Higher Geodesy: 02 Newton's Law of Gravitation PDF

Higher Geodesy 02 Newton‘s Law of Gravitation Annette Eicker Annette Eicker 1 Higher Geodesy: Physical Geodesy  Higher Geodesy Physical Geodesy Mathematical Geodesy „Ph...

Higher Geodesy 02 Newton‘s Law of Gravitation Annette Eicker Annette Eicker 1 Higher Geodesy: Physical Geodesy  Higher Geodesy Physical Geodesy Mathematical Geodesy „Physical shape“ of the Earth Coordinate systems (2D and 3D) and computations on the sphere/ellipsoid => gravity field, height systems, etc. Annette Eicker 2 Higher Geodesy: Physical Geodesy  Outline (Physical Geodesy) Introduction Fundamentals of mechanics motion of point masses, Newton‘s Law of Gravitation today conservation of energy, conservative force field, gravitational potential non-inertial reference frames, pseudo forces three-body problem, tidal forces The gravity field of the Earth representation of the gravity field: spherical harmonics, geodetic boundary value problems gravity field models gravimetric measurements Earth models, normal gravity field functionals of the gravity field: disturbing potential, gravity disturbances, gravity anomalies Height systems physical vs. geometrical heights ellipsoidal heights, normal heights, orthometric heights, normal-orthometric heights Annette Eicker 3 Higher Geodesy: Physical Geodesy  Let‘s start with some basic m mechanics… z r Motion of point masses y x Annette Eicker 4 Higher Geodesy: Physical Geodesy  Example: Motion in a circle Position vector: y  R cos(ω t )    r (t ) =  R sin(ω t )   0  angular velocity   r = x2+ y2 + z 2 = R r R sin(α ) α x R cos(α ) α =ω t Annette Eicker 5 Higher Geodesy: Physical Geodesy  Example: Motion in a circle Position vector: y  R cos(ω t )    r (t ) =  R sin(ω t )    0  angular velocity  r r = x2+ y2 + z 2 = R Velocity: r R sin(α )  − Rω sin(ω t )  α   x r (t ) =  Rω cos(ω t )  R cos(α )  0    r = x 2 + y 2 + z 2 = Rω Temporal derivative: dr r (t ) = dt Annette Eicker 6 Higher Geodesy: Physical Geodesy  Example: Motion in a circle Position vector: y  R cos(ω t )    r (t ) =  R sin(ω t )    0  angular velocity  r r = x2+ y2 + z 2 = R Velocity: r R sin(α )  − Rω sin(ω t )  α r   x r (t ) =  Rω cos(ω t )  R cos(α )  0    r = x 2 + y 2 + z 2 = Rω Acceleration:  − Rω 2 cos(ω t )    r(t ) =  − Rω sin(ω t )  2    0  r = x2 + y2 + z2 = Rω 2 Annette Eicker 7 Higher Geodesy: Physical Geodesy  Example: Linear motion Example Acceleration: r = 0 r Velocity: integration r (t ) = r0 = const integration Position: r (t ) = ∫ r (t )dt = r0t + r0 a  d   a ⋅t + d        r (t) =  b  ⋅ t +  e  =  b ⋅ t + e  c  f  c ⋅t + f        ⇒ constant, rectilinear motion (= in a straight line) ⇒ uniquely defined apart from 6 constants of integration Annette Eicker 8 Higher Geodesy: Physical Geodesy  Example: constant acceleration Example Constant acceleration: r=g  z Velocity: integration r r (t= ) ∫ r dt= gt + r0 integration Position: r= (t ) ∫ r = dt ∫ gt + r 0 g x 1 2 = gt + r0t + r0 2 ⇒ motion is a parabola ⇒ uniquely defined apart from 6 constants of integration Annette Eicker 9 Higher Geodesy: Physical Geodesy Philosophiae Naturalis Principia Mathematica Isaac Newton (1643-1723) Annette Eicker 10 Higher Geodesy: Physical Geodesy  Newton‘s Laws of Motion First law: Law of inertia „In an inertial reference frame, an object either dr remains at rest or continues to move at a = r= const constant velocity, unless acted upon by a force.“ dt Force-free motion. Coordinate system moving with constant velocity along a straight line: Defines the reference frames in which the laws of motion are valid: z => All inertial reference frames are in r = const a state of constant, rectilinear motion with respect to one another Our Earth-fixed coordinate system is not an y inertial system! It moves along the sun‘s orbit in a „circle“ and the Earth rotates! x (We will come back to this later.) Annette Eicker 11 Higher Geodesy: Physical Geodesy  Newton‘s Laws of Motion First law: Law of inertia „In an inertial reference frame, an object either dr remains at rest or continues to move at a = r= const constant velocity, unless acted upon by a force.“ dt Conservation of momentum: F = 0 ⇒ p = 0 Second law dp „The acceleration of an object as produced by a net = p = F force is directly proportional to the magnitude of the dt net force and in the same direction as the net force.“ with momentum p = mr d d Equation of motion: inserted: F = p = p = (mr ) for constant mass dt dt F = mr = mr + m r m = 0 with m: inertial mass „force = mass x acceleration“ Annette Eicker 12 Higher Geodesy: Physical Geodesy  Newton‘s Laws of Motion First law: Law of inertia „In an inertial reference frame, an object either dr remains at rest or continues to move at a = r= const constant velocity, unless acted upon by a force.