Curriculum For ExcellenceAdvanced Higher PhysicsAstrophysicsCfE Advanced Higher Physics – Unit 1 – Astrophysics0. Introduction: A brief history of our understanding of the UniverseGRAVITATION1. Gravitational field strengtha. Newton’s initial work on gravitation based on information available to himb. Inverse square nature of the gravitational fieldc. Weighing the Earthd. Gravitational field patterns and lines of equipotentiale. Variations in the value for “g”f. Satellites and kinetic energyg. Consequences of gravitational fields2. Gravitational potential and potential energya. Nature of potential with increased distance from bodyb. Conservative fields3. Escape velocitya. Derivation of escape velocityb. Effects on atmospherec. Escape velocity and photons (reference to black holes and gravitationalredshift)GENERAL RELATIVITY4. Equivalence principle and its consequences5. Space-time and space-time diagramsa. Curvature of space-time due to massb. Behaviour of motion of mass due to curvature of space-time6. Black holesa. Schwarzschild radius and the event horizonb. Time dilation and gravitational redshiftc. Consequences and evidence of curvature of space-timei. Gravitational lensingii. Recession of Mercury’s orbitSTELLAR PHYSICS7. Properties of starsa. Structure, size and surface temperatureb. Blackbody radiationc. Peak wavelength and temperatured. Masse. Brightness, luminosity and magnitudef. Detectiong. Classificationh. Solar activity8. Hydrogen and helium fusion reactions – production of deuterium, helium 3, helium 4,positrons, neutrinos and gamma raysa. Stellar equilibrium9. Stellar Evolution10. The Hertzsprung-Russell (H-R) diagram (included in stellar evolution section)1Compiled and edited by F. KasteleinSource - Robert Gordon's CollegeBoroughmuir High SchoolCity of Edinburgh Council
Curriculum For ExcellenceAdvanced Higher PhysicsAstrophysicsHistorical IntroductionThe development of what we know about the Earth, Solar System and Universe is a fascinatingstudy in its own right. From earliest times Man has wondered at and speculated over the ‘Natureof the Heavens’. It is hardly surprising that most people (until around 1500 A.D.) thought thatthe Sun revolved around the Earth because that is what it seems to do! Similarly most peoplewere sure that the Earth was flat until there was definite proof from sailors who had venturedround the world and not fallen off!It may prove useful therefore to give a brief historical introduction so that we may set this topicin perspective. For the interested student, you are referred to a most readable account ofGravitation which appears in “Physics for the Inquiring Mind” by Eric M Rogers - chapters 12to 23 (pages 207 to 340) published by Princeton University Press (1960). These pages includeastronomy, evidence for a round Earth, evidence for a spinning earth, explanations for manygravitational effects like tides, non-spherical shape of the Earth, precession and the variation of‘g’ over the Earth’s surface. There is also a lot of information on the major contributors overthe centuries to our knowledge of gravitation. A briefer summary can be found in theIntroduction of “A Brief History of Time”, by Stephen Hawking. An even more concise historyof gravitation is included as an introduction to these notes.Claudius Ptolemy (A.D. 120) assumed the Earth was immovable and tried to explain thestrange motion of various stars and planets on that basis. In an enormous book, the “Almagest”,he attempted to explain in complex terms the motion of the ‘five wandering stars’ - the planets.He suggested that the universe consisted of nested spheres, one within the other and that eachindependent spheres motion was responsible for the independent motion of certain objects, withthe stars on the outmost sphere, their relative positions “fixed”. This is known as a geocentric(Earth centred) model.Nicolaus Copernicus (1510) insisted that the Sun and not the Earth was the centre of, not onlythe solar system, but the universe. He was the first to really challenge Ptolemy. He was thefirst to suggest that the Earth was just another planet, centred only within the lunar sphere. Hisgreat work published in 1543, “On the Revolutions of the Heavenly Spheres”, had far reachingeffects on others working in gravitation. This is known as a heliocentric (Sun centred) model.Tycho Brahe (1580) made very precise and accurate observations of astronomical motions.He did not accept Copernicus’ ideas. His excellent data were interpreted by his student Kepler.Johannes Kepler (1610) Using Tycho Brahe’s data he derived three general rules (or laws) forthe motion of the planets. He could not explain the rules.Galileo Galilei (1610) was a great experimenter. He invented the telescope and with it madeobservations which agreed with Copernicus’ ideas. His work caused the first big clash withreligious doctrine regarding Earth-centred biblical teaching. His work “Dialogue” was bannedand he was imprisoned. (His experiments and scientific method laid the foundations for thestudy of Mechanics).Isaac Newton (1680) brought all this together under his theory of Universal Gravitationexplaining the moon’s motion, the laws of Kepler and the tides. In his mathematical analysishe required calculus - so he invented it as a mathematical tool!2Compiled and edited by F. KasteleinSource - Robert Gordon's CollegeBoroughmuir High SchoolCity of Edinburgh Council
