Gauss Formulation of the gravitational forces

Transcription

1 Chapter 1 Gauss Formulation of the gravitational forces 1.1 ome theoretical background We have seen in class the Newton s formulation of the gravitational law. Often it is interesting to describe a conservative field depending on the inverse of the distance from the source of the field in an alternative formulation often refered to as the Gaussian formulation. You have probably already seen this formulation in physics when you have studied electrostatic and it is part of the Maxwell equation in the case of static charges. Let s start by the definition offlux: The flux of a vectorial field through a surface is the integral of the scalar product of the field in a given place time the infinitesimal oriented surface int that point. For example the velocity of a fluid time the x-section area of a pipeline give the flux of the fluid in the tube. In mathematical term we can write: ˆ g da (1.1) where g is the vectorial field, the full surface and da the infinitesima oriented surface. Often we are interested in the flux through a close surface and we write it as: g da (1.2) The flux can be locally related to the divergence of the given field. To understand this concept let s try to compute the flux of a generic field f ( )x through an infinitesimal cube. 2

2 Figure 1.1: To make the math easier let s take a field that is locally directed as ˆx 3 (we can always look for such reference frame for a given point). On the lower face of the cube since the normal of the surface is parallel to x 3 the scalar product f da will be equal to the projection of f (x 1, x 2, x 3 ) in the direction of x 3 by definition f 3 time the surface x 1 x 2. On the top surface of the cube the normal is now directed in the opposite direction. From the previous analysis we can write the flux in the direction x 3 as: φ 3 = (f 3 (x 1, x 2, x 3 + x 3 ) f 3 (x 1, x 2, x 3 )) x 1 x 2 (1.3) where I have defined the flow out from the cube as positive. For very small x 3 by definition of dervivative we can write f (x1, x 2, x 3 + x 3 ) f 3 (x 1, x 2, x 3 ) = f 3 (x 1, x 2, x 3 ) x 3 (1.4) that give the expression for the flux in the x 3 direction: φ 3 = f 3 (x 1, x 2, x 3 ) x 3 x 1 x 2 (1.5) imilarly we have: φ 2 = f 2 (x 1, x 2, x 3 ) x 2 x 2 x 1 x 3 (1.6) φ 1 = f 1 (x 1, x 2, x 3 ) x 1 x 1 x 3 x 2 (1.7) That give the full flux through the cube: ( f3 (x 1, x 2, x 3 ) + f 2 (x 1, x 2, x 3 ) + f ) 33 (x 1, x 2, x 3 ) x 3 x 1 x 2 (1.8) x 2 3

3 or for the limit of the limit of the volume V = x 3 x 1 x 2 0 φ V = f 3 (x 1, x 2, x 3 ) + f 2 (x 1, x 2, x 3 ) + f 33 (x 1, x 2, x 3 ) (1.9) x 2 and for the definition of divergence f = f3(x1,x2,x3) + f2(x1,x2,x3) x 2 + f33(x1,x2,x3) lim φ V 0 V = f (1.10) From [1.10] we can see that the divergence is equivalent to the limit for the volume going to 0 of the flow per unit volume in the surrounding of the point. Figure 1.2: Let s take now the flow through a generic surface (Figure 1.2a) expressed in the form [1.2]. If now we divide the volume in 1.2a as in Figure 1.2b, the flow through the surface D is positive for one of the 2 volume is positive and the other is negative, thus the flow can be written as: j φ 1 + φ 2 = f da f da (1.11) Independently by the number of divisions we can write: and if we call V i the volume within the surface i we can write: N φ i (1.12) φ i = 1 f da (1.13) V i V i i or N N φ i φ i = = V i N 1 V i f da = i f da (1.14) For the limit of N the volume V 0 for [1.10] we have: lim N that is the THEOREM OF DIVERGENCE OR OF GAU. N ˆ φ i = f dv = f da (1.15) V i V 4

4 1.2 The Newton s law From the Newton s law we know that the gravity force can be express as: F = G m 1 m 2 r 2 ê r (1.16) where F is the force between the 2 masses, G thr universal gravity constant, r the distance between the 2 masses, m 1 and m 2 the 2 point masses and ê r the unit vector in the direction of the lines connecting the 2 masses. Given only the point mass m we create a field that surround it for which every unit mass in any point of the space will feel a gravitational acceleration toward m given by: g = G m r 2 êr (1.17) Putting [1.17] in the definition [1.2], the flux of the field g through a sphere of radius R centered on the mass m is given by: g da = G m ˆ R 2 e r da (1.18) But for a sphere we can write da = R 2 ê r dω where dω is the solid angle from the center of the sphere pointing to the small area da. Introducing this expression in [1.18] we obtain 1 Gm R 2 êrr 2 ê r dω = Gm dω = 4πGm (1.19) The expression [1.19] can be derived for generic close surface (not necessary a sphere) and for a generic distribution of masses. In this later case we can substitute m with the sum of all the masses included within the surface M. This espression is often called the FIRT FORMULATION OF THE NEWTON LAW IN THE GAU FORMAT: g da = 4πGM (1.20) In general we can say that for every field dependent as the inverse of the square of the distance the flux through a close surface is proportional to the sum of the sources within the surface. For the theorem of divergence [1.15] we can also express [1.20] as: g da = ˆ V ˆ g dv = 4πG ρdv (1.21) V that give the ECOND GU FORMULATION OF THE GRAVITATIONAL FIELD : g = 4πGρ (1.22) 5

5 1.3 ome applications Let s look at some applications of the Gauss formulation of the Newton s law. The gravitational field is also conservative that means it exist a potantial V dependent only by the position (x 1, x 2, x 3 ) such that g = V (1.23) If we substitute [1.23] into [1.22] we obtain: g = V = 2 V = 4πGρ (1.24) The potential of the gravitational field is related to the mass distribution by the Laplace equation. We can use the Gauss formulation of the Newton s law to compute the gravity anomaly due to a burried sphere of anomalous density ρ. Let s assume that the sphere of radius a is burried at depth z. (Figure 1.2). Figure 1.3: Let s take a spherical surface of radius R centered on the sphere. For reason of simmetry the gravitational attraction of the anomalous sphere on the surface will be directed to the center of the sphere along the direction of the radius R. The geometry of the spherical surface tell us that it is always perpendicular to the radius at every point. Furthermore, since on the surface the distance from the center of the sphere is constant, also g is constant. This means that g da is nothing else than the value of the field at a given distance R time the surface of the sphere. ince the surface of a sphere of radius R is 4πR 2 and the field g is always perpendicular to to the surface of the sphere, the flux will be given by: g da = 4πR 2 g (1.25) 6

6 (Note, this analysis is true for a sphere if we use different simmetry or geometric shape we need to change the computation of the area as well!!). But for [1.20] we know that: g da = 4πGM = 4π 2 G 4 3 a3 ρ = 4πR 2 g = 4π 2 G 4 3 a3 ρ (1.26) that gives: g = 4 3 πga3 ρ R 2 (1.27) In reality the perturbation with respect to the gravitational attraction of the planet the perturbation of the anomaly is small. ince by definition the gravitational attraction of the planet is the vertical we want only the z component of the anomaly: g z = g cosθ = g z R = 4 3 πga3 ρz R 3 = 4 z 3 πga3 ρ (x 2 + z 2 ) 3 /2 (1.28) that is the formula that give the gravity anomaly above a burried a homogeneous sphere of radius a and anomalous mass ρ at depth z when we measure it at a distance x from the center of the sphere. 7