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Numeric Solutions (Transfer Orbit of Mars Direct)

(x_f1-x_e1)^2+(x_f2-x_e2)^2=
d^2+(x_s1-x_e1)^2+(x_s2-x_e2)^2-2 d ||(x_s1,x_s2)-(x_e1, x_e2)||
(x_f1-x_m1)^2+(x_f2-x_m2)^2=
d^2+(x_s1-x_m1)^2+(x_s2-x_m2)^2-2 d ||(x_s1,x_s2)-(x_m1, x_m2)||

The above equations describe all possible says a rocket can go from, a point on earth orbit, (x_e1,x_e2) to a point in another orbit (x_m1,x_m2). There are two equations and three unknowns. The unknowns are the second focus (x_f1,x_f2) (note the first focus is the sun) and the constant d which is defined so that the sum of the distance from each focus to a point on the ellipse is d. These equations could be solved numerically or algebraically. I would like to solve the problem numerically first to put some meaning behind these equations. The first problem I would like to solve is to find the transfer orbit used in mars direct. To solve this problem I need to know the coordinates of the earth when the rocket leaves earth and the coordinates of mars when the rocket arrives in mars. Does anyone know these coordinates?

Orbital Intersections (Finding The Plane)

Consider the problem of finding a transfer trajectory between two orbits. The transfer trajectory will be elliptical or hyperbolic. Therefore the transfer trajectory must lie on a plane. Three points must be in the plane. The intial point, the final point and the sun. A plane is defined by the equation a1x1+a2x2+a3x3=c or in [http://mathworld.wolfram.com/Vector.html]vector form: a [http://mathworld.wolfram.com/DotProduct.html]dot x = c. Let the point on the sun be given by x_s=[x1_s x2_s x3_s] the point of intersections with the earth orbit be x_e=[x1_e x2_e x3_e] and the point of intersection with the other orbit be x_m=[x1_m x2_m x3_m]. The vector a is given by the system of equations:
a dot x_S=c
a dot x_e=c
a dot x_m=c

or in matrix form

Xa=c
where:
X=[x1_s   x2_s    x3_s]
     [x1_e   x2_e   x3_e]
     [x1_m x2_m   x3_m].
c=[c]
     [c]
     [c]

Once the equation of the plane is found we can describe the transfer orbit with only two coordinates:

b1=[1            ]
       [0            ]
       [(c-a1)/a3]

b2=[      0      ]
       [      1      ]
       [(c-a2)/a3]

All points in the plane are equal to come constant x multiplied by b1 plus y multiplied by b2. The vectors b1 and b2 form a set of [http://mathworld.wolfram.com/VectorBasis.html]basis vectors  for the plane of the transfer orbit, and x b1+y b2 is called a [http://mathworld.wolfram.com/LinearCombination.html]linear combination of the basis vectors b1 and b2.

I choose, Laser Interferometer Space Antenna over Hubble. I think the existance of gravitional waves is a very important thing to verify.

It seems to me that if solar flares are such a problem on the moon the will be just as big a problem during the trip to mars. This is especially true if the crew is spending 500 days in space. Who is right? What kind of shielding is necessary.

Say for some reason we already have a fully fueled chemical stage in orbit that can dock to another stage.

From [http://www.marssociety.org/news/2004/0123.asp]the mars societies statement on the Bush initiative.

?Without dismissing the important value of NEP for outer solar system robotic missions and other missions involving large velocity changes undertaken across extended time frames, we note that the size of NEP units required to supply propulsion for human exploration missions are on the order of 10,000 kilowatts. In contrast, when used to produce chemical propellants on planetary surfaces, the required reactor size to support human exploration is reduced to about 100 kilowatts. This is because a much smaller reactor stationed on a planetary surface making propellant can emit energy over a long period of time prior to flight, store it as chemical propellant, which then can release the energy as fast as it is needed under flight conditions. The mission mass leverages achieved by such ISRU supported chemical propulsion options are greater than those offered by NEP, while for inner solar system missions, the flight times are less (two orders of magnitude less for Lunar applications).?

There doesn?t seem to be a good way to provide a linked reference to another post. If I use the quote button as a link, then when someone clicks on it does the same thing as if they were replying to the post I am trying to reference.

Bill, your idea sounds very good. Unfortunately, I don?t know much about the shuttle C. How quickly can one be deployed, what are the development costs? I really like the idea of launching other payloads at the same time of the shuttle, so that more components can get assembled each shuttle mission. Unfortunately, I don?t know how long the shuttle can stay in space, how long the crew can work effectively in assembling the space station before returning to earth and how long each component will take to assemble. If your approach is the best course of action, I hope NASA implements it soon.

BTW if we finish the ISS ahead shedual by such a sceam, I am all in favor of sending one last shuttle mission to service the Hubble.

As has been said repeatedly we all agree that if our only goal is to go to mars then there is no point at stopping on the moon. Still, let?s stop thinking about why the moon is a hindrance to mars and let?s look at how it can be helpful if for no other reason as a pure academic exercise. The mars direct trajectory takes 6 moths and uses 4.3 km/s of delta V. How much delta V is required for 5 months, 4 months 3 months 2 months and 1 month?

If we look at the numbers again it takes 9.7 km/s of delta V to go from earth to low earth orbit yet it only takes 1.6 km/s to go from lunar orbit to low earth orbit. Clearly if we were making rockets on the moon, there would be know question that we would have much more fuel left over launching from the moon to low earth orbit then from earth. If the rockets launched from the moon docked with the rocket carrying a crew from earth, how much more delta V would we have to go from LEO to mars. Further, what if the rockets were launched from the moon with a mass driver and reached LEO by either tethers or solar tugs, then how much more delta v do we have to go from LEO to MARS. How much can this cut down on the trip time?

Don?t insult me be saying surly you don?t think it is worth building all this infrastructure to go to mars? I didn?t say it was. But if as a result of the political environment we go to the moon first and we build some of this infrastructure, how does it help us get to mars. Remember this is an academic exercise and not a practical engineering decision.

The point is if you have more fuell you don't need to aerobreak, or you can slow down before you aerobreak.

It depends on what kind of life I will have when I get there and who will be there with me. I know my girlfriend wouldn?t come to mars with me.

I'd choose Martin Fogg, or Neilius Borr

Feed back from the members. This is very positive

Lets consider this statement, an offense against sciece? Maybe, I am not convinced. An officne against civilization? Now this is a stretch. What is an offince against civilization? It must obviously be different then a crime against humanity. Shurly the distruction of the hubble cannot be compared to a mass geneside on earth. I think someone representing the mars society should be a little more carefull with the words the choose. A departure from the pioneering spirt? More rettoric.

Now it is a crime against science? I doubt the cancellation of the Hubble is anywhere?s comparable to the cancellation of some of the big particle accelerators that have been planed in the past. We have already learned much from the Hubble and there are newer better telescopes coming. I don?t here the mars society lobbying for particle accelerators. I have also not been adequately convinced that the grounds for deserting the Hubble are irrational. Also to accuse people of moral cowardice is nothing more then mudslinging.

Now this seems like a needless attack on the only leaders? scenes the Apollo days to show some vision towards space. How do these attacks further our objective of getting to mars.

This is quite an accusation. I hope Zubrin has some good evidence to support these accusations. Otherwise he is creating enemies and perhaps risking a liable suite.