EXPEDITION TO EARTH
EXPEDITION TO EARTH ESS 314
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This 13 page Class Notes was uploaded by Miss Jeanette Keebler on Wednesday September 9, 2015. The Class Notes belongs to ESS 314 at University of Washington taught by John Booker in Fall. Since its upload, it has received 25 views. For similar materials see /class/192650/ess-314-university-of-washington in Earth And Space Sciences at University of Washington.
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Date Created: 09/09/15
ESS 314 Space Physics by John Booker Introduction Earth is immersed in an environment that is not empty space despite what we call the region outside its atmosphere Our sun is constantly losing large amounts of matter to its surroundings sometimes extremely violently The surface of Earth would not be a very hospital to life if it were not for a variety of physical phenomena that protect the surface from most of these effects This chapter discusses these phenomena Since this is the starting point for our journey to the planet s surface it will also be the starting point for a review of basic physics Matter There are four basics forms of matter Gas 7 a loose configuration of molecules that interact only by collision This means that a gas will expand forever unless confined Liquid 7 a configuration of molecules that stick to one another but which can be easily broken apart by external forces Like gases liquids deform permanently under the in uence of forces that do not change their volume Unlike gases they can form free surfaces that can confine them Solid 7 a tight configuration of molecules that require a threshold force in order to break them apart Solids deform under the in uence of external forces but return to their original configuration if the force is removed and does not exceed the threshold for tearing the molecules apart Plasma 7 is a gas whose molecules have been energized sufficiently to strip many if not all electrons from their atomic nuclei Thus there are no molecules Under the in uence of electromagnetic fields a plasma reacts very differently from a gas because positively and negatively charged pieces of the plasma experience different forces At the low densities typical of interplanetary space collisions between plasma particles are rare or nonexistent and they interact primarily through electromagnetic forces N E 4 Many common substances such as water exist in the first three forms at temperatures typical of our environment and are therefore very familiar Gasses and liquids are both termed uids because they share the property of deforming forever under the in uence of forces that do not try to change their volume Plasmas exist inside devices such as orescent lights in lightning and other electrical sparks and are the primary form of matter in space So this chapter is primarily about the physics of plasmas Basic Physical Laws Fundamental laws of classical dynamics are 1 Conservation of mass 7 Einstein generalized this to include the equivalence of energy and mass and the possibility of conversion of mass to energy This is important to us because N E 4 it is going on inside the Sun but it is not important to the other phenomena that we will consider in this course Thus we will assume that mass is strictly conserved Conservation of linear momentum 7 If m is the mass of an object assumed constant and u is its velocity the quantity mu is called its momentum This law says that the in the absence of force the momentum does not change with time More generally F mu ma where F is the vector sum of all forces acting on an object and a is its acceleration This is known as Netwton s 2quotd Law In a uid this must be rewritten in terms of quantities per unit volume Dir M a Dr f pa p p where 1 is the instantaneous position of a uid particle and p is the uid density A tricky point is that the volume used in this law is attached to the molecules of the uid No mass is ever allowed to cross its surface The shape of the volume can deform as the uid moves but the mass inside the volume is constant To remind us that we are discussing a volume that moves with the uid we replace d in the time derivative by D when discussing uids Conservation of angular momentum 7 Angular momentum is a generalization of linear momentum for motion in a circle and is mass times velocity times the radius of the circle of curvature It is conserved in the absence of the rotational equivalent of force which is called torque Conservation of energy 7 Energy comes in several avors kinetic potential heat radiation and mass We have already agreed not to worry about the equivalence of energy and mass Taking the dot product of the lst equation above with the velocity gives d d l 2 d 11 F u dtmu dt2mu thKE The quantity E K in the brackets on the right is called kinetic energy In the absence of force its conservation is obviously a consequence of the conservation of momentum