# 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 '19 at 19:34
• so the spacecraft would slow down – Westin Ritter Jun 7 '19 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 '19 at 3:55
• Impulse of the solar wind would be a larger effect than the change in mass due to charging the battery. Also, why didn't you ask about the decrease in mass due to expending battery energy in running the instruments? – Bill N Jul 30 '19 at 22:45

It gains mass and spirals in slowly

In relativity every object/particle/etc has a 4 momentum, which is a 4-element vector made by concatenating the object's total mass/energy onto the ordinary momentum. Collisions, whether elastic or inelastic, conserve 4 momentum in all reference frames. We divide the spatial components of the 4 momentum by the time component to get the ordinary velocity, however the spatial components of the 4-momentum aren't that useful to look at because the sun's gravity can give or take momentum as well as it's light.

Suppose the probe starts out in a circular orbit and has a tiny cross section but then suddenly unfurls it's 100% efficient nano-film panels to charge up it's antimatter battery. When the photons are absorbed by the probe it gains their 4-momentum, which both adds to its mass/energy and give it a sideways push. The continuous sideways push cancels out a portion of the suns gravity. This would set the probe into a slightly eccentric orbit that takes it farther and closer to the sun than it's initial orbit.

Over a period of time, the probe is gaining mass as per the increasing energy component of the 4 momentum (it is moving "slowly" so it's total mass-energy is very close to it's rest mass). Because the photons are coming in perpendicular in the sun's frame, the angular momentum is constant. Per unit mass the angular momentum is dropping, so the orbit shrinks over time (smaller orbits have less angular momentum than larger ones). Dust grains face a similar problem, even though they don't store energy they lose momentum as they re-radiate it. In orbital mechanics, drag forces cause us to speed up because we fall deeper into the sun's gravity well.

The velocity would not change. All objects fall at the same speed, no matter what the mass is.

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.

• Photons very definitely have momentum. There are also protons being expelled from the sun. – Bill N Jul 30 '19 at 22:46