Which Solar System Body is Hardest to Reach and What Does It Take to Get There?

[elorc]

Homework Statement

(a) In terms of energy expenditure:
1. Which planet in the solar system is easiest to reach?
2. Which major body in the solar system is hardest to reach? Why, and what shape orbit would be needed to get there?
3. Which of the following scenarios requires more energy?
- Leaving Earth and landing on Mars, or
- Leaving Earth and doing a flyby of Jupiter

Homework Equations

N/A

The Attempt at a Solution

This is what I'm thinking...

(a) 1. Venus. Because it is the closest planet to Earth and is between Earth and the Sun, less velocity is required due to the GSI of the Sun pulling us in the proper direction. It also requires the fewest transit orbits.

2. This one tripped me up. He specifically words it as "major body" and gives a hint that the answer may not be on the list of planets he provided. Because of that, I'm thinking the Sun. I'm not entirely sure why, though. The only thing I can imagine is that it's because of how absolutely massive the Sun is, even though it's easier to move in that direction, it's a lot harder in terms of energy use to make sure you don't smack into it.

3. I imagine it's difficult to get to Jupiter in terms of energy use, but landing on a planet surely requires the most energy (again, so you don't crash into it or overshoot it) to make sure you are in the correct positions at the correct time, and to place the craft onto the surface. So I'm rolling with landing on Mars.


It's worth noting that this class doesn't get into any of the actual math (although I think it'd be easier for me if it did) behind these problems, or orbits in general, which I think is why I'm having a harder time with it.

 

[gneill]

Homework Statement

(a) In terms of energy expenditure:
1. Which planet in the solar system is easiest to reach?
2. Which major body in the solar system is hardest to reach? Why, and what shape orbit would be needed to get there?
3. Which of the following scenarios requires more energy?
- Leaving Earth and landing on Mars, or
- Leaving Earth and doing a flyby of Jupiter

Homework Equations

N/A

Surely the required $$\Delta$$V's in order to reach each object are important?

The Attempt at a Solution

This is what I'm thinking...

(a) 1. Venus. Because it is the closest planet to Earth and is between Earth and the Sun, less velocity is required due to the GSI of the Sun pulling us in the proper direction. It also requires the fewest transit orbits.

You should recheck your thinking here. There's no "easier going downhill" when it comes to moving around the solar system. It costs energy to make the required $$\Delta$$V's no matter which direction they take you.

2. This one tripped me up. He specifically words it as "major body" and gives a hint that the answer may not be on the list of planets he provided. Because of that, I'm thinking the Sun. I'm not entirely sure why, though. The only thing I can imagine is that it's because of how absolutely massive the Sun is, even though it's easier to move in that direction, it's a lot harder in terms of energy use to make sure you don't smack into it.

Again, what makes you think that it's easier to move sunward? It's all about the required $$\Delta$$V's, which determines the energy expenditures.

3. I imagine it's difficult to get to Jupiter in terms of energy use, but landing on a planet surely requires the most energy (again, so you don't crash into it or overshoot it) to make sure you are in the correct positions at the correct time, and to place the craft onto the surface. So I'm rolling with landing on Mars.


It's worth noting that this class doesn't get into any of the actual math (although I think it'd be easier for me if it did) behind these problems, or orbits in general, which I think is why I'm having a harder time with it.

That would make it difficult. Do you have any information from previous classes about the energy of orbits?

 

[elorc]

That's the problem, these Hohmann orbits are entirely new to me (and I'm embarrassed to admit that I'm having trouble with this because it seems as though it should be pretty simple). I don't know the math behind them and it's not presented in the class. The most I have is a table of the planets and their respective travel times by Hohmann orbit and Hohmann orbit speeds at 1 AU. This was provided as part of the class. I'm not sure how to incorporate that into figuring the questions out, though.

I think the point of the class is to deal more in terms of generalities (as it's an introductory course), but my mind has an easier time grasping this sort of material when I know the math that explains it. Complicating matters is that this is a 4-week online course between semesters, so there's no classroom/face time and progress is a bit rushed.

The chart I referenced is this (not sure how to format it best here):

Planet    TravelTime   Hohmann Orbit Speed @ 1AU
Mercury   105 days     22.3 km/s 
Venus     146 days     27.3 km/s 
Mars      259 days     32.8 km/s 
Jupiter   2.7 years    38.6 km/s 
Saturn    6.0 years    40.1 km/s 
Uranus    16.0 years   41.1 km/s 
Neptune   30.6 years   41.5 km/s 
Pluto     45.2 years   41.6 km/s

I find the material (and physics in general) to be immensely interesting but I'm just having a hard time working with this under the circumstances. I'm most likely missing something mind-blowingly simple. Any help/suggestions/guidance/clarification/whatever you can offer would be appreciated. :)

It goes beyond the scope of the class but I would like to know more about the calculations involved. What exactly is $$\Delta$$V representing? Is it just the change to current velocity required to carry out each transit orbit?

