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Astrodynamics
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| Question | Answer |
|---|---|
| (AxB)xC = B(A*C) - C(A*B) | False. The order of the parenthesis matters. Ax(BxC) = B(A*C)-C(A*B) |
| When propagating orbits using the f&g series (or functions), the initial states r0 and v0 are not required to be orthogonal vectors. | True. They do not need to be orthagonal. |
| A particle travelling along a straight line may have non-zero angular momentum. | True. If the particle does not pass through the reference pt then L = r xmv is not equal to zero. Otherwise, r and v are parallel. Think of a reference pt off a train track and the r and v. Position is determined based on the reference point. |
| r0 and v0 provide the necessary constants to solve the 2-body motion problem. | True 6 constants are needed. |
| How to find the inverse of a dcm? | (C_AB)^-1 = (C_AB)^T = C_BA |
| How to find the negative rotation of a dcm? | Can make the angle negative which will change the sign of sine since cosine doesn't matter if angle is negative or positive. |
| DCM: C1 | Rotation about x. [1 0 0; 0 C S; 0 - S C] |
| DCM: C2 | Rotation about y. [C 0 -S; 0 1 0; S 0 C] |
| DCM: C3 | Rotation about z. [C S 0; -S C 0; 0 0 1] |
| C_BA | Denotes the transformation from basis A to basis B. Read from right to left. |
| A^( ) | Denotes a vector quantity coordinatized in basis A. |
| Orthanormal vs orthagonal? | Orthonormal means that they are unit vectors that are orthagonal. |
| Coriolis effect and Coriolis velocity. | The Coriolis effect is an apparent deflection of a moving particle when it is observed in a rotating frame of reference. V = 2(w x v_relative). In his notes its V |
| Centrifugal velocity | w x (w x r). When you are rotating you want to move away from the center in a rotating frame. Think about turning in a car. |
| Tangential Velocity | Velocity tangent to the rotation. (w_dot x r) = 0 since the angular velocity is constant. |
| Relative Velocity | Relative velocity is v double dot. It is the velocity of one object as observed from the frame of reference of another object. |
| r1 * r2 x r3 = 0 is a requirement of the Gibbs orbit determination method. | True. The three position vectors needs to be coplanar to then be simplified to writing one in terms of two so we only need 6 constants. |
| Kepler's eqn provides a relationship btw eccentric and true anomalies. | False. Kepler's equn provides a relationship btw eccentric anamoly and time. |
| The areal velocity is constant for all orbits. | True. h = 2dA/dt = r^2 df/dt = sqrt(P*mu) = sqrt(mu *a(1-e^2)) |
| Kepler's 2nd law states that the motion is described by conic section. | False. Kepler's 2nd law states that an object orbits equal areas in equal times. dA/dt = constant. |
| What are Kepler's three laws? | 1. Law of orbits. Orbits are ellipses, Sun at a focus. r = (a(1-e^2))/(1+ecos(f)) = P/(1 + ecos(f)) = a(1-ecosE). 2. Law of areas. dA/dt = const 3. Law of Periods. P^2 = a^3 |
| All DCM columns are orthonormal . T/F | True |
| If a * b x c = 0, then constants, alpha, beta, and chi exists such that alpha (a) + beta (b) +chi (c) = 0. | True |
| In general, r dot = v which is not equal to the magnitude of v vector. | True |
| r * v < 0 implies a negative flight path angle, phi i.e. phi < 0. | True |
| The angle between the velocity vector and the local horizonal direction is referred to as the flight path angle and can be either positive or negative. | True. Top it is positive, bottom it is negative. |
| Apoapsis plane changes require less delta v than periapsis plane changes. | True. A plane change maneuver requires the change of the inclination /orientation of an orbit. DV = 2v_f sin(i/2). Orbital velocity is lowest at apoapsis which requires a lower DV. |
| The obliquity of the ecliptic is the angle btw the equatorial and ecliptic planes. | True |
| All two impulse coplanar cotangential transfers are Hohmann transfers. | False. Other types of cotangential transfers exist where the orbits can be circular or elliptical as long as they are on the same plane. Hohmann require burns at peri/apo but other transfers can happen at different points. Bi-elliptic, bi-parabolic. |
| The vector property |a*b| < ab is known as the triangle inequality. T/F | False. This is the schwartz inequality. The triangle inequality is c = a + b. c < a+b. |
| At the apoapsis ( or periapsis) r * v = 0, thus the flight path angle is zero. | True |
| The equatorial plane is the fundamental plane for heliocentric orbits. | False. It is the ecliptic plane. Equatorial is true for geocentric orbits. |
| Definition of a coordinate system requires the following: a fundamental plane and a principal direction in the plane. | True. |