“ dt Second law dp „The acceleration of an object as produced by a net = p = F force is directly proportional to the magnitude of the dt net force and in the same direction as the net force.“ with momentum p = mr Third law: Action-Reaction Law “When one body exerts a force on a second body, the second body simultaneously exerts a force Factio = −Freactio equal in magnitude and opposite in direction on the first body.” Annette Eicker 13 Higher Geodesy: Physical Geodesy  Law of Gravitation Positions: (Both bodies are „point masses“.)  x1   x2      r1 =  y1  r2 =  y2  z  z  F12  1  2 m2 Distance: r2 − r1 = ( x2 − x1 ) 2 + ( y2 − y1 ) 2 + ( z 2 − z1 ) 2 r2 z m1 r1 y x Annette Eicker 14 Higher Geodesy: Physical Geodesy  Law of Gravitation Gravitational force is proportional (Both bodies are „point masses“.) - to the inertial mass of the two bodies ~ m1 , ~ m2 - to the square of the reciprocal distance m2 F12 ~ 1 l2 with l = r2 − r1 - acts along the line of sight r2 − r1 − e12 = − r2 r2 − r1 (unit vector) z m1 Force of gravitation: e12 r1 l2 r2 − r1 F12 = −Gm1m2 3 r2 − r1 y with the gravity constant m3 x G = (6672 ± 4 ) ⋅10 -14 s 2 kg Annette Eicker 15 Higher Geodesy: Physical Geodesy  Law of Gravitation Law of gravitation (Both bodies are „point masses“.) r2 − r1  m F12 = −Gm1m2 N = kg  s 2  3 r2 − r1 m2 F12 Law of Motion m2r2 = F Setting them equal: r2 r2 − r1  m z m1 m2r2 = −Gm1m2  N = kg s 2  r2 − r1 3 r1 gravitational acceleration y x Annette Eicker 16 Higher Geodesy: Physical Geodesy  Law of Gravitation Law of gravitation The(Both gravitational bodies areacceleration forms a „point masses“.) r2 − r1 vector field  m F12 = −Gm1m2 N = kg  s 2  3 r2 − r1 m2 F12 Law of Motion m2r2 = F Setting them equal: r2 r2 − r1 m z m1 r2 = −Gm1 r2 − r1 3  s 2  r1 gravitational acceleration y x Annette Eicker 17 Higher Geodesy: Physical Geodesy  Excursion: Vector field Acceleration  rx (r )  r2 − r1   r = −Gm1  3 =  ry (r )  r2 − r1  r (r )   z  A vector field is a function, which, assigns a vector to every point in space A scalar field is a function, which assigns a number to every point in space  rx   rx      r(1,4,0) =  ry  r(2, 4, 0) =  ry   r   r   z  z Annette Eicker 18 Higher Geodesy: Physical Geodesy  Excursion: Vector field Acceleration  rx (r )  r2 − r1   r = −Gm1  3 =  ry (r )  r2 − r1  r (r )   z  A vector field is a function, which, assigns a vector to every point in space A scalar field is a function, which assigns a number to every point in space Annette Eicker 19 Higher Geodesy: Physical Geodesy  Law of Gravitation Law of gravitation (Both bodies are „point masses“.) r2 − r1  m F12 = −Gm1m2 N = kg  s 2  3 r2 − r1 m2 K12 Law of Motion m2r2 = F Setting them equal: r2 r2 − r1 m z m1 g = r2 = −Gm1 r2 − r1 3  s 2  r1 gravitational acceleration Now: y mass m1 = M => mass of the Earth mass m2 = m => mass of, e.g. a satellite x Annette Eicker 20 Higher Geodesy: Physical Geodesy  Law of Gravitation Law of gravitation m r2 − r1  m F12 = −Gm1m2 N = kg  s 2  3 r2 − r1 Law of Motion r2 m2r2 = F M r1 Setting them equal: z r2 − r1 m g = r2 = −Gm1 3  s 2  r2 − r1 gravitational acceleration y Place the origin of the coordinate system at the center of the Earth : r1 = 0 x Annette Eicker 21 Higher Geodesy: Physical Geodesy  Law of Gravitation Law of gravitation m z r2 − r1  m F12 = −Gm1m2 N = kg r2 − r1 3  s 2  r Law of Motion r =r y m2r2 = F M x Setting them equal: r2 − r1 m g = r2 = −Gm1 3  s 2  r2 − r1 gravitational acceleration r Equation of r = −GM 3 motion r Place the origin of the coordinate system at the center of the Earth : r1 = 0 => defines the (simplified) motion of a satellite Annette Eicker 22 Higher Geodesy: Physical Geodesy  Law of Gravitation Simplifications so far: m z Earth assumed to be a point mass No other planets or other forces r Reality: r =r y Earth has a non homogeneous M density distribution, changing with space and time x Earth is non-uniformly rotating other planets have to be considered … r = f (t , r, r ) r Equation of r = −GM 3 motion Idea: If mass distribution is known for a r specific time, gravitational acceleration can be computed as sum of tiny point => defines the (simplified) motion of a satellite masses (=> next lecture) Annette Eicker 23 Higher Geodesy: Physical Geodesy  Questions What do Newton’s three “Laws of Motion” describe? What is a (linear) momentum and how does it relate to a force? What does the “Equation of Motion” describe? (=> F = mr) What does the gravitational force depend on? What does the “Law of Gravitation” say? What is an inertial reference frame? What is a vector field / scalar field? Annette Eicker 24 Higher Geodesy: Physical Geodesy