# How much will Bennu rotate faster and faster with mirrors facing the Sun on its left side?

Several answers to the question "Why paint only one-half of Bennu" describe the Yarkovsky effect as an important force acting on the asteroid 101955 Bennu.

Basically radiation pressure consists of two parts: the pressure caused by the incident light wave and the pressure of the recoil due to a partially reflective surface. In the case of a perfect reflector the total pressure doubles compared to the pressure on a totally absorbing (black) body.
If the incident and the reflected wave are at an angle to the normal to the surface, the pressure will be proportional with the square of the cosine of that angle.

Now imagine 12 mirrors placed along the equator of Bennu, 142 meter apart from each other, with a circumference of the asteroid of about 1700 meters.

For simplicity of calculation each mirror has an area of 100 X 100 m$^2$ and can rotate along its horizontal axis and is facing eastward.
Since Bennu has a retrograde rotation, the left side of it turns away from the Sun, so to get the asteroid rotating faster by radiation pressure only on the left side the mirrors will have to reflect the sunlight.

On the right side of the asteriod the mirrors should be turned in such a way that almost no sunlight will fall on them to prevent radiation pressure being exerted upon them then.

Will indeed Bennu rotate faster and faster this way and can it be roughly calculated how much ?

To make the calculations somewhat easier, only the force on one mirror at any time is asked for, namely during the time interval when the angle of the incident light wave with the normal of the position of the mirror on the surface changes from 60$^0$ to 90$^0$.

• Not a duplicate, but related: Can a satellite work like a radiometer? The physics here is really interesting; I'd recommend you modify the question and leave what to put on each side open, and ask for what would work best. The three simplest coatings would be white, black, and mirror, but then there are the materials with very different albedos/emissivities in the visible wavelengths versus the thermal wavelengths. (e.g. Why are RTGs different colors?) – uhoh Jul 4 '18 at 14:29
• @uhoh It's not like a radiometer at all. Do you recommend the black coating for the backside of the mirrors to exert less pressure on the right side of the asteroid ? But the question is about the acceleration , a=k/m,and if a is constant, then velocity v=a.t will grow more and more by time ? – Cornelisinspace Jul 4 '18 at 16:47
• @uhoh Thank you, i appreciate your comments, only the first one caused me to compare Bennu with a radiometer.Besides, i could ask this question because i got to know Bennu by your question concerning this asteroid. – Cornelisinspace Jul 4 '18 at 22:18
• @uhoh I've changed the question somewhat to get a more practical answer. – Cornelisinspace Jul 5 '18 at 8:07
• I think this is a calculation you should just do in that case. The only number you need is the radiation pressure at e.g. Earth's orbit, which is 9 uN/m2 (9x1E-9 newtons per square meter) C.f. en.wikipedia.org/wiki/Radiation_pressure – Bob Jacobsen Jul 5 '18 at 14:45

Making this quantitative is hard, but the basic answer is yes, such an asymmetry can make a torque, and that can spin the body.

The separate radiometer question and answers shows the complications: The panels are not just reflecting/absorbing sunlight, but also emitting. That emission depends on their temperature and emissivity. Absorbing surfaces are also generally better at emitting. Temperature depends on what their back sides are doing (cooling?), whether they're conducting heat, etc.

Does the speed keep increasing forever? At some point, other considerations come into play. Earth satellites do feel (a tiny amount of) drag, though that's really small for an asteroid. Pieces can start to fly off and change the inertia tensor, hence rotation axis. Etc.

• Thank you for your answer, but i've changed the question somewhat to get a more practical answer. – Cornelisinspace Jul 5 '18 at 8:05
• Which @Conelisinspace hopefully will not do very often, as changing questions after answers are posted can get one in hot water. – uhoh Jul 5 '18 at 8:19
• @uhoh In the text i already hinted on "How much" and luckily there was still only one answer. – Cornelisinspace Jul 5 '18 at 10:00
• I've added a rather complex sentence at the end, i hope you can unravel it ! – Cornelisinspace Jul 5 '18 at 13:12

With a radiation force of about 9 microNewton per m$^2$at the distance of Bennu from the Sun, the pressure will be about 8 microNewton with a 90% reflectivity of the mirror.
But on the right side of the asteroid the incident light wave will also exert 50 % of the total on the surface and with an albedo of 4.5% one can add up 5% of reflectivity there.
So the net radiation force will be about 9 x (90-55)% = 3.1 microNewton. Since at any time one mirror of 10$^4$ m$^2$ faces the Sun, this will produce a force of 3.1 x 10$^-$$^2 Newton. Because the distance to the centre of the mass of Benno is about 250 m, the torgue, the rotational force, will be 250 x 3.1 x 10^-$$^2$ = 7.8 N.m.

The moment of inertia of a solid sphere of constant density about an axis through its center of mass is $^2$/$_5$ mR$^2$.
With Bennu having a mass m of about 7.10$^1$$^0 kg and a radius R of about 250 m. its moment of inertia will be about 4.4 x 10^1$$^5$ kg.m$^2$.

Now the moment of inertia together with the torque determines the angular acceleration similar to how mass together with force determines the acceleration.

So the angular acceleration of Bennu because of the reflectivity of the mirrors will be the torque value of 7.8 N.m divided by the moment of acceleration value of 4.4 x 10$^1$$^5 kg.m^2 which gives about 1.8 x 10^-$$^1$$^5$ rad/sec$^2$.

Since a year has more than 31 million seconds the angular acceleration of Bennu owing to the mirrors will then be about 1.8 rad. which is really small compared to the 12.800 rad. that Benno has already achieved in a year.