How to calculate the drag force of a Bussard ramjet?

[Nivek Writer]

Hello everyone,

I am making a paper on the possible applications for nuclear fusion.
One of these possible applications is a Bussard ramjet. I'm trying to calculate if Bussard ramjets
are feasible, based on their thrust and the drag they experience. A Bussard ramjet would
use a very big electromagnetic field to collect hydrogen from its surroundings.
These hydrogen atoms are compressed by the electromagnetic field until the atoms fuse.
The energy of the nuclear fusion can then be used to accelerate the spaceship.

I have a about Bussard ramjets. First, I know that Bussard ramjets probably
won't work due to the drag they experience from collecting hydrogen and accelerating the hydrogen atoms
to the speed of the spaceship. I calculated with the equation a = π r2 ρ η c² / M from http://www.dangermouse.net/gurps/science/ramjet.html that a spaceship with a mass of 9.8 * 10⁶ kg and a electromagnetic field radius of 3000 km would have an acceleration of 1g.
For η I used the factor 0.00753 (0.753%) of hydrogen fusion and for ρ I used the value 5 * 10^-21 kg/m³ based on 0.3 hydrogen atoms per cm³ in outer space.

Using F = m*a, I calculated that this spaceship would have a thrust of 9.6 * 10⁷ N. Furthermore, I calculated (with an equation from that website) that the minimum speed for nuclear fusion to be possible has to be 0.06c. Now, I want to calculate the drag force of the hydrogen atoms on the spaceship when it has a speed of 0.06c. My question is, how do you calculate the drag force of hydrogen atoms on a Bussard ramjet when you know all these other values?

P.S.: I calculated the mass for the spaceship (9.8 * 10⁶ kg) with the equations from dangermouse and a picture from http://orbitalvector.com/Interstellar%20Flight/Bussard%20Ramjet/Bussard%20Ramjet.htm which states that a ramjet with a scoop radius of 3000 km and an acceleration of 1g would have to go at least 0.06c in order for the fusion reaction to function.

 

[Astronuc]

and a electromagnetic field radius of 3000 km would have an acceleration of 1g.
For η I used the factor 0.00753 (0.753%) of hydrogen fusion

That's an awfully broad magnetic field. Compare to the radii of Earth and Mars.

The fusion efficiency has no basis. What fusion reaction would one propose? It certainly can't be p+p, since the cross-section is rather low.

 

[Nivek Writer]

That's an awfully broad magnetic field. Compare to the radii of Earth and Mars.

The fusion efficiency has no basis. What fusion reaction would one propose? It certainly can't be p+p, since the cross-section is rather low.

Hello Astronuc,

Thanks for responding. I also thought that the equation I used seemed strange, since it didn't incorporate the cross section of the fusion reaction.
Let's say I have these values:

- A spaceship with a mass of 1000 tons
- An electromagnetic field with a radius of 200 km
- A hydrogen density in space of 5 * 10^-21 kg/m³
- The nuclear fusion in the spaceship takes place at a temperature of 100 million Kelvin (8.62 keV)
- The cross section of p+p fusion at this temperature is 3.5438*10^-41 cm³/s = 3.5438*10^-47 m³/s using page 20 of http://www.fisicanucleare.it/documents/0-19-856264-0.pdf
- P-p fusion generates 5.91 * 10¹⁴ J per kg of fused hydrogen atoms

How do I use these values to calculate the thrust and the drag of this Bussard ramjet? Which equations do I use?
I know the ramjet I described above wouldn't work, because of the extemely low cross section of the p-p fusion, but I would
like to show this by calculating the thrust and drag of the ramjet. How do I use these values to calculate the thrust and the drag of this Bussard ramjet at different speeds? Have I forgotten any values? I Which equations do I use?