Nuclear Waste Disposal into Sun or outside Solar System?by osxraider Tags: delta v requirements, nuclear waste, orbit determination 

#19
Dec812, 02:30 PM

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#20
Dec812, 02:50 PM

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[tex] V_o = \sqrt{\frac{GM}{r}}[/tex] where M is the mass you are orbiting (in this case the Sun) and r is the radius of your orbit (assuming a circular orbit) Escape velocity is found by [tex] V_e = \sqrt{\frac{2GM}{r}}[/tex] Note that the only difference is the 2. Thus escape velocity is 1.414... times greater than the orbital velocity at any given distance from the Sun. In order to hit the Sun, you have to, as explained by DH, put the rocket in a orbit that grazes the Sun. To work out what it talks to do this, you use the visviva equation: [tex]V = \sqrt {Gm \left ( \frac{2}{r} \frac{1}{a} \right )}[/tex] Here, a is the semimajor axis, which is found by taking the Sum of the radius of the Sun and the radius of the Earth's orbit (r) and dividing it in half. This will give you the velocity that you would have to slow the rocket to in order to have it skim the Sun's surface. 



#21
Dec812, 02:59 PM

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With no planetary encounters involved, there is a much cheaper way of hitting the Sun compared to canceling 87% of the Earth's orbital velocity. Make a prograde burn instead of this huge retrograde burn. This will put the vehicle on an elliptic orbit with a perihelion equal to Earth's orbital radius and an aphelion well beyond Earth's orbit. At aphelion it is fairly cheap to transfer to another elliptical orbit that will take the vehicle just inside the surface of the Sun at perihelion. Note that this intermediate orbit has to take the vehicle well beyond Earth's orbit to make this two burn approach cheaper. 



#22
Dec1412, 08:01 PM

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The visviva equation is exactly what I employed and yes, I have the spacecraft grazing the Sun on at aphelion of the Hohmann transfer ellipse. This is the simple way. The project calls for seeing if any planetary transfers would reduce this delta v. I was simply assuming that along the transfer ellipse, we could encounter Mercury/Venus or both and have them slow down the craft a bit so less delta v is required but the problem I seem to have is when we do fly by's, we don't really get to control the post fly by trajectories do we. It could end up being anything but I suppose if it is in a smaller orbit after doing fly by with let's say Mercury, then it would be easy to then do another Hohmann transfer from this smaller orbit to then graze the Sun.
My value for delta v using Hohmann transfer directly from Earth to surface of the Sun is 24.18 km/s. Does that sound right? 



#23
Dec1412, 08:07 PM

P: 26

Also, in the other case of leaving Solar system, should I simply do Hohmann transfer to Jupiter or Saturn and then have a fly by with one of these planets and get it into a larger Orbit post flyby and then just do escape velocity for Solar system to have it leave.
At this point, it is due tomorrow so I couldn't do too much detail anyway because if I wanted to do multiple fly bys's, I would have to calculate where the new post fly by trajectories intersect other planets and then calculate fly bys with them , then find the new post flyby trajectories. Is that practical if the reduction to delta v is only slight? I would rather just have the satellite head to SUn on Hohmann and graze or have the Sun head to Mercury/Venus, get a slow slow down that puts it into new smaller orbit and then do Hohmann again to graze Sun. In CASE 2: I am planning a Hohmann to Jupiter or Saturn and then get it into new larger orbit and then do an escape velocity to leave the Solar system. 



#24
Dec1412, 08:11 PM

P: 26

Also, thanks a lot D H and Janus. You guys have been very helpful!
I just think at this point, I am finding it hard to understand a lot of stuff you are saying for example doing prograde burn and then going beyond earth, then transferring into elliptical orbit which passes through Sun. The only thing I know from the context of this course is to d Hohmanns, calculate extra delta v imparted due to fly by's as a function of Hyperbolic excess speed and turn angle. Also, does the mass the planet's have an effect on the delta v for gravity assist? There doesn't seem to be an equation for that although it might have been accounted for in the turn angle itself. I'm so sorry, I feel like problems would have been solved in 5 mins if we could directly interact. 



#25
Dec1612, 04:34 PM

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I realize that your assignment is already turned in, but I suggest you try starting from a low Earth orbit, for example, a 200 km altitude orbit. You should find that it is a good deal less than your 24.18 km/s Δv. Now add in the largish Δv just needed to get to a geostationary orbit and you'll see why interplanetary missions start from LEO rather than from geostationary altitude. This is essentially the Oberth effect. You oftentimes get a bigger bang for the Δv buck when you perform the burn close to a planet (or the Sun) compared to at a more remote distance. 



#26
Dec1612, 04:54 PM

P: 26

Yes, I should have check what it might have been from LEO. I'll look into the Oberth effect but is this essentially simply because the circular velocity in LEO larger than GSO?
Also, I used GSO's in both cases so at least comparatively, hopefully, I still got the right answer as to which is cheaper. I did turn it in and frankly, I could have done a much better job but it was still a great learning experience. The entire course, we have been given which orbit to which orbit and parameters were all specified but that's not how real missions are. It was very interesting to design the orbits themselves. What's funny is that I struggled just with doing Hohmann transfers in the same plane. In reality, there is inclination, solar radiation pressure and wind and so many more optimizations to think about. I do have one question though. When I did patched conics for planetary fly bys, we always have to convert the vectors from heliocentric to local and then back. I don't quite understand this. I know that the Planet's themselves have velocities but I thought this was account for in the Hyperbolic Excess velocity equations. For example, when the spacecraft on a transfer ellipse reaches periapsis in the fly by hyperbola, we should we able to to apply any delta v that will either allow it to be captured, continue on the same transfer ellipse (no delta v) or move into larger ellipse. This is not how its done in the book. They use the turn angle which then goes into lots of complex vector equations. If the planet is providing a boost based on delta v depending on the turn angle which is dependent on the eccentricity of fly by hyperbola, which is dependent on fly by altitude, that should just be it....a delta v which then gets added to the spacecraft and we assume that this happens at perigee. So it should be like any other Hohmann transfer or impulsive burn. I will maybe post an example with the solution after all my exams but to conclude, I did fairly well on the final and throughout the class. This project is 10% of my grade though. I will be taking more orbital classes in the future (next year). Also, this forum has been awesome! There is so many other questions I have in general. I would have to start new threads though :) I hope this thread is useful to others who might stumble upon it in the future :) 



#27
Dec1712, 01:16 AM

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