Orbit calculations in the solar system (journey to Mars)

In summary, the conversation discusses difficulty in understanding a solution for a problem involving Keplerian orbits. The solution provided is described as terse and skipping steps, and the name of the transfer involved is mentioned as "Hohmann-Transfer Orbit". The individual requests a more detailed explanation and acknowledges that they may be violating any rules. They are advised to attempt solving the problem themselves, and are given a suggestion to write down what they know about Keplerian orbits.
  • #1
theanswerai
4
1
Homework Statement
The ship starts out in a circular orbit around the sun very near the Earth and has a goal of
moving to a circular orbit around the Sun that is very close to Mars. It will make this transfer
in an elliptical orbit as shown in bold in the diagram below. This is accomplished with an
initial velocity boost near the Earth ∆v1 and then a second velocity boost near Mars ∆v2.
Assume that both of these boosts are from instantaneous impulses, and ignore mass changes
in the rocket as well as gravitational attraction to either Earth or Mars. Don’t ignore the
Sun! Assume that the Earth and Mars are both in circular orbits around the Sun of radii RE
and RM = RE/α respectively. The orbital speeds are vE and vM respectively.
i. Derive an expression for the velocity boost ∆v1 to change the orbit from circular to
elliptical. Express your answer in terms of vE and α.
ii. Derive an expression for the velocity boost ∆v2 to change the orbit from elliptical to
circular. Express your answer in terms of vE and α.
iii. What is the angular separation between Earth and Mars, as measured from the Sun, at
the time of launch so that the rocket will start from Earth and arrive at Mars when it
reaches the orbit of Mars? Express your answer in terms of α.
Relevant Equations
centripetal forces, angular momentum, Kepler's 3rd Law
I cannot understand the solution at https://www.aapt.org/physicsteam/2015/upload/E3-2-5-solutions.pdf, because the solution is terse and skip steps (at least i think so). I figured out that the name of this transfer is "Hohmann-Transfer Orbit". A detailed walkthrough would be appreciated. If I am violating against any rules please comment, as this is my first post :-).
 
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  • #2
We cannot simply give you the answer. You need to make an attempt at solving the problem. Try writing down what you know about Keplerian orbits. For example, what would be the velocity of the ship at the beginning and at the end?
 

1. How do you calculate the orbit of a spacecraft traveling to Mars?

To calculate the orbit of a spacecraft traveling to Mars, scientists use a combination of Kepler's laws of planetary motion and Newton's laws of motion. They also take into account the gravitational pull of other celestial bodies in the solar system, such as Earth and the Sun. This allows them to determine the trajectory and timing of the spacecraft's journey to Mars.

2. What factors influence the duration of a journey to Mars?

The duration of a journey to Mars is influenced by several factors, including the distance between Earth and Mars at the time of launch, the speed of the spacecraft, and the alignment of the two planets. Additionally, the type of spacecraft propulsion system and the chosen flight path can also impact the duration of the journey.

3. How do scientists account for the varying orbits of Earth and Mars?

Earth and Mars have elliptical orbits around the sun, which means their distance from each other can vary greatly. To account for this, scientists use mathematical models to predict the positions of the two planets at any given time. They also use precise measurements and observations to refine these calculations and ensure the spacecraft will arrive at the desired location.

4. Can a spacecraft's orbit be adjusted during the journey to Mars?

Yes, a spacecraft's orbit can be adjusted during the journey to Mars. This is known as a mid-course correction and is often necessary to ensure the spacecraft stays on the correct trajectory and arrives at Mars at the desired time. Scientists use data from the spacecraft's sensors and communication with Earth to make these adjustments.

5. How do scientists ensure the safety of a spacecraft during its journey to Mars?

Scientists take several precautions to ensure the safety of a spacecraft during its journey to Mars. This includes thorough testing and simulations before launch, redundant systems to prevent malfunctions, and constant monitoring and communication with the spacecraft during its journey. Scientists also carefully plan the trajectory to avoid potential hazards, such as collisions with other objects in space.

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