105-2019201 Practice

How to break free from the Sun (The journey of Voyager 1 part 1)

Mariner-Jupiter-Saturn 1977 Spacecraft Artwork, 1975 NASA and JPL initially referred to what became the Voyagers as the Mariner-Jupiter-Saturn 1977 Project. The two Voyagers were advanced versions of the Mariner-class spacecraft that JPL had flown successfully to Venus, Mars, and Mercury. Shown here is a 1975 JPL artist's rendering of Voyager after encountering Jupiter and, after a gravity assist, approaching Saturn.

Question 1

Delta-v (literally "change in velocity"), symbolised as ∆v and pronounced delta-vee, as used in spacecraft flight dynamics, is a measure of the impulse per unit of spacecraft mass that is needed to perform a maneuver such as launch from, or landing on a planet or moon, or in-space orbital maneuver. It is a scalar that has the units of speed. As used in this context, it is not the same as the physical change in velocity of the vehicle.

When rocket thrust is applied in short bursts the other sources of acceleration may be negligible, and ∆v becomes

$\Delta V=\int _{t_{0}}^{t_{1}}\left|{\vec {a}}\right|dt$

As a simple example, take a conventional rocket which achieves thrust by burning fuel. Delta-v is the change in velocity that can be achieved by burning that rocket's entire fuel load.

(a)

Is it possible to calculate delta-v requirements of an orbital maneuver from conservation of energy by considering only the total energy of the vehicle in the initial and final orbits?

(b)

Orbit maneuvers are made by firing a thruster to produce a reaction force acting on the spacecraft. The size of this force will be

$F_{T}=v_{\text{exh}}\rho$

where $v_{\text{exh}}$ is the velocity of the exhaust gas in rocket frame and $\rho =-{\frac {dm}{dt}}$ is the propellant flow rate to the combustion chamber.

Assuming $v_{\text{exh}}$ to be a constant not depending on the amount of fuel left prove the ∆v is given by Tsiolkovsky rocket equation.

$\Delta v=v_{\text{exh}}\ln \left({\frac {m_{0}}{m_{1}}}\right)$

where $m_{0}$ is the initial mass of the spacecraft (before the thrust), and $m_{1}$ is the final mass of the spacecraft (after the thrust).

Question 2

What is the escape velocity from the solar system starting from earth?

Question 3

In orbital mechanics and aerospace engineering, a gravitational slingshot, gravity assist maneuver, or swing-by is the use of the relative movement (e.g. orbit around the Sun) and gravity of a planet or other astronomical object to alter the path and speed of a spacecraft, typically to save propellant and reduce expense.

Gravity assistance can be used to accelerate a spacecraft, that is, to increase or decrease its speed or redirect its path. The "assist" is provided by the motion of the gravitating body as it pulls on the spacecraft.[1] The gravity assist maneuver was first used in 1959 when the Soviet probe Luna 3 photographed the far side of Earth's Moon and it was used by interplanetary probes from Mariner 10 onwards, including the two Voyager probes' notable flybys of Jupiter and Saturn.

Let's take the planet's direction of motion as the x axis, and the perpendicular direction (in the orbital plane) as the y axis. The probe is initially moving with a speed v relative to the solar reference frame, in a direction approaching the oncoming planet at an angle θ. Two views of this are shown below, one with respect to the planet's rest frame, and the other with respect to the solar reference frame.

solutions