 

[gneill]

It goes beyond the scope of the class but I would like to know more about the calculations involved. What exactly is $$\Delta$$V representing? Is it just the change to current velocity required to carry out each transit orbit?

Yes, the $$\Delta$$V represents the required change in velocity in order to place a craft, currently traveling on a circular orbit at one distance from the Sun, onto the transfer orbit that will reach the desired orbit of the destination. When it reaches that distance there will be a another speed change required to match the circular orbit speed at that distance.

You've got a table of the Hohmann orbit speeds for various destinations as they would be departing from the vicinity of the Earth's orbit (1AU). Suppose your craft were already in a circular orbit at 1AU (same as the Earth is, more or less; the Earth's orbit is very slightly elliptical). It would already have some starting velocity corresponding to this initial orbit (Do you have this number?). The required $$\Delta$$V is the difference between the current orbit speed and the speed needed in order to be on the transfer orbit.

 

[elorc]

Well I had a reply written all up and it got blown away since my session timed out. Bah. :) I'll do my best to retype it as I had it.

First, I want to thank you gneill for taking the time to help me so far. It's greatly appreciated!

We were not provided an initial orbit in this assignment. The only information we got beyond the table that I posted is that we are to assume all planets have a circular orbit and that all orbits are on the same plane.

I had picked the Sun because it seemed like it would require the most use of thrusters to get our craft to it successfully. The planets all seemed to be easier targets because we're actually using the Sun's gravity to alter our course and set us out on the correct approach.

With Venus, my assumption was that it would in general require less energy to travel to the planets closest to the Sun. The difference between Mercury and Venus being the fact that with Venus, we can use aerobraking to reduce the need for thruster use. Mercury has no atmosphere, so aerobraking is not an option with it. With the outer planets, we would need to consume more fuel to get into the required Hohmann orbit (or so it seemed to me).

That was may train of thought based on the information I have and my understanding of the material.

The most I know to do with the numbers on the table is that using the Hohmann orbit time, we can determine the launch window. So in Venus' case, I believe it looks like this (again, assuming circular orbits on the same plane... correct me if I'm wrong):

Launch Window = 180° - ((Hohmann Orbit Time / Orbital Period) x 360°)

So if Venus' Hohmann orbit time is approximately 146 days and it's orbital period is 225 days...

Launch Window = 180° - ((146/225) x 360°)
Launch Window = 180° - 234°
Launch Window = -54°

So our launch window would be when the Earth is ahead of Venus by 54 degrees. I'm hoping I have at least that much down. :)

I tried to use the tex tags to format it out all pretty with the fractions and such, but it came out a mess (and kept insisting on throwing the delta symbol in) so I just typed that out as-is. heh.

 

[gneill]

Presumably you can find the speed of the Earth on its orbit somewhere in your course materials. Or you could calculate it from its period (the sidereal year) and its orbit radius. (Once you’ve done that you’ll find that it’s about 29.79 km/sec).

If you can also find a table of the distances of the planets from the Sun, preferably in Astronomical Units (AU), then using Kepler’s laws you can find out the orbital speeds of all the other planets. (How?)

Armed with this information you can find the total delta-V required to go from an orbit at the Earth’s distance from the Sun to the transfer orbit, then from the transfer orbit to one that’s circular at the destination radius. Energy required is proportional to the square of the total delta-V (from the formula for kinetic energy: KE = (1/2) m*v²).

In order to get to the Sun from the Earth, because it’s such a small target (about a half degree on the sky as seen from Earth), you’d need to essentially drop straight down on it. That is, your transfer orbit would be a nearly straight line extending from 1AU down to the Sun. That means killing all the forward velocity from your initial circular orbital speed – your delta-V is equal to the Earth’s orbital speed, about 30 km/sec.

Calculating the energy required to land on a planet is another matter. You’ve already thought about many of the factors involved, such as aerobreaking and so forth. Estimating the magnitude of the effects would be difficult. I have my doubts that the course intends you to do so.

 

[lpetrich]

It's rather easy to calculate. To go from distance r1 to distance r2, one goes into an orbit with a major axis equal to (r1+r2)/2. The changes in velocity (delta-V):

Departing from r1: sqrt((r2/r1)*(2GM/(r1+r2))) - sqrt(GM/r1)
Arriving at r2: sqrt(GM/r2) - sqrt((r1/r2)*(2GM/(r1+r2)))

The planets, with departure, arrival, and total delta-V's in km/s.

Mercury -7.53 -9.61 -17.14
Venus -2.49 -2.71 -5.20
Mars 2.94 2.65 5.59
Jupiter 8.79 5.64 14.43
Saturn 10.29 5.44 15.73
Uranus 11.28 4.66 15.94
Neptune 11.65 4.05 15.71
Pluto 11.81 3.69 15.50

A minus sign means going backward relative to a planet's orbital motion.

So Venus is the easiest to get to, followed by Mars. The other planets are much more difficult.

The speed from a planet's surface is sqrt(v_escape² + v_i-p²)

For an orbit, sqrt(2*v_orbit² + v_i-p²) - v_orbit

i-p -- interplanetary