz instead of MHz satellite clock errors are periodically corrected by Control Segment

PHYS Astronomy Light

Light Light as a wave f  = c Light as a particle E = hf photon
PHYS Astronomy Light What is light? - A vibration in an electromagnetic field through which energy is transported. The dual nature of light or wave-particle duality: Light as a wave f  = c Light as a particle E = hf photon

PHYS Astronomy Properties of Waves Period: time to complete one cycle of vibration - from crest to crest or trough to trough Frequency (f): number of crests passing a fixed point per second Frequency= 1/period Amplitude (a): maximum displacement from equilibrium Wave length (λ): distance between successive crests Speed (of a wave) (s)= wave length x frequency s= λ x f

PHYS Astronomy Light as a Wave Light is a vibration in an electromagnetic field through which energy is transported - a transverse wave - vibration or oscillation is perpendicular to direction of propagation of wave (vs. longitudinal wave - vibration or oscillation is in the same direction as propagation of wave) So electrons can be manipulated by light. Electrons wiggle up and down as light passes by.

For a wave, its speed: s =  x f
PHYS Astronomy For a wave, its speed: s =  x f But the speed of light is a constant, c. For light:  x f = c The higher f is, the smaller  is, and vice versa. Our eyes recognize f (or ) as color. Visible light ranges through 7 major colors from long wavelengths (low frequency - red) to short wavelengths (high frequency - violet) - Red, orange, yellow, green, blue, indigo, violet (Roy G Biv)

Light as a Particle (Photon)
PHYS Astronomy Light as a Particle (Photon) Light propagates as quanta of energy called photons Photons move with speed of light have no mass are electrically neutral Energy of a photon or electromagnetic wave: E = hf = h c/  where h = Planck’s constant f = frequency of a light wave - number of crests passing a fixed point in 1 second c = velocity of light  = wavelength of a light wave Higher frequency/shorter wavelength - higher energy

The Electromagnetic Spectrum
PHYS Astronomy The Electromagnetic Spectrum Most wavelengths of light can not be seen by the human eye. The visible part of the electromagnetic spectrum lies between ultraviolet and infrared light (between about 400 and 700 nm). The higher the frequency (shorter the wavelength), the higher the photon energy. Radio waves are at the long wavelength end of the spectrum and gamma rays are at the short wavelength end of the spectrum.

Light as Information Bearer
PHYS Astronomy Light as Information Bearer We can separate light into its different wavelengths (spectrum). Spectrum of a distant object - a spectrum is the amount of energy or intensity at different wavelengths. By studying the spectrum of an object, we can learn its: Composition Temperature Velocity

Electron Energy Levels
PHYS Astronomy Electron Energy Levels Electrons can not have just any energy while orbiting the nucleus. Only certain energy values are allowed. Electrons may only gain or lose certain specific amounts of energy. Each element (atom and ion) has its own distinctive set or pattern of energy levels - holds the key to studying of distant objects in the universe. This diagram depicts the energy levels of Hydrogen. 1 eV (electron volt) = 1.6 X J Electron jumps to higher energy levels can only occur with addition of the particular amounts of energy representing differences between possible energy levels. Energy levels are quantized - study of electron energy levels called quantum mechanics. Atom gains this energy either from KE of another atom colliding with it or from absorption of energy carried by light - falls to lower energy level by emitting light or transfer of energy by collision.

PHYS Astronomy Absorption and Emission. When electrons jump from a low energy shell to a high energy shell, they absorb energy. When electrons jump from a high energy shell to a low energy shell, they emit energy. This energy is either absorbed or emitted at very specific wavelengths, which are different for each atom. When the electron is in a high energy shell, the atom is in an excited state. When the electron is in the lowest energy shell, the atom is in the ground state.

The Hydrogen Atom. The hydrogen atom is the simplest of atoms. Its
PHYS Astronomy The Hydrogen Atom. The hydrogen atom is the simplest of atoms. Its nucleus contains only one proton which is orbited by only one electron. In going from one allowed orbit to another, the electron absorbs or emits light (photons) at very specific wavelengths. Note - wavelength is often written as  and the unit used is an angstrom (A) = 10-8 m

Interaction of Light with Matter
PHYS Astronomy Interaction of Light with Matter So each electron is only allowed to have certain energies in an atom. Electrons can absorb light and gain energy or emit light when they lose energy. It is easiest to think of light as a photon when discussing its interaction with matter. Only photons whose energies (colors) match the “jump” in electron energy levels can be emitted or absorbed. Hydrogen Emission Spectrum Absorption Spectrum So visible emission spectrum is created when a gas is heated and collisions in gas continually bump electrons to higher energy levels - emit photons of specific wavelength as they fall back to lower levels. Absorption spectrum is produced when white light is passed through cloud of cool gas. Photons of specific wavelengths absorbed as electrons jump to higher energy levels.

Emission Spectra Orion Nebula in Ultraviolet
PHYS Astronomy Orion Nebula in Ultraviolet The atoms of each element have their own distinctive set of electron energy levels. Each element emits its own pattern of colors, like fingerprints. If it is a hot gas, we see only these colors, called an emission line spectrum.