| An analytic solution can be found for the restricted 3 - body problem. T/F | False. While we find the lagrangian points for stability, we are not finding analytic solutions to this problem. The body problem does have analytic solutions. a = -mu/r^3 r_vec |
| The time between successive conjunctions of the Earth-Venus system is known as syzygy. | False. The time between successive conjunctions is called the synodic period |
| Lagrangian points are equilibrium points for the restricted 3-body problem. | True. |
| A successful Hohmann transfer from an inner planet to an outer planet requires that the inner planet is behind the outer planet at the initiation of the maneuver. | True |
| For a given DV, a maneuver at the apoapsis yield the largest plane change. | True. |
| Dynamical systems with only gyroscopic damping can be asymptotically stable. | False, because gyroscopic damping alone cannot provide asymptotic stability to a dynamical system. A system is asymptotically stable if it remains bounded and eventually returns to equilibrium. The gyroscopic terms transfer energy from one state to anothe |
| Motion in the z-direction about a lagrangian point is always unstable. | False. L4 and L5 are stable in all directions, as long as the mass requirement is satisfied. The rectilinear points are always unstable including the z-direction. |
| Jacobi's constant is an 11th constant of motion for the 3-body problem. | False, we only need 6 constants and can have at most 10 constants. |
| An orbital maneuver that affects only the inclination of the orbit must be performed at the intersection of the line of nodes and the orbit. | True. Lines of nodes is the intersections line between the orbital plane and a reference plane. Defines ascending node and descending node. The change must be applied at a point on the line of nodes so that the orbital elements are not altered. |
| A positive tangential DV applied at the periapsis raises the apoapsis altitude. | True since increasing the velocity at the periapsis makes the elliptic orbit get larger. |
| The L4 Lagrangian point is an equilibrium pt of the restricted 3-body system and as the system rotates it is located at a point that is behind the larger mass. | True. |
| The following planetary alignment is an example of a superior opposition: Earth-Sun-Mars. | False, there is no such thing as superior opposition. This would be superior conjunction. |
| During a planetary flyby vectors v- = v+. | False, the speeds are the same but the vectors change. r- = r+. |
| Plane changes performed at the apoapsis require less DV than those performed at the periapsis. | True. Less speed correlates with a smaller DV. |
| probe at a lagrangian point has zero relative velocity, thus its inertial velocity is zero. | False |
| Escape from an elliptic orbit is most fuel efficient if performed at the apoapsis. | False, it is the most fuel at the periapsis because your speed is already high so to reach the escape velocity, the burn would be less than at periapsis. |
| f_infinity | true anamoly at infinity |
| v-_infinity vector | previous velocity at infinity |
| v_rel | relative speed not velocity since it is not a vector |
| E | Eccentric anamoly |
| Delta | Aim radius |
| mu_sun | Gravitation constant of the sun |
| h vector | angular momentum vector |
| ohm | Right ascension of the ascending node = acos(n_vec * i_hat/n) where n = k x h |
| omega | Argument of periapsis |
| e vector | eccentricity vector = (v x h)/ mu - r_vec/ r |
| r_p | position at periapsis = a(1-e) |
| epsilon | energy = -mu/2a = v^2/2 - mu/r |
| n | mean motion = sqrt(mu/a^3) |
| n_vector | nodal vector - intersection of planes = k x h |
| a | semi-major axis |
| b | semi-minor axis |
| f | true anamoly |
| M | mean anamoly. = n(t-tp) where t is the current time, tp is time of periapsis passage, n = mean motion = sqrt(mu/a^3) = 2pi/TP |
| phi | Flight path angle. r * v = cos (phi). Positive at top negative at bottom. |
| In general rdot = v = |v_vec| | False. |
| The rows of all DCMs are orthonormal | True. |
| Five orbital elements define the size, shape, and orientation of a Keplerian orbit. | True. |
| Using the f&g functions (or series) for orbit propagation regquires the r0 and v0 are orthogonal vectors. | False. |
| If a is a fixed length vector than a * a_dot =0 | True |
| If f <= pi, then E <= pi. | True, same half. |
| The eccentricity vector and the periapsis position vector are collinear. | True. |
| Orbit propagation using the f&g functions does not require solution of Kepler's equation. | False you have to find n and the change in time. |
| C | Capital C is center of an ellipse. |
| The reference plane for geocentric orbits is the ecliptic plane. | False. It is the equatorial plane. |
