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2 Physics This document contains brief tutorials on some physical and mathematical concepts you may find helpful in understanding the AMS. A more general and complete source of information on these topics can be found in any elementary physics text. Topics include: Vectors Newton's Law Orbital Parameters Perilune/Apolune Adjustments Plane Change Maneuvers Vectors A vector is a shorthand way to represent a quantity that ┌──────────────────┐ has both a magnitude and a direction, e.g., a velocity of │VECTOR NOTATION: │ 5 m/s toward the north. A vector is written as a symbol │ v <=> {i,j,k} │ with a "~" underneath and corresponds to an ordered │ ~ │ triplet of real numbers, each of which gives the magnitude │ w <=> {x,y,z} │ of the vector in one of three orthogonal directions, │ ~ │ e.g., {north,east,up}. └──────────────────┘ Operations involving vectors include addition and ┌─────────────────────────┐ subtraction, multiplication by a scalar, the dot │VECTOR ADDITION │ product, and the cross product. │ /SUBTRACTION: │ │ v ± w = {i±x,j±y,k±z} │ Addition/Subtraction: The sum (difference) of two │ ~ ~ │ vectors is the sum (difference) of their └─────────────────────────┘ components. ┌───────────────────────┐ │SCALAR MULTIPLICATION: │ Scalar Multiplication: Multiplication of a vector by │ b·v = {bi,bj,bk} │ a scalar changes its magnitude by multiplying each of │ ~ │ its components. └───────────────────────┘ Vectors (continued) Dot Product: The dot (or scalar) product is found by ┌──────────────────────┐ summing the products of corresponding components. The │DOT (SCALAR) PRODUCT: │ result can also be written as the product of the │ v · w = ix+jy+kz │ magnitudes of the two vectors times the cosine of the │ ~ ~ = v·w·cos(Θ) │ angle between them. Two vectors at right angles to └──────────────────────┘ one another have a dot product of zero, since ┌──────────────────────┐ cos(90°) = 0. The magnitude of a vector is equal to │VECTOR MAGNITUDE: │ the square root of the dot product of a vector with │ v = sqrt(v·v) │ itself. A unit vector has a magnitude of one. Body │ ~ ~ │ axes are sets of three orthogonal unit vectors │ = sqrt(i²+j²+k²) │ {Face,Left,Up} used to define the orientation of the └──────────────────────┘ LOL, LM, PLSS, and LRV. The LOL Face axis points north, the ┌─────────────┐ Left axis points toward the Earth, and the Up axis is │UNIT VECTOR: │ perpendicular to the other two. Body axes for the LM, PLSS, │ u · u = 1 │ and LRV are referenced to the Pilot. The LM main engines are │ ~ ~ │ located below the Pilot in the LM so that, when ignited, the LM └─────────────┘ is accelerated in the direction of the Pilot's head. Vectors (continued) Cross Product: The cross product of two ┌───────────────────────────────┐ vectors is another vector perpendicular to │CROSS PRODUCT: │ the plane formed by the two vectors. The │ v x w = {jz-ky,kx-iz,iy-jx} │ result can also be written as the product of │ ~ ~ = v·w·sin(Θ) u │ the magnitudes of the two vectors times the │ ~ │ sine of the angle between them. The └───────────────────────────────┘ direction of the resulting vector can be established using a right-hand rule. The orthogonality of the cross product to one of its constituents is demonstrated as an example of vector manipulation: v · ( v x w ) = {i,j,k} · {jz-ky,kx-iz,iy-jx} ~ ~ ~ = ijz - iyk + xjk - ijz + iyk - xjk = 0 . Newton's Law Newton's Law states that the acceleration of an object "a" is ┌──────────────┐ proportional to the applied force "F" and inversely │NEWTON'S LAW: │ proportional to its mass "m". Forces acting on the LM include │ a = F / m │ gravity, main engines, and RCS thrusters. For example, a │ ~ ~ │ fully loaded LM has a mass of 17248 kg and the RCS thrusters └──────────────┘ at medium throttle have a thrust of 197 nt. The resulting acceleration is 0.0114 m/s². The acceleration due to gravity is independent of the ┌──────────────────┐ mass of the object. It