Curriculum For ExcellenceAdvanced Higher PhysicsAstrophysicsGravitationConsideration of Newton’s HypothesisIt is useful to put yourself in Newton’s position and examine the hypothesis he put forward forthe variation of gravitational force with distance from the Earth. For this you will need thefollowing data on the Earth/moon system (all available to Newton).Data on the Earth“g” at the Earth’s surfaceradius of the Earth, REradius of moon’s orbit, rMperiod, T, of moon’s “circular” orbittake 19.8 m s-26.4 x 106m3.84 x 108 m27.3 days 2.36 x 106 s.60Assumptions made by Newton All the mass of the Earth may be considered to be concentrated at the centre of the Earth.The gravitational attraction of the Earth is what is responsible for the moon's circularmotion round the Earth. Thus the observed central acceleration can be calculated frommeasurements of the moon's motion:v2arHypothesisNewton asserted that the acceleration due to gravity “g” would quarter if the distance from thecentre of the Earth doubles i.e. an inverse square law.1Acceleration due to gravity, a g 2 rv24π2 RCalculate the central acceleration for the Moon: use aor a.rT2Compare with the “diluted” gravity at the radius of the Moon’s orbit according to the1hypothesis, viz.x 9.8 m s-2.(60)2ConclusionThe inverse square law applies to gravitation.Astronomical DataPlanetorsatelliteMass/kgDensity/kg m-3Radius/mGrav.accel./m s-2Escapevelocity/m s-1Mean distfrom Sun/mMean distfrom Earth/mSun1.99 x 10301.41 x 1037.0 x 1082746.2 x 105--1.5 x 1011Earth6.0 x 10245.5 x 1036.4 x 1069.811.3 x 1031.5 x 1011--Moon7.3 x 10223.3 x 1031.7 x 1061.62.4 x 103--3.84 x 108Mars6.4 x 10233.9 x 1033.4 x 1063.75.0 x 1032.3 x 1011--Venus4.9 x 10245.3 x 1036.05 x 1068.910.4 x 1031.1 x 1011--3Compiled and edited by F. KasteleinSource - Robert Gordon's CollegeBoroughmuir High SchoolCity of Edinburgh Council
Curriculum For ExcellenceAdvanced Higher PhysicsAstrophysicsInverse Square Law of GravitationNewton deduced that this can be explained if there existed a Universal Constant ofGravitational, given the symbol G.We have already seen that Newton’s “hunch” of an inverse square law was correct. It alsoseems reasonable to assume that the force of gravitation will be dependent on both of the massesinvolved.F m, F M and F F 1giving F r2Mmr2GMmwhere G 6.67 x 10-11 N m2 kg-2r2Consider the Solar SystemM Ms, the mass of the Sunandm is mp, the mass of any planet.GM mS P(r distance from Sun to planet)Force of attraction on a planet is: Fr2Now consider the central force if we take the motion of the planet to be circular.GMS mPGravitational force Fr2Central ForcesoFmPGMS mPr2GMS mPrearranging%T2r2%mP #&MS'!mPforce of gravity supplies this central force.and2πr 2 Trgivingv !"GMS mPr2mP 4π2 r 2rT2Kepler had already shown that T2 was a constant for each planet and, since Ms is a constant, itfollows that G must be a constant for all the planets in the solar system (i.e. a universalconstant).Notes: We have assumed circular orbits. In reality, orbits are elliptical. Remember that Newton’s Third Law always applies. The force of gravity is an actionreaction pair. Thus if your weight is 600 N on the Earth; as well as the Earth pullingyou down with a force of 600 N, you also pull the Earth up with a force of 600 N. The Gravitational force is very weak compared to the electromagnetic force (around1039 times smaller). Electromagnetic forces only come into play when objects arecharged or when charges move. These conditions only tend to occur on a relativelysmall scale. Large objects like the Earth are taken to be electrically neutral.4Compiled and edited by F. KasteleinSource - Robert Gordon's CollegeBoroughmuir High SchoolCity of Edinburgh Council
Curriculum For ExcellenceAdvanced Higher PhysicsAstrophysics“Weighing” the EarthObtaining a value for “G” allows us to “weigh” the Earth i.e. we can find its mass.Consider the Earth, mass ME, and an object, mass m, on its surface. The gravitational force ofattraction can be given by two equations:GME mFF W mg2REwhere Re is the separation of the two masses, i.e. the radius of the earth.thusmgGME mR2EMEgR2EG9.8-.6.40-106 /26.67-10-11 6.02 x 1024 kgThe Gravitational FieldIn earlier work on gravity we restricted the study of gravity to small height variations near theearth’s surface where the force of gravity could be considered constant.ThusAlsoFgrav m gEp m g hwhere g constant ( 9.8 N kg-1)When considering the Earth-Moon System or the Solar System we cannot restrict ourdiscussions to small distance variations. When we consider force and energy changes on a largescale we have to take into account the variation of force with distance.Definition of Gravitational Field at a point.Defined as the force