Absorption Spectra Hydrogen
PHYS Astronomy Absorption Spectra Hydrogen If light shines through a cool gas, each element will absorb those photons whose colors match their electron energy levels. The resulting absorption line spectrum has all colors minus those that were absorbed. We can determine which elements are present in an object by identifying emission and absorption lines.

Temperature and Thermal Energy
PHYS Astronomy Temperature and Thermal Energy Temperature - measure of the average kinetic energy of the particles in a substance - particles in box on right have higher temperature - higher velocity = more KE = higher temperature Both boxes have same temperature - particles have same average velocity/KE - box on right has more thermal energy - energy contained in a substance - more particles Why does water burn your skin so much quicker than air? Why is falling into a 32º F lake more dangerous than standing outside naked on a 32º F day?

PHYS Astronomy This diagram compares three common temperature scales. The Fahrenheit scale is used in the United States, but nearly all other countries use the Celsius scale. Scientists prefer the Kelvin scale because O K represents absolute zero, the coldest possible temperature.

PHYS Astronomy Thermal/Blackbody Radiation Photons are produced whenever charged particles are accelerated - A moving charge gives rise to a magnetic field, and if the motion is changing (accelerated), then the magnetic field varies and in turn produces an electric field - electromagnetic radiation - photons In an opaque object or dense gas cloud, photons can’t easily escape - they “bounce around” in the object. This randomizes their radiative energies and resulting photon energies depend only on the body’s temperature - produces a continuous spectrum called a thermal radiation or blackbody spectrum. Blackbody - a hypothetical body that completely absorbs all wavelengths of thermal radiation incident on it - does not reflect light - appears black if temperature low enough so as not to be self-luminous. - all blackbodies heated to a given temperature emit thermal radiation with the same spectrum - required by thermal equilibrium - distribution of blackbody radiation as a function of wavelength - the Planck law, cannot be predicted using classical physics. - the first motivating force behind the development of quantum mechanics

Key Features of a Blackbody Spectrum
PHYS Astronomy Key Features of a Blackbody Spectrum - a dense object produces light at all possible wavelengths if the object is above absolute zero. - since everything in the universe is above 0 K, all dense objects (solids, liquids, thick gases) will produce a thermal spectrum. - the shape of a continuous spectrum depends on only the temperature of the object not its chemical composition. - as the temperature of an object increases, more light is produced at all wavelengths as the temperature of an object increases, the peak of thermal spectrum curve shifts to shorter wavelengths (higher frequencies) -cool things appear red or orange, hotter things appear yellow or white, and very hot things blue or purple.

Temperature (K) of Black Body
PHYS Astronomy Temperature (K) of Black Body Wavelength (max) at Which Most Radiation is Emitted Type of Radiation 3 0.1 cm Radiowaves 300 0.001 cm "Far" Infrared 3,000 1000 nm "Near" Infrared 4,000 750 nm Red Light 6,000 500 nm Yellow Light 8,000 375 nm Violet Light 10,000 300 nm "Near" Ultraviolet 30,000 100 nm "Far" Ultraviolet 300,000 10 nm "Soft" X-Rays 1.5 million 20 nm "hard" x-rays 3 billion 0.001 nm Gamma rays

Hotter objects emit more total radiation per unit surface area.
PHYS Astronomy Hotter objects emit more total radiation per unit surface area. E = T4 ( = 5.67 x 10-8 watts/m2 K4) - Stefan-Boltzmann Law Hotter objects emit photons with a smaller wavelength (higher average energy.) max (nm) = x 106 nm-°K/T 106 / T(K) [nm] - Wien’s Law

Derivation of the Stefan-Boltzmann Law
PHYS Astronomy Derivation of the Stefan-Boltzmann Law

PHYS Astronomy Solid Angle The solid angle  subtended by a surface S - the surface area  of a unit sphere covered by the surface's projection onto the sphere. This can be written as where n is a unit vector from the origin, da is the differential area of a surface patch, and r is the distance from the origin to the patch. Written in spherical coordinates with  the colatitude (polar angle) and  for the longitude (azimuth), this becomes Solid angle is measured in steradians, and the solid angle corresponding to all of space being subtended is 4 steradians.

PHYS Astronomy

PHYS Astronomy Consider the solid angle subtended by one face of a cube of side length centered at the origin. Since the cube is symmetrical and has six sides, one side obviously subtends 4/6 steradians.

PHYS Astronomy

PHYS Astronomy

- the hotter the star, the more energy radiated per square meter
PHYS Astronomy So, the luminosity of a star depends on temperature and size (surface area). Amount radiated from every square meter equals σT4 (Stefan-Boltzmann Law) - the hotter the star, the more energy radiated per square meter Total amount radiated (luminosity): L = 4R2σT4 R is star's radius, surface area = 4R2 Stellar luminosities generally given in number of solar luminosities: If we measure L and T, we can estimate R T can be determined using Wein’s Law

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