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← Revision 34 as of 20191207 17:59:45 ⇥
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 <5:> '''coordinates in km, no decimal places   hour  Bennu X  Bennu Y  Moon X  Moon Y  separation   16.7951262589  70339  377910  70339  377910  4.95E10   16.7951262589  70339  377910  70339  377910  4.80E10  
 <5:> '''coordinates in km, no decimal places   hour  Bennu X  Bennu Y  Moon X  Moon Y  separation   16.79512625890850  70339  377910  70339  377910  4.95E10   16.79512625890849  70339  377910  70339  377910  4.80E10  
Intercepting Bennu
A work in progress ...
101955 Bennu is a 500 meter diameter Near Earth Asteroid, likely to crash into the Earth sometime in the next 20,000 years, and possibly sooner than 200 years from now. The impact energy would be equivalent to a 100 megaton bomb, or 4 times the explosive energy of the Mount Saint Helens eruption. That won't be a global catastrophe, but it could kill millions of people if it impacted in the wrong place.
Bennu is an Apollo asteroid, with a semimajor axis of 1.1264 AU and an eccentricity of 0.20375. It will make a close approach of 750,000 km on 23 September 2060, and another of 300,000 ± 200,000 km on 25 September 2135. The uncertainty results from small variations of arrival distance creating larger variations in the scattering angle after the 2060 encounter.
I am unable to calculate an accurate model of Bennu's orbit, nor whether it will arrive "inbound" or "outbound", or which side of the Earth the miss will occur. However, I have enough information to calculate the relative velocity (5 km/s) and the angle from the Sun ( 13 degrees ). How?
The 2060 and 2135 encounters are both in late September, about 100 days before Earth's perihelion on January 4th 2061 and January ?? 2136. That lets us calculate Earth's radial distance from the Sun (1.0029 AU), hence the radius of Bennu at that close approach. Given the radius and Bennu's orbit, we can compute Bennu's angle from perihelion, 1.1858 radians ( 67.94 degrees ).
At that angle, Bennu's radial velocity (sunwards or outwards?) is ±5.143 km/s, and the circumferential velocity is 30.860 km/s, relative to the Earth's radial velocity of 0.490 km/s (sunwards) and circumferential velocity of 29.703 km/s. So, the radial velocity difference is 0.490∓5.143 km/s, either 5.903 km/s inwards () or 4.923 km/s outwards. I'd like to know, but the results will be similar either way.
The circumferential velocity difference will be the same; Bennu will approach from "behind" the Earth with a relative velocity of 30.860  29.703 or 1.156 km/s in the direction of Earth's (and Bennu's orbit), though it might miss to the front (orbital east) or miss to the back (orbital west) and probably at a considerable distance north or south (not enough information).
The vector sum of those velocities is either 6.016 or 5.057 km/s, which is the "v∞" of the encounter. The angle from inwards towards the Sun would be 11 degrees, if the trajectory was outwards, 13 degrees from solar zenith. The trajectory would be deflected somewhat towards the Earth, and so the outgoing angle would differ a little from those numbers, depending on the side of the Earth that the close approach happens.
If this led to the collision, the Earth's 11.18 km/s escape velocity would be vector summed with v∞; the result would be an impact velocity of 12.271 or 12.696 km/s. For 70 million tonne Bennu, that is a kinetic energy equivalent of a 106 or a 103 MT bomb.
But Bennu is likely to miss in both 2060 and 2135, and all I know about the miss is that it will pass through a circle perpendicular to Bennu's path (at the given miss distance), radius 750,000 km in 2060 and 300,000±200,000 km in 2135 (perhaps 63 Bennu orbits later).
Bennu Close Approach Calculations
Earth 
Bennu 


7.00E+10 
kg Mass 
1 
1.1264 
AU Semimajor axis 
149597900 
168507075 
km Semimajor axis 
1 
1.1954 
year Period 
365.2422 
436.64 
day Period 
0.0167086 
0.20375 
eccentricity 
29.790 
28.666 
km/s v0 
1.0167 
1.3559 
AU aphelion 
0.9833 
0.8969 
AU perihelion 

100 
degrees from perihelion, close approach earth orbit angle 
1.00290 
1.00290 
AU Close approach radius 

1.07651 
1+ecos() 
1.7453 
1.1858 
radians at close approach 
29.703 
30.860 
km/s circumferential velocity 
0.490 
±5.413 
km/s radial velocity 
if Beppu orbits 


outwards 
inwards 

1.156 
1.156 
km/s circumferential difference 
5.903 
4.923 
km/s radial difference 
11.08 
13.22 
degrees velocity angle from radial 
6.016 
5.057 
km/s total relative velocity 
11.180 
11.180 
km/s Earth escape velocity 
12.696 
12.271 
km/s Earth impact velocity 
4.44E+17 
4.29E+17 
Joules impact kinetic energy 

4.18E+15 
Joules per megatonne TNT 
106 
103 
MT TNT equivalent 
Hitting the Moon?
Before I did this calculation, I had hoped that the 2135 near miss could be arranged to impact the Moon, which will be three days after full on 25 September. The Moon (in a 384,000 km orbit) will be about 37 degrees or 200,000 km east of zenith at that time.
It turns out ...
That if the nominal 2135 closest approach is somehow adjusted to 141191 km, and the Moon is exactly 36 hours after full Moon at that closeapproach time, then 16.8 hours before the close approach, the positions of Bennu and the Moon will coincide exactly  a collision.
That is, if Bennu is coplanar with the position of the Moon at the time of impact.
Of course, Bennu is unlikely to be coplanar, and the unperturbed radial miss could be in any radial direction from the Earth normal to the velocity vector. I will need a better description of Bennu's actual three dimensional trajectory to make this claim. Still, it shows that it may be possible (with some high precision artificial deflection before the 2060 close approach) to deliver Bennu into the side of the Moon, preferably the far side as illustrated, so less ejecta is thrown directly towards the Earth.
That is illustrated below as Bennu Descending ... towards the Sun.
download MoonIntercept.png . . . . . drawing for illustration, not to scale.
36 
hours after full moon 

384400 
Moon Radius 
141191.276781674 
km Bennu closest approach 

655.73 
Moon orbit period 
4125 
km/h Bennu eastbound 

139618.824888302 
km X Bennu at t0 
21250 
km/h Bennu radial inbound 

21013.3380510765 
km Y Bennu at t0 
0.1493838462 
radians Bennu 

0.0095819702 
radians/hr Moon 

coordinates in km, no decimal places 

hour 
Bennu X 
Bennu Y 
Moon X 
Moon Y 
separation 
16.79512625890850 
70339 
377910 
70339 
377910 
4.95E10 
16.79512625890849 
70339 
377910 
70339 
377910 
4.80E10 