and not a separate law The right hand side of the above relation is the rate of change of what is often called work although it is in a different form than you may have seen it before The standard definition of the amount work done moving an object a distance 5x against a force F is 3W F 5x Dividing this by a small time interval 5t gives zp 6t 6t If this time interval is made very small and F is actually the component of a vector force in the direct of the displacement this becomes dW dx d d F F u FuuF E E dt dt dt pg dt m So the change in kinetic energy must be balanced by a change in work or potential energy E 12 Again this is a consequence of the conservation of momentum and not a separate law This brings us to heat Kinetic energy depends on the magnitude of the velocity and not on its direction The kinetic energy just considered is of an object in which all parts are moving in the same direction However this is not the only kind of kinetic energy All substances that are not at a temperature of absolute zero have internal kinetic energy that is random in nature Because this motion is random in direction its average contribution to the largescale movement of the object cancels out and does not contribute to the kinetic energy that is conserved as a consequence of the conservation of momentum This thermal energy is however conserved and can be passed on to adjacent objects through collisions or electromagnetic forces in a plasma A hot solid for instance has rapidly vibrating molecules When this solid is placed next to a cool solid with slower moving molecules the rapidly vibrating ones in the hot substance will pass some of their momentum to the lower ones at points where they touch This is a oneway process A fast molecule can speed up a slow one but a slow one cannot speed up a fast one Thus heat which is the average random kinetic energy of a substance ows from hot regions to cold regions We will discuss this in more detail later Finally electromagnetic radiation can transfer energy from one place to another In contexts that interest us radiation becomes of interest when it is converted to heat V39 Inverse square laws for mass and charge 7 There are two fundamental forces between small pieces of matter Gravitational Fg Gm1m2 L2 lrl l f Electrostatic F2 91 2 47239s0 M where m1 and m are the masses Kg G 66742gtlt103911 NmzKg ql and q are the charges oftwo particles Coulombs and 80 88542gtlt103912 CzNmz r is the distance between the two particles and f is a unit vector pointing from particle one to particle two The gravitational equation is called Newton39s lst Law Except for the constants G and 1 47180 gravitation and electrostatic forces have the same form The electrostatic force is much stronger than the gravitational force however For two electrons m 91094gtlt10 31 Kg q 16022gtlt10 19C 417x10 where 10 2 rneans 1 followed by 42 zeros a very blg number Another major dlfference ls that llke eharges repel eaeh other whlle opposlte eharges attraet For gravrty rnasses always attraet eaeh other If we de ne the eleetrle andrnagnehe elds g andE by gsz E r s H the above force equahons beeorne 2 and negatwe eharge plaeed elose to one anotherls ealled a dlpolequot andlooks llke ths times the eharge separahon Companng the gravrtatlonal force equanon ln lts eld form wlth Newton s 2 Law we see that A L A on the surface of aplanet sueh as Earth we olten wnte the force perunltvolurne rrogi because the direction of the vertical unit vectori is almost always de ned to be the direction of gravity Another important phenomenon occurs when electric charges move they generate a magnetic eld A collection of moving charges is called an electric current Most currents are due to moving electrons By convention a positive current is in the opposite direction to the electron velocity The magnetic eld of a straight line current looks like this Iun Its magnitude decays like r71 away from the current and its direction is determined by the right hand rule point your thumb in the direction of the current and your ngers will curve in the direction of the magnetic eld If the current travels in a circle the magnetic eld inside the loop will be perpendicular to the plane of the loop Here are a side and angle view Outside the loop at a distance large compared to the diameter of the loop the magnetic eld looks like a dipole due to a positive and negative magnetic charges or monopoles located just above and below the plane of the loop Magnetic charges have never been isolated in nature and are generally thought to be impossible Thus magnetic elds are always the result of moving electric charges TNhPr were no The spaual separahon ofthe eld hhes IS propomonat to the magmmde ofthe f1eldwh1ch 15 clearly stronger as you approach the poles whae the surface eld 15 vemcal We W111 remm w the effect that the Sun has tater Va har tt FM q uXB electromagaeuc force can be wntten FF qEuxB The x tauvtsud FF m c Hut 