experienced per unit mass in a gravitational field. i.e. gFm(N kg-1)The concept of a field was not used in Newton’s time. Fields were introduced by Faraday inhis work on electromagnetism and only later applied to gravitation.Note that g and F above are both vectors and whenever forces or fields are added this must bedone by appropriate vector addition (taking into account direction as well as magnitude!).Field Patterns (and Equipotential Lines)(i) An Isolated ‘Point’ MassNotes: (1)(2)(3)(ii) Two Equal ‘Point’ Massesequipotential lines are always at right angles to field lines.the cross represents the centre of mass of the systemviewed from far enough away, any gravitational field will begin to look likethat of a point mass.5Compiled and edited by F. KasteleinSource - Robert Gordon's CollegeBoroughmuir High SchoolCity of Edinburgh Council
Curriculum For ExcellenceAdvanced Higher PhysicsAstrophysicsVariation of g with height above the Earth (and inside the Earth)An object of mass m is on the surface, radius r, of the Earth (mass M). We now know that theweight of the mass can be expressed using Universal Gravitation.mgThusgGMmGMr2r2(r radius of Earth in this case)(note that g 1r2above the Earth’s surface)However the density of the Earth is not uniformand this causes an unusual variation of g with radiiinside the Earth. For a uniform density we wouldsee the following graph for g at radii both withinand without the earthVariation of “g” over the Earth’s SurfaceThe greatest value for “g” at sea level is found at the poles and the smallest value is found atthe equator. This is caused by the rotation of the earth.Masses at the equator experience the maximum spin of the earth. These masses are in circularmotion with a period of 24 hours at a radius of 6400 km. Thus, part of a mass’s weight has tobe used to supply the small central force due to this circular motion. This causes the measuredvalue of “g” to be smaller.Calculation of central acceleration at the equator:av2randv2πrTObserved values for “g”:givea#23 2 4ar4π2 rat poles 9.832 m s-2at equator 9.780 m s-2difference is 0.052 m s-2T24π2 -6.4-106.24-60-60/20.034 m s -2Most of the difference has been accounted for. The remaining 0.018 m s-2 is due to the nonspherical shape of the Earth. The equatorial radius exceeds the polar radius by 21 km. Thisflattening at the poles has been caused by the centrifuge effect on the liquid Earth as it cools.The Earth is 4600 million years old and is still cooling down. The poles, nearer the centre ofthe Earth than the equator, experience a greater pull.In Scotland “g” lies between these two extremes at around 9.81 or 9.82 m s-2. Locally “g” variesdepending on the underlying rocks/sediments. Geologists use this fact to take gravimetricsurveys before drilling. The shape of underlying strata can often be deduced from the variationof “g” over the area being surveyed. Obviously very accurate means of measuring “g” arerequired.6Compiled and edited by F. KasteleinSource - Robert Gordon's CollegeBoroughmuir High SchoolCity of Edinburgh Council
Curriculum For ExcellenceAdvanced Higher PhysicsAstrophysicsSatellites in Circular OrbitThis is a very important application of gravitation. The central force required to keep thesatellite in orbit is provided by the force of gravity.mv 2Thus:v7GMr2πrTrGMm7r2GMrvand T 2πr7r2π7rGM2πrT3GMr2π7GM.Thus a satellite orbiting the Earth at radius, r, has an orbit period, T3This is Kepler’s 3rd Law of planetary motion (or any circular orbital motion, for that matter).Energy and Satellite MotionConsider a satellite of mass (m) a distance (r) from the centre of the parent planet of massM where M m.Rearranging:mv 2rGMmr2 9mv 2We find that the kinetic energy of a satellite, EK, is given by EKGMm2rGMm2rNote that EK is always positive.The gravitational potential energy of the satellite in this system is EPNote that Ep is always negative.Thus the total energy isEtotalEtotalEtotalEK EPGMm-2rGMm #--GMmrGMmr 2rCare has to be taken when calculating the energy required to move satellites from one orbit toanother to remember to include both changes in gravitational potential energy and changes inkinetic energy.7Compiled and edited by F. KasteleinSource - Robert Gordon's CollegeBoroughmuir High SchoolCity of Edinburgh Council
Curriculum For ExcellenceAdvanced Higher PhysicsAstrophysicsConsequences of Gravitational FieldsThe notes which follow are included as illustrations of the previous theory.Kepler’s LawsApplied to the Solar System these laws are as follows: The planets move in elliptical orbits with theSun at one focus, The radius vector drawn from the sun to aplanet sweeps out equal areas (A) in equaltimes (t). The square of the orbital period of a planet isproportional to the cube of the semi-major axisof the orbit T 2 r 3 .TidesThe two tides per day that we observe are caused by the unequal attractions of the Moon (andSun) for masses at different sides of the Earth. In addition the rotation of the Earth and positionof the Moon also has an effect on tidal patterns.The Sun causes two tides