# A spacecraft orbiting the sun charges a battery which becomes a tiny bit more massive. How does this effect the velocity of the spacecraft

According to E=mc² the spacecraft which stores the Energy from the sun becomes a tiny bit more massive. how does this effect the impulse of this spacecraft. Would it grow because P = (m + extra mass) * v and the speed stay the same. Or would the spacecraft slow down as : impulse(start) + impulse of extra mass(as it has no velocity) = total impulse and the total impulse / (starting mass + extra mass) = velocity. And is this a stupid question?
The spacecraft has solar panels which are constantly facing the sun. The velocity in question is the one with which the spacecraft travels around the sun.

• Note that one has to consider 2 conservation laws: conservation of energy and conservation of momentum. A photon with energy E has (size of ) momentum p=E/c=h/lambda. – Poutnik Jun 7 at 19:34
• so the spacecraft would slow down – Westin Ritter Jun 7 at 19:55
• Slowing down means higher orbit and higher energy. But the gravitational effects of other bodies are usually many orders more significant. – Poutnik Jun 8 at 3:55

## 2 Answers

Since the velocity of the spacecraft is perpendicular (at 90 degrees) to the photons from the sun, as one of the comments says, the most important effect would be the impulse from the photons against the solar panel. Every time a photon was absorbed by the solar panel, the satellite would get an impulse $$I = h/\lambda$$, very slightly increasing the satellite's momentum away from the sun. So actually, the only noticeable effect would be that the satellite's orbit starts to (very very) slowly move out to larger distance from the sun.

The effect due to an increase in mass is minuscule, though I suppose correct. You could use:

$$E / c^2 = Power * Time/c^2 = Mass$$

So if the power of photons incident on the spacecraft is ~1000W, and you had an infinite capacity for storing the energy as electrical potential, it would in theory take around 2850 years to increase the mass of the spacecraft by 1 gram. So yeah, minuscule, and in reality would not have an effect.

HOWEVER, the momentum $$p$$ in the direction the spacecraft is initially travelling in stays constant, so after a photon strikes it the tiny increase in mass must be counteracted to keep $$Momentum = const. = Mass * Velocity$$ constant. So I think you're right, the velocity will decrease. The point is the decrease in velocity would be too small to measure, even if you waited thousands of years.

• so you're saying the increase in mass of the satelite wouldn't effect its velocity? even at a subatomic barley measurable scale? Aka the impulse of the spacecraft would increase? – Westin Ritter Jun 7 at 20:38
• The spacecraft has a constant momentum in the direction it's initially travelling in, since there are no forces acting on it in that direction. Say it gains a tiny bit of mass when it stores the energy of the photon: its momentum is still constant, so by p = mv, if the mass has increased, the velocity must decrease to keep p constant. The point is that this decrease in velocity is far too small to measure. – alex1stef2 Jun 7 at 21:01
• thank you for clarifying – Westin Ritter Jun 7 at 21:04
• edited to include your question – alex1stef2 Jun 7 at 21:05
• There is a force due to aberration of the incoming light. – Bob Jacobsen Jun 7 at 22:00

The spacecraft would slow down as the extra mass had no momentum in the direction in which the spacecraft is moving when it it was in the form of light.