1 thstead the force per umt volume tsr j x13 and the total electromagaeuc forces per umt volume are IMF332ij h LorenLZ force Orbital motion If we attach a mass m to the end of a string we can whirl it in a circle as shown in this gure 5n m 111 111 I 112 l I l I I 1 5n I 5t The velocity of the mass changes all the time as the mass goes around because its direction changes Thus it is accelerated The direction of this acceleration is towards the circle center We can use the properties of the cross product to calculate what this acceleration must be Let Q be a vector whose length is Q and whose direction is perpendicular to the plane in which the mass circulates Let R be the position vector of m from the center of circulation It is then obvious from this gure 9 and the properties of the cross product that 11 Q X R and that the magnitude this velocity is QR If 9 is constant the vector acceleration is ad uiQXRQX QXH ch ch ch Note that the right hand rule for the cross product implies that Q gtltu points towards the center of circulation as we expect The magnitude of the acceleration is a Qu QQR 92R Note that in this gure a force39 vector f is drawn in the opposite direction to the actual force ma that must be applied to the mass to keep it in the circular path This f is the tensile force in the string which must just balance the force applied to the mass if the length of the string is to remain constant This f is sometimes called the centripetal force while ma is usually called the centrifugal force The fact that the two forces are equal in magnitude and opposite in direction for the equilibrium situation is an example of Newton39s 3rd Law for every action there is an equal dand opposite reaction which is not a separate law at all but another direct consequence of his 2quot Law The period of a satellite The string in the last example is an arti ce that is not required if the body circulates under the in uence of gravitational or electric elds The gravitational eld outside of a planet to a very good approximation the eld of a point mass M with the same mass as the planet For a small mass m to orbit at a constant distance R from the center of the planet the forces on it must balance thus G M szR R2 If we solve this for 9 we easily see that the orbital period of a satellite is which is independent of the mass of the satellite Physically one can think of a satellite as constantly falling towards the planet but always missing because of its orbital velocity This relation allows one to determine the mass of the planet from the period of a satellite arti cial or natural Rearranging the last equation gives M 472392R3 GT2 This also allows one to determine the mass of our Sun using the orbital period of any one of its planets and the radius of the orbit Earth s distance from the Sun is about 150 million km Thus the mass of the Sun is approximately 472392 gtltl5gtlt108 gtlt1033 4gtlt3375gtlt1033 m 2X1030K 6667gtlt103911 X 7239 gtlt1072 67 gtlt103 g This theory is not the whole story because there is another equally valid solution in which the path of the satellite is an ellipse with the planet at one of the focii Earth s orbit about the Sun is actually an ellipse which varies slowly with time This will be discussed later when we discuss climate variation Furthermore a circular or elliptical orbit is only approximate for very massive satellites such as Earth s Moon In this case both Earth and Moon orbit around the center of mass of the EarthMoon system Gyro motion of a charged particle in a magnetic eld An electron moving with velocity u perpendicular to a uniform magnetic eld B experiences a constant force q 11 X B that is perpendicular to the velocity and the magnetic eld Just as for the circulating mass on a string or the satellite in a radial gravitational eld this force will cause the electron to follow a curved path B A l quot quot Note qe lt 0 Note that the charge on an electron is negative Thus the direction of ma is opposite to what you get from applying the right hand rule to the cross product of the velocity and the magnetic eld The only forces acting on the electron in this case are the centripetal force and the electromagnetic force Since they must balance they both must be radial to the center of curvature Balancing their magnitudes gives q uB q QR szzR and thus Q q me B which depends only on the strength of the magnetic field and the charge to mass ratio of the electron This is called the gyro frequency by plasma physicists and the cyclotron frequency by atomic physicism The fact that radius does not enter this relation means that the elliptical solution for a satellite does not exist in this case and the electron orbit will always be circular Substituting this expression for Q in the force balance above and rearranging terms leads immediately to the gyroradius mam 2 quot2E1ltE R qngl qngl which depends on the momentum or the energy of the electron A proton would also