Problem 4 Estimate the Av requirement for a mission to Venus, departing from a 400km (altitude) circular orbit around the Earth, and arriving into a 300x10,000 km (altitude) capture orbit around Venus Problem 5 The Erven mothership is currently on a safe orbit around the Sun with Keplerian elements: [a, e, i, 0, w, 0, EA, M] = [1.36034, 0.36276, 2.16893, 228.65885, 282.57331, 3.14159, 3.14159, 3.14159), where a is in AU, i, 2, and w, are in degrees and the anomalies are in radians. Since the mothership will require years of repair until it is able to maneuver freely, the Erven's prepare their scoutships for use. A likely looking planet, third from the Sun is identified as a target. Launching from the Erven's decending node, identify: a) The Av needed to correct the inclination to zero. b) After correcting their inclination, the Erven scouts drift to perihelion where they adjust their aphelion to match the target orbit (1 AU). What is the Avrequired.4. Step 1: Launch from Earth into a Transfer Orbit (AV,) We'll calculate the velocity required to transfer from Earth to Venus using a Hohmann Transfer Orbit. Parameters: . Radius of Earth orbit E = 6778 km (from Earth's radius and 400 km altitude). . Radius of Venus capture orbit v = 3, 006, 052 km (from Venus's radius and the 3,000,000 km altitude). . Semi-major axis of the transfer orbit a: TETTV 6778 + 3, 006, 052 a = = 1, 506, 415 km 2 2Use the Vis-Viva Equation: The vis-viva equation helps calculate orbital velocities at any point along the orbit: V = where / is the gravitational parameter of the Sun Mo = 1.327 x 10 l km3/s2. Calculate velocity at Earth's orbit (periapsis): 2 NE = 1 1.327 x 1011 ~ 42.1 km/s 6778 1, 506, 415Calculate velocity at Earth's orbit (periapsis): 2 VE = 1 1.327 x 1011 ~ 42.1 km/s 6778 1, 506, 415, Calculate Earth's orbital velocity in a circular orbit: For comparison, the velocity in a circular orbit around the Sun at Earth's distance (1 AU) is: 1.327 x 1011 Earth orbit ~29.8 km/s TE 6778Av, for Earth departure: The spacecraft must increase its velocity to enter the transfer orbit: Av1 = VE - Earth orbit = 42.1 km/s - 29.8 km/s = 12.3 km/s Step 2: Capture at Venus (AV2) Upon arriving at Venus, we need to slow down to be captured into the desired orbit. Calculate velocity at Venus orbit (apoapsis): Using the vis-viva equation at Venus's capture orbit: 2 1 Uv = 1/1.327 x 1011 ~ 11.7km/s 3, 006, 052 1, 506, 415Velocity for a circular orbit at 3,006,052 km around Venus: To be captured into a circular orbit around Venus, the spacecraft must slow down to match the circular orbital velocity: UV Ucapture= rv where My is Venus's gravitational parameter My = 3.248 x 10' km3 / $2. Let's compute that: 3.248 x 105 capture = ~ 0.33 km/s 3, 006, 052AV2 for Venus capture: To enter the circular orbit, the spacecraft must reduce its velocity: Av2 = UV - Vcapture = 11.7km/s - 0.33 km/s = 11.37km/s Total Av for the Mission: Now, sum up the Av values for departure and capture: Avtotal = Avi + Av2 = 12.3km/s + 11.37km/s = 23.67 km/s Final Answer for Problem 4: The total Av requirement for the mission to Venus is approximately 23.67 km/s.Problem 5: Erven Mothership's Orbital Maneuver We are given the following parameters for the Erven mothership's orbit around the Sun and asked to calculate the required Av for two parts: Part (a): Correct the inclination to zero. The orbital elements provided are: Semi-major axis a = 1.36034 AU Eccentricity e = 0.36276 Inclination 7 = 2.16893 + Other elements: (2, w, and anomalies (all expressed in degrees or radians). We need to determine the Av required to reduce the inclination to zero. Key concept: The Av required to change inclination is given by the following equation: A Av = 2vsin (71) where: - v is the orbital velocity at the point where the maneuver is performed (usually done at the node, where the plane changes). A7 is the change in inclination (in this case, 2.16893). Step 1: Calculate orbital velocity at the node We'll use the vis-viva equation again to calculate the velocity v at the node, using the semi-major axis a and eccentricity e. Convert the semi-major axis from AU to kilometers (since 1 AU = 149,597,870.7 km): a = 1.36034 AU = 1.36034 x 149, 597, 870.7 km = 203, 438, 000 km The velocity at the node is calculated using the vis-viva equation: 0 = VM -! ) where Mo = 1.327 x 10ll km3 /s2.Since we are at the node, ~ a, so: 1.327 x 10! Y= \\/ 203, 438, 000 s Step 2: Calculate Av to correct the inclination We know the change in inclination is Ai = 2.16893, so: 2.16893 =) Convert 2.16893 to radians: 2.16893 = 0.03785 radians Av:2>