per day and the Moon causes two tides every 25 hours. When thesetides are in phase (i.e. acting together) spring tides are produced. When these tides are out ofphase neap tides are produced. Spring tides are therefore larger than neap tides. The tidalhumps are held ‘stationary’ by the attraction of the Moon and the earth rotates beneath them.Note that, due to tidal friction and inertia, there is a time lag for tides i.e. the tide is not directly‘below’ the Moon. In most places tides arrive around 6 hours “late”.Gravitational PotentialVwork donemassWe define the gravitational potential (V) at a point in a gravitational field to be the work doneby external forces in moving a unit mass, m, from infinity to that point.We define the theoretical zero of gravitational potential for an isolated point mass to be atinfinity. (Sometimes it is convenient to treat the surface of the Earth as the practical zero ofpotential. This is only valid when we are dealing with differences in potential.)Gravitational Potential at a distanceThis is given by the equation below, the units of gravitational potential are J kg-1V-GMrThe Gravitational Potential ‘Well’ ofthe EarthThis graph gives an indication of howsatellites are ‘trapped’ in the Earth’sfield. Imagine a 3 dimensional,frictionless, curved funnel with amarble rolling around inside. Always“attracted” to the centre, yet neverfalling in since no energy is lost to thefriction. This is a stable orbit.8Compiled and edited by F. KasteleinSource - Robert Gordon's CollegeBoroughmuir High SchoolCity of Edinburgh Council
Curriculum For ExcellenceAdvanced Higher PhysicsAstrophysicsConservative fieldThe gravitational force is known as a conservative force because the work done by the force ona particle that moves through any round trip is zero i.e. energy is conserved. For example if aball is thrown vertically upwards, it will, if we assume air resistance to be negligible, return tothe thrower’s hand with the same kinetic energy that it had when it left the hand.An unusual consequence of this situation can be illustrated byconsidering the following path taken in moving mass on a round tripfrom point A in the Earth’s gravitational field. If we assume thatthe only force acting is the force of gravity and that this actsvertically downward, work is done only when the mass is movingvertically, i.e. only vertical components of the displacement need beconsidered.Thus for the path shown below the work done is zero.By this argument a non-conservative force is one which causes the energy of the system tochange e.g. friction causes a decrease in the kinetic energy. Air resistance or surface frictioncan become significant and friction is therefore labelled as a non-conservative force.Escape VelocityThe escape velocity for a mass escaping to infinity from a point in a gravitational field is theminimum velocity the mass must have which would allow it to escape the gravitational fieldi.e. from a point at radius r to infinity.GMAt the surface of a planet the gravitational potential is given by: V - r .The potential energy of mass m is given by V x m (from the definition of gravitationalpotential).GMmEp rThe potential energy of the mass at infinity is zero. Therefore to escape completely from theGMmsphere the mass must be given energy equivalent in size to r .To escape completely, the mass must just reach infinity where its Ek reaches zero(Note that the condition for this is that at all points; Ek Ep 0).1at the surface of the planetve2ve2mve2 2GM7GMmr 0m cancelsr2GMror greater to escapeFor the Earth the escape velocity is approximately 11km s-1.No regard is given here for the presence of an atmosphere and so no energy is lost to airresistance.9Compiled and edited by F. KasteleinSource - Robert Gordon's CollegeBoroughmuir High SchoolCity of Edinburgh Council
Curriculum For ExcellenceAdvanced Higher PhysicsAstrophysicsAtmospheric Consequences:vrms of H2 molecules 1.9 km s-1vrms of O2 molecules 0.5 km s-1(at 0 C)(at 0 C)When we consider that this is the r.m.s. of a range of molecular speeds for hydrogen molecules,and that a small number of all H2 molecules will have a velocity greater than ve , it is notsurprising to find that the rate of loss of hydrogen from the Earth’s atmosphere to outer spaceis considerable. In fact there is very little hydrogen remaining in the atmosphere. Oxygenmolecules on the other hand simply have too small a velocity to escape the pull of the Earth.The Moon has no atmosphere because the escape velocity (2.4 km s-1) is so small that gaseousmolecules will have enough energy to escape from the moon.Black Holes and Photons in a Gravitational FieldA dense star with
Curriculum For Excellence Advanced Higher Physics Astrophysics 2 Compiled and edited by F. Kastelein Boroughmuir High School Source - Robert Gordon's College City of Edinburgh Council Historical Introduction The development of what we know about the Earth, Solar System and Universe is a fascinating study in its own right. From earliest times .
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