EXPEDITION TO EARTH
EXPEDITION TO EARTH ESS 314
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Date Created: 09/09/15
E55 314 Space Physics by John Booker and Gerard Roe 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 con guration of molecules that interact only by collision This means that a gas will expand forever unless con ned Liquid 7 a con guration 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 con ne them Solid 7 a tight con guration 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 con guration 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 suf ciently 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 rst three forms at temperatures typical of our environment and are therefore very familiar Gasses and liquids are both termed fluids 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 N E 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 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 d E 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 Dzr M f 3 E p th 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 at 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 Name Expression Equivalent name Equivalent expression linear linear rotation rotation Distance X Angle Speed u dXdt Angular speed 00 dGdt Velocity u dxdt Angular velocity 9 dedt see figure Acceleration a dzxdt2 Angular acceleration a dzedt2 Mass m Moment of inertia I mr2 Force F Torque T 1 x F Table showing properties of linear motion and the rotational equivalent For every property of linear momentum there is an equivalent property in rotational physics See the table Also the laws of linear motion have their rotational equivalent So linear momentum mv the angular momentum 1S2 Newton s second law for linear motion is given by F ma The rotational equivalent is given by TIu A quot39 Figure shows righthandrule The fingers curl in the sense of rotation and the thumb points in the direction of the vector 9 4 Conservation of energy 7 Energy comes in several avors kinetic potential heat radiation and mass We have already agreed not to worryabout the equivalence of energy and mass Taking the dot product of the lSt equation above with the velocity gives d d 1 d u39F uEmua mu2 7EKE 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 wor 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 3 W F 5x Dividing this by a small time interval 5t gives 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 7F7F F 39F7E 7E dt dt I u dt PE dt KE So the change in kinetic energy must be balanced by a change in work or potentia energy E FE 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 5 Inverse square laws for mass and charge 7 There are two fundamental forces between small pieces of matter f Grav1tational Fg Gmlm2 W I Electrostatic39 F 1 q q f e 4 0 1 2 M2 where m1 and m are the masses Kg G 66742x103911 NmzKg ql and q are the charges of two particles Coulombs and 0 88542x103912 CzNmz r is the distance between the two particles and f39 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 47reo gravitation and electrostatic forces have the same form The electrostatic force is much stronger than the gravitational force however For two electrons me 91094x103931 Kg q9 16022x103919 C i417x1042 F E where 1042 means 1 followed by 42 zeros a very big number Another major difference is that charges can be positive or negative while mass can only be positive as far as we know Because of the negative in front of the constant in the electrostatic force equation like charges repel each other while opposite charges attract For gravity masses always attract each other If we define the electric and magnetic fields g and E by IGm M the above force equations become Fens 1111 2 and negative charge placed close to one anothens eaned a dpolequot andlooks hke this times the charge separation Companhg the ghamtatmha1 force equation m its eld form with Newton s 2 d Law we see that oh the surface of aplahet sueh as Earth we o en wnte the force perumtvolume Mi gramty eld A m w iv v velomty The magnetic eld of a straght hhe euhehtlooks hke this B I2qu 7 Diagram showing the relationship between a current I represented by a sum of moving charges and the magnetic eld they generate The sense of the eld is another application of the righthandrule 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 electnc charges If Lhae were no The spaual separatxon ofthe eld 1 clearly strongerasyou approach m opomonal to the magmtude ofthe f1eldwh1ch I mes IS pr 5 e polesquot whae the surface eld xsva ucal We W111 return to the effect that the Sun has late TM Mr R tt 17 11an r th ttat eleckomagqeuc force can be wnttm F FFqnnxn The x mm 1 u FF rt 1 m t ad I 4 L Mun volumeare feflpeE ij where pe is the electric charge density The last term on the right is commonly called the Lorentz 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 111 111 112 I I I l l l 6n a St 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 u Qx R and that the magnitude this velocity is QR If 9 is constant the vector acceleration is a i xR Qx xu dt dt dt Note that the right hand rule for the cross product implies that qu points towards the center of circulation as we expect The magnitude of the acceleration is a Qu QQR 92R u QxR 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 3r Lawquot 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 mM mQZR R 2 If we solve this for 9 we easily see that the orbital period ofa satellite is 3 27722 L Q GM 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 artificial or natural Rearranging the last equation gives 4n2R3 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 441392 x15x10quotx103 3 4x3375gtlt1033 2x103 K 6667x1039 xmlt1072 67x103 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 field B experiences a constant force q uxB that is perpendicular to the velocity and the magnetic field 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 T N a quot x 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 field 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 l l M q QR quot19211 and thus 9 i 3 m 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 physicists 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 R ms I39ll VzmsEKE q IBI q IBI 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 eld I creates no electromagnetic force and thus ullremains constant as the electron orbits in the magnetic eld The acceleration on the electron is entirely perpendicular to the direction of the magnetic eld 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 uuquotuLuquot xR The kinetic energy of this electron is 1 2 m 2 EKE Emelul 79 ulll which must be conserved This means that if QR increases qu 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 B2 meQIRl2 meQZRZZ Simple manipulation reduces this relation to and thus M 52 QIRIY 91R1 9192 91 The gyrofrequency relation above then gives If lel gt 31 We must have that Qsz gt QIRI So We nally conclude that the conservation of kinetic energy requires that unmust decrease as 3 increases To sum up As the magnetic field 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 field In fact IIH can actually decrease to 0 One might think that the electron Will just stop its motion along the field When the growing orbital kinetic energy has robbed all the kinetic energy of the motion along the magnetic field Ho ver i is not th ase The orce on e 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 of the helix as you can see in this figure Thus for converging magnetic field lines there is always a component ofthe force TH that is in the direction offield quot 439 quotquotquot r 39 39 39 and the field convergence and does not go to zero When ungoes to zero Thus the electron Will start to accelerate back out of the region of converging magnetic field lines This is called a magnetic bounce Because Earth s magnetic field is a dipole With northern and southern polar regions of magnetic field convergence electrons can have bounce points in both hemispheres and be trapped like this Basic Camponents 0f Particle Marian bounce gyraziiin and drift Flux Tube Nimh Tra ped Particle Trajectmy Pizen angle 90 Magnetic Field ine xaw Magnetic Canjugate Paint Th 15 the a a a L n enoughthat an L mum ann dnm W Tn HM mu V mhm LoLhepage p M Mr radm of wah n the pamcle dn squot mm mm cnnslam y nLhe unward sxde The dn however1mphesLhaLevenLually the planet gure above mjected mto Eanh s eld 0 s trapped pamcles are found at all longxtudes
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