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# Week 1: Introduction to Orbital Mechanics 230

UVM

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This 4 page Class Notes was uploaded by Adrian Notetaker on Thursday September 8, 2016. The Class Notes belongs to 230 at University of Vermont taught by Darren L. Hitt in Fall 2016. Since its upload, it has received 13 views. For similar materials see Orbital Mechanics in Mechanical Engr Department at University of Vermont.

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Date Created: 09/08/16

ME 230 Orbital Mechanics Adrián Lasa Week 1: Introduction to Orbital Mechanics 1. Summary 1.1. Newtonian potential The Newtonian potential (Φ) is a magnitude that measures the intensity of gravitational field in a concrete point A. IMPORTANT: It isn’t a vector. | | ⃗ Where is the gravitational constant. is the mass of the body who produces the gravitational force. is the distance between the point mass and a particular point. The negative value is taken by convection. Gravitational potential vs radio 5E+11 0 ???????? 0 2000 4000 6000 8000 ???? ???????? -5E+11 Δ???? -1E+12 ???? ???????? Graph: The red point indicated the gravitational potential in the Earth’sEsurface (R ). The variation of potential energy (ΔΦ) is equal to the work done between the two orange points. 1.2. Gravitational force The gravitational force is defined like: | | |⃗ | Where is the mass of the body 2. 1 ME 230 Orbital Mechanics Adrián Lasa Week 1: Introduction to Orbital Mechanics ⃗ is the distance between the two bodies (they must be taken that points mass). In order to use the vectorial shape we can use the two next expressions. First expression: We can use a unit vector ⃗⃗ . ⃗ ⃗⃗ |⃗ | Second expression: We use a unit vector, but this time it is represented such a coefficient of the vector ⃗ and its magnitude. ⃗ ⃗⃗⃗ ⃗⃗ |⃗ | |⃗ | |⃗⃗ | Both expressions are the same. The difference is that we don’t need to calculate the unit vector, ⃗⃗ , in the second because this calculus are incorporated in the equation (less probability to have a mistake). 1.3. Simplification of point mass First, if we want to consider a body like a point mass, the most important feature is the scale. The next two examples show that. D S-E h dE ???????? 2 ???????? ≈ ≈ 5???????? ???? 2 ???????? ℎ ???????? ???? ≈ 8 ≈ 4 ???????? ???? ???? ???? ???????? We can suppose the Earth a point of mass We can assume the Earth a point mass because of huge difference in the distance because the difference in the distance between the bodies and the Earth’s side. between the bodies and Earth’s side is small. In short, the smaller the side of the body is (what produce the gravity) compared with the distance between bodies, the more accurate is this simplification. However, if the body has a homogeneous density (it is composed for the same material) and a spherical shape, the body is equivalent to point mass at its center of gravity with the total mass of the body ( ). d ∫ ρ 2 ME 230 Orbital Mechanics Adrián Lasa Week 1: Introduction to Orbital Mechanics In the reality, the space bodies aren’t spherical due to rotation and they haven’t uniform density. 1.3.1. Principle of superposition We want to know the gravitational potential in a particular point A. We divide the body in infinitesimal elements. Each element produces a gravitational potential in P equal to: |⃗| ???????? ???? ???????? |⃗| So, the total gravitational potential is: ∫ ∫ |⃗| If we have a body which is equivalent to point mass, we have the next equivalence using the principle of superposition, is that, the sum of all the infinitesimal contribution are equal to the contribution of a point mass. ???? ????⃗ ⃗ ???? ∑ |⃗| (⃗ | |⃗| |⃗|) |⃗| ⃗ ???? 2. Introduction to N-body problem In order to obtain the force produce in a body due to other, we need to use the principle of superstition to the gravitational forces. The truth problem is calculated the trajectory of this body, but we will only study the two body problem. ???? 4 ???? ????⃗ ⃗ 4 ???? ⃗ ⃗ ⃗ ⃗ ???? ???? ???? ???? ???? ???? ???? ???? ???? 3 ???? 4 ???? |⃗ | ⃗ ???? |⃗ |3 ⃗3 ???? ????⃗ |3 ????⃗3 ⃗ 3 3 3 ???? 3 Where ⃗ is the force in the body i. ⃗ is the force in the body I due to the body j. 3 ME 230 Orbital Mechanics Adrián Lasa Week 1: Introduction to Orbital Mechanics 3. Notes The moon modifies rotation of Earth because of its progressive orbit (same rotation than the Earth) Earth’s rotation decreases because the generated equilibrium force. Geostationary orbit: it’s an orbit with the same period of rotation than the Earth. It’s the perfect orbit to satellites. They seem immobile from the Earth’s surface, is that, their latitude is constant and they only change the longitude. Asteroids haven’t enough mass, so the generated force in the rotation is too small to become spherical asteroids. 4

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