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## Nuclear Waste Disposal into Sun or outside Solar System?

 Quote by osxraider Janus, you provided some direct numbers. I'm curious as to how you came up with them. Could you give me a brief walk through?
If the vehicle cancels all of its tangential velocity relative to the Sun it will fall straight into the Sun. You don't have to go quite that far because all you need to do is have the vehicle just graze the surface of the Sun. It's similar to how reentry burns are calculated. Simply make the periapsis inside the planet's atmosphere and let the atmosphere do the rest. Reentry isn't quite that simple; one has to worry whether the entry is too deep or too shallow. But you can ignore that for this problem. Make perihelion equal to the Sun's radius and voila! Your waste is disposed of.

The problem is that the delta V needed to go from a circular orbit at 1AU to an elliptical orbit with a 1.39 million km perihelion is extremely expensive.

 Hohmann transfer are supposed to be the most efficient? correct?
If you don't use gravity assists, and if you only use two burns, yes. Gravity assists change everything. You can also sometimes do better using more than two burns. Bi-elliptic transfers can be more efficient than Hohmann transfers, for example. That huge Δv needed to just graze the Sun from Earth orbit can be made considerably smaller if you start off in the wrong direction.

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 Quote by D H If the vehicle cancels all of its tangential velocity relative to the Sun it will fall straight into the Sun. You don't have to go quite that far because all you need to do is have the vehicle just graze the surface of the Sun.
Granted, and I'll admit that my estimate of nearly all the orbital velocity was a bit overstated. It still works out to having to shed ~87% of the Earth's orbital velocity.

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 Quote by osxraider Also, the post about just letting the craft decay is incorrect. The craft will still remain in orbit around the Sun trailing Earth. To get it to decay, you would have to slow it down to 0. Since it is following Earth in it's orbit, its speed must be the same as that of the Earth for that orbital radius around the Sun which means that your delta v would have to be the negative of Earth's velocity which is HUGE! That would definitely not be the minimum delta v. One thing to remember about why delta v is so important is that the craft will have to carry 7-9 times the amount old compared to cargo depending on specific impulse. Also, I am not worrying too much about which one is lesser right now. Either way, I have to solve both and then simply find out so which one is lesser is actually of little relevance in the end as the motivation for the project is simply to design the orbits and perform the calculations. In the end, the conclusion would simply compare the 2 different delta v's and make a recommendation. I am still very interested in Janus's numbers! Was it really that easy to calculate? Also, I apologize for not quoting. I'm a newbie here.
Orbital velocity is found by

$$V_o = \sqrt{\frac{GM}{r}}$$

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

$$V_e = \sqrt{\frac{2GM}{r}}$$

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 vis-viva equation:

$$V = \sqrt {Gm \left ( \frac{2}{r}- \frac{1}{a} \right )}$$

Here, a is the semi-major 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.

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 Quote by Janus Granted, and I'll admit that my estimate of nearly all the orbital velocity was a bit overstated. It still works out to having to shed ~87% of the Earth's orbital velocity.
I get a Δv equal to a bit less than 89% of the Earth's orbital velocity, so yeah, we're in the same ball park. (Starting from LEO, it's only 71.3%, thanks to the Oberth effect.)

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.
 The vis-viva 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?
 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 fly-by 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 fly-by 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.
 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.

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 Quote by osxraider 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?
That sounds right assuming you are starting from a geostationary orbit.

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.
 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 :)

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 Quote by osxraider 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?
In this case, yes, but it can be even more dramatic with highly elliptical orbits. Suppose the vehicle applies Δv at periapsis that just brings it to escape velocity. Apply that same Δv at apoapsis and the vehicle doesn't escape; it's not even close.

 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.
And atmospheres, and non-impulsive burns, and perturbations from other planets, etc. Take all of those other things into account and you have a problem for which there is no closed form solution. You have to resort to numerical integration.

 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.
The hyperbolic excess velocity equations are planet-centric. In a patched conic approximation, you ignore the influence of the Sun while the vehicle is within the planet's sphere of influence. From the perspective planet-centered inertial frame, the vehicle's speed is the same on entering and on leaving the sphere of influence. All that changes from this perspective is the direction of the velocity vector. From the perspective of a heliocentric frame, both magnitude and direction are changed.

 Tags delta v requirements, nuclear waste, orbit determination

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