is directed toward the center of │GRAVITATION: │ the Moon and inversely proportional to the square of the │ g = - G r / r^3 │ distance to the center of the Moon. The magnitude of the │ ~ ~ │ gravitational acceleration at the surface of the Moon is │ G = 4.9075x10^12 │ given by └──────────────────┘ g = - G / R² = - (4.9075x10^12) / (1738300)² = -1.624 m/s² . where "R" is the Lunar radius and the symbol "^" means "raised to the power". Orbital Parameters Important orbital parameters include angular momentum, energy, perilune, apolune, and orbit period. Angular Momentum: The angular momentum "L" is a constant of motion, given by L / m = v x r , ~ ~ ~ where "m" is the mass, "v" is the velocity, and "r" is the position relative to the Moon of the spacecraft. The direction of the angular momentum vector can be displayed on the INS display. Energy: The total energy of a spacecraft "E", another constant of motion, is the sum of kinetic and potential energies: E / m = v² / 2 - G / r . If the total energy is less than zero, the spacecraft is in a closed, elliptical orbit. If the energy is greater than zero, the spacecraft is in a hyperbolic orbit and will eventually escape the Moon. If the energy equals zero, the spacecraft is in a parabolic escape trajectory. Orbital Parameters (continued) Perilune/Apolune: Assuming that the spacecraft is in a closed elliptical orbit, the eccentricity and semimajor axis of the ellipse are e = sqrt [ 1 + 2 (E/m) (L/m)² / G² ] and s = - G / (2E/m) , respectively. The perilune and apolune are given by { perilune / apolune } = s ( 1 ± e ) - R , where "+" gives the apolune and "-" gives the perilune. Period: The orbital period is found from T = sqrt [ 4 π² s^3 / G ] . Orbital Parameters (continued) Circular Orbits: For the case of a circular orbit, the magnitude of the velocity is related to the distance to the center of the Moon. In order to maintain constant altitude, the gravitational force must exactly balance the centrifugal acceleration, i.e., G / s² = v² / s => v = sqrt [ G / s ] . For example, a 100 km circular orbit implies a velocity magnitude of 1633.9 m/s. The orbital period is 7069 s for the same orbit. Perilune/Apolune Adjustments The velocity change required to raise or lower the perilune or apolune can be approximated using the Orbital Parameters discussion. Using the equation s = - G / (2E/m) = R + P/2 + A/2 , where "P" is the perilune and "A" is the apolune, perilune and apolune changes can be found by differentiation: ds 1 dP 1 dA G d 1 G d(E/m) Gv ── = ─ ── = ─ ── = - ─ ── (───) = ─────── ────── = ────── . dv 2 dv 2 dv 2 dv E/m (2E/m)² dv (2E/m)² Therefore, dP dA Gv ── = ── = ───── . dv dv (E/m)² For example, at DOI in a 100 km circular orbit, v = 1633.9 m/s² and E/m = -1334793 (m/s)² so that dP/dv = 4500 s. A 1 m/s velocity change results in a 4.5 km change in perilune. An 80 km change requires about 18 m/s. Plane Change Maneuvers The velocity change required to modify the orbital plane can also be approximated using the Orbital Parameters discussion. The magnitude of the angular momentum for a spacecraft in a circular orbit is L / m = v r = sqrt [ G r ] . The torque "N" applied by a main engine or RCS burn with thrust "T" at right angles to the orbital plane adds a component perpendicular to the angular momentum vector with magnitude d(L/m) ────── = (N/m) = (T/m) r = a r , dt where "a" is the acceleration of the spacecraft due to the applied thrust. Both sides can be divided by the acceleration, and the expression "a dt" can be replaced by "dv" so that d(L/m) ────── = r . dv Plane Change Maneuvers (continued) The angular change (in radians) can be approximated by dividing both sides by (L/m) = sqrt[Gr], dΦ 1 ── = sqrt [ r / G ] = ─ . dv v where "v" is the circular orbit velocity magnitude. For example, a 1 m/s velocity change in a 100 km orbit results in an angular change of 0.612 milliradians or 0.035°. A 1° angle change requires a velocity change of about 29 m/s.