One thing you're missing seems to be Oberth's effect. To go from LEO to solar system escape velocity, you have to counter Earth's escape velocity, but after that, you get an additional multiplier by doing the burn at a higher initial velocity (at LEO).
Your method here also has a problem:
Hence an additional 11.2 km/s - 9.5 km/s = 1.7 km/s is needed to escape the gravitational field of the Earth.
To get to LEO, the 9.5 or 10 km/s is the delta v you need delivered by the engines. But that doesn't mean you're that much closer to escape from Earth's gravity. That's because air drag and gravity drag are both "wasted" impulse. Ultimately, they just go to friction. So if you're in LEO:
- It took 9.5 km/s to get there (we'll say)
- It will take an additional 11.2-7.9 = 3.3 km/s to escape the gravity well
Now, going from LEO to hyperbolic orbits is a bit more difficult. I'll use the energy balance, because I find it easiest to understand. The specific orbital energy is:
$$ \epsilon={v^2\over2}-{GM\over{r}} $$
After it's escaped Earth's sphere of influence, the energy balance will be simply:
$$ \epsilon = \frac{ v_{\infty}^2 }{2 } $$
For either escape or dropping into the sun, we have a final velocity in mind. This is at 1 AU from the sun, after we get out of Earth's sphere of influence. Earth is moving at 29.78 km/s. so we need:
- To get to the sun, we need 0 km/s net velocity, so moving 29.78 km/s relative to Earth
- To get out of the solar system, we need 42.1 km/s in the direction of Earth's motion, so 42.1-29.78 = 12.32 km/s
Now we need to use those above energy equations to get those velocities after getting out of Earth's sphere of influence. Now let's imagine we're half way through the burn, and are at LEO altitude with escape velocity. So we've spent exactly 9.5+3.3 = 12.8 km/s so far. We need to figure out how much more we need in this same burn to shoot for our destination.
$$ v_{\infty} = 29.78 \text{ km/s or } 12.32 \text{ km/s} = \sqrt{ 2 \left( \frac{v^2}{2} - \frac{GM}{r} \right) } $$
Solve this for both cases in terms of $v$ now. For completeness, I use $r=6,354.82 \text{ km}$. Everything else is known. Now the results are:
- for getting to the sun v = 31.8 km/s
- for getting out of the solar system v = 16.65 km/s
These are the numbers for the total velocity you need at LEO altitude. In the story I'm telling, you're at 11.2 km/s at the end of the previous leg, so subtract that number in order to calculate the final burn. Again, the trip is broken up into 3 segments according to my organization, but the last 2 are really the same burn. Let me focus on the escape from our solar system. The three legs are:
- A launch that requires a 9.5 km/s burn
- The first part of the burn at LEO to get to escape velocity is 3.3 km/s
- Continuing that same burn, an additional 16.65-11.2=5.45 km/s to get the solar system's escape velocity after you're out of Earth's sphere of influence
The total of all these comes out to 18.25 km/s. If your propellant exhaust velocity is 4 km/s, then your ultimate mass fraction on the launchpad will be about 96-to-1. So a million pound rocket could get 10,436 pounds out of the solar system with this method (I'm not saying it's a good method for this purpose).
I hope this clears up the "from lower Earth orbit" part. It's not as simple as adding things up, because you're trying to get the velocity to escape the sun's gravity well, while you're still in Earth's gravity well. To do that, you have to include the Oberth effect due to your location within Earth's potential well. I hope I've demonstrated that correctly.
EDIT: here is a different set of numbers that starts with the radius of the Earth, instead of the "11.2" and "7.9" numbers, which I only used because they were in prior discussion.
- Base Earth radius 6378.1 km
- LEO altitude 300 km
- LEO radius 6678.1 km
- V at LEO 7.725529305 km/s
- Escape V from LEO 10.92554832 km/s
- Burn from LEO to escape 3.200019015 km/s
- V needed at LEO to get 29.78 km/s 31.81648047 km/s
- Extra burn needed past escape V 20.89093216 km/s
- Oberth ratio 1.425498861 unitless
- V needed at LEO to get 12.32 km/s 16.64999789 km/s
- Extra burn needed past escape V 5.724449572 km/s
- Oberth ratio 2.152171985 unitless