orbit in a similar way but would have a much lower frequency and a much larger gyroradius because of is larger mass It would also circulate in the opposite direction The result just derived has interesting consequences If the electron39s initial velocity is not perpendicular to the magnetic field is path will be a helix like this This is because the component of velocity along parallel to the magnetic field uH creates no electromagnetic force and thus uH remains constant as the electron orbits in the magnetic field The acceleration on the electron is entirely perpendicular to the direction of the magnetic field and must be balanced by the centripetal acceleration about the center line of the helix Trapping of charged particles in a planetary magnetic eld Things get even more interesting if lB increases in the direction of B An electron spiraling along a eld line has a velocity parallel to the eld line and a velocity of circulation around the eld line and hence perpendicular to the eld line Thus uuH ui uH QgtltR The kinetic energy of this electron is E ou r QR2 KE l 2 2 m2 2 which must be conserved This means that if QR increases luH l must decrease The relation between QR and lB can be deduced from the conservation of angular momentum Angular momentum is a generalization of linear momentum and is mass times velocity times the radius of the circle of curvature Thus as a particle moves from position 1 to a position 2 where the magnetic eld strength has changed from lBll to M m2 QIRIZ MZQZRZZ Simple manipulation reduces this relation to and thus 2 2 2 2 92a QIRI 2 91 R1 91 QZ 91 The gyrofrequency relation above then gives If le gt lBl we must have that 2sz gt QlRl So we nally conclude that the conservation of kinetic energy requires that uH must decrease as lBl increases To sum up As the magnetic eld gets stronger the orbital velocity and the orbital part of the kinetic energy increase This is done at the expense of the velocity component parallel to the magnetic eld In fact uH can actually decrease to 0 One might think that the electron will just stop its motion along the eld when the growing orbital kinetic energy has robbed all the kinetic energy ofthe motion along the magnetic field However this is not the case The force on the electron is perpendicular to both the magnetic field and the velocity and when the field lines are converging this force is not perpendicular to the center ofthe helix as you can see in this figure Thus for converging magnetic field lines there is always a component of the force TH that is in the direction of field line divergence This component is determined by the gyrovelocity and the field convergence and does not go to zero when uH goes to zero Thus the electron will start to accelerate back out of the region of converging magnetic field lines This is called a magnetic u c use Earth magnetic field is a dipole with northern an southem polar regions of magnetic field convergence electrons can have bounce points in both hemispheres and be trapped like this Basic Campdments gr Particle Marian bounce gyrazian and drift As long as an electron spiraling along Earth39s field bounces high enough that it is unlikely to collide with an air molecule the electron will bounce back and forth many times This is the reason for the existence of the Van Allen radiation belts If the bounce point is low enough that the electron hits the upper air it will cause the gas to emit light that is called the Aurora There is one further effect that we can explain with the same physics This is eastwest drift of trapped particles If an electron is circulating in a region where the magnetic eld magnitude changes in the direction perpendicular to the magnetic eld the radius of gyro motion will change as the electron goes around its orbit In this illustration the magnetic field perpendicular to the page It is weak at the top and so a particle of a given energy will have a large radius of gyration In the lower gray area the magnetic field is strong and so a particle with the same energy has a much smaller radius of gyration A particle that spends part of its time in the weak magnetic field and part of its time in the strong magnetic field will have an orbit that has a different radius of gyration in the two regions The result is that its orbits are not closed and so the particle drifts Faltlcle melatlnn 39 con stant r Constant h cyclic mat an d dri Constant Palticle gyration Strung constant r field 0 This happens around Earth because the magnetic field decreases with distance from the center of the planet Thus trapped particles drift east or west depending on their charge see the next to last figure above The biggest source of trapped particles is the Sun and they are preferentially injected into Earth s field on the sunward side The drift however implies that eventually trapped particles are found at all longitudes
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