| The areal velocity is constant only for close orbits. | False. It is constant for all orbits. |
| The time between successive conjuctions of the Earth-Venus system is known as syzygy. | False. You can have inferior and superior conjunction. Syzygy is only when their relative alignment in the orbit is the same. |
| A darkside planetary flyby always results in a positive deltaV. | False. This does not indicate if you are going behind or in front of the planet. |
| A planetary flyby after a heliocentric Hohmann transfer from an inner planet to an outer planet results in a decrease in the heliocentric speed. | False. Your speed increases because you pass behind the planet. |
| For a heliocentric Hohmann transfer between planets, the hyperbolic excess velocity at the planet's sphere of influence is collinear with the planet's velocity. | True. |
| For all Keplerian orbits, the position and velocity vectors are coplanar. | True. Two body orbit, the motion is confined to a single plane. This results in the conservation of angular momentum since h = r x v. |
| For all orbits, the Laplace vector point in the direction of minimum radius. | True. Laplace vector and eccentricity vector point in the same direction. |
| Escape from a circular orbit requires that the speed be increased by sqrt(2) v_cir | True. If increased means times then this is true since v_esc = sqrt(2mu/r) |
| Gauss' equation provides a relationship between true and eccentric anomalies. | True. tan E/2 = ... |
| Kepler's first law is a statement of conservation of angular momentum. | False. His first law is that orbits are ellipses. His second law is related to angular momentum because equal areas are sweeped at equal intervals of time. |
| An example of syzygy for the Earth-Venus system is the time between successive superior conjunctions of the planets. | True. |
| A planetary fly-by that results in an increase of the heliocentric speed passes behind the planet | True. |
| The rectilinear Lagrangian points are always unstable. | True. |
| A spacecraft at the bottom of the lactus rectum has f = 3pi/2 | True. |
| The L2 point always lies between the two primary bodies and is always closer to the smaller of the two bodies. | False |
| The Laplace vector, eccentricity vector, and periapsis vector are collinear. | True. |
| The columns (or rows ) of a direction cosine matrix are orthonormal. | True. |
| With a second rotation angle of pi, asymmetric Euler sequences are singularities. *** | False. Symmetric sequences have singularities at pi, asymmetric have singularities at pi/2. |
| f_0 | Epoch position |
| Plane changes at the apoapsis are more fuel efficient than those performed at the periapsis. | True. |
| A planetary flyby after a heliocentric Hohmann transfer from an outer planet to an inner planet results in the decrease in the heliocentric speed. | True |
| Kepler's 3rd law states that the square of the orbital period is proportional cube of the mean distance. | True. Mean distance means semi-major axis since it is the average value of the distance over one orbital period. |
| Hohmann transfer orbits are minimum deltaV transfer orbits between coplanar circular orbits whose radii satisfy R2/R1 < 11.96. | True. |
| The Clohessy-Wiltshire (CW) equations are linearized equations that can be used for orbit determination. | False. They are used for relative motion analysis. If they were modified so it was v inertial instead of v rel then it would be true. |
| Name that assumptions of the n-body problem and their significance. | (1) Bodies are spherically homogenous. This means that the CM is at the geometric center. The bodies are reduced to points. (2) Only gravitational forces exist. Other non-gravitational influences are neglected. |
| Name what the 10 constants are composed of. | Linear momentum 6 Angular momentum 3 Energy 1 |
| Linear momentum from n-body | mRc = tc1 + c2 these are vectors so we have 6 constants. |
| Angular momentum from n-body | Hn = Cn these are vectors so we get 3 constants. |
| Energy equation from n-body | The energy constant is found by taking the dot product of r_dot into the ith EOM and summation of this. |
| What does the cross product of r and the EOM give? | The angular momentum is constant. |
| What does the dot product of rdot into the EOM give? | The energy is constant. |
| Name that assumptions of the 3-body problem and their significance. | 1. m1 & m2 >> m3. The CM lies along this line. 2. m1 and m2 move in circular orbits about their COM. This describes the motion and tells us there is an angular velocity. |
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ncam0720
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