What is the largest mass the sun can have without destroying the Earth?

[haiku11]

This is one of the questions on a small exercise but I don't understand it.

I thought that the Earth is in orbit because the mass of the Earth and the Sun combine perfectly such that the orbital velocity is exactly the same as the pull of the sun so the Earth is forever falling towards it. If this is the case, then any significant change to the mass would (I'm actually not sure) alter the magnitude of the pull, causing the Earth to either slowly spiral into the Sun or away from it, kind of like a satellite orbiting the Earth. So wouldn't any change to the Sun's mass kill us?

That was my first thought, then it occurred to me that the Sun has solar flares which expel tons and tons of mass. So technically the Sun should be getting less massive and it's been doing this for at least 4.5 billion years cause that's how old the Earth is. So with this is mind, I have no idea how the Earth isn't spiralling away from the Sun as we speak.

These are the 2 thoughts I had on this question and I really don't know how to answer it.

 

[Nik_2213]

IIRC, the Earth's orbital radius is increasing, but very slowly, as the Sun's mass-loss is a very, very tiny fraction of the remaining mass.

Bad news is that such increase is unlikely to save us when the Sun goes red-giant...
;-(

 

[DaveC426913]

I thought that the Earth is in orbit because the mass of the Earth and the Sun combine perfectly such that the orbital velocity is exactly the same as the pull of the sun so the Earth is forever falling towards it. If this is the case, then any significant change to the mass would (I'm actually not sure) alter the magnitude of the pull, causing the Earth to either slowly spiral into the Sun or away from it, kind of like a satellite orbiting the Earth. So wouldn't any change to the Sun's mass kill us?

Yes and no.

1] If the sun's mass increased, the Earth would not spiral inward to its death. It would change to a highly elliptical path whose apogee is right where the Earth is now, but whose perigee would be proportionally closer to the sun. The new orbit would be stable.

2] This is true if you don't allow for a change in the orbital velocity to compensate. I suspect that the question assumes you would.

What you might want to read up on is Roche Limits. It may not be the answer, but it is an answer.

 

[BobG]

I'm not even sure what the question means. Does it mean to destroy all life on Earth or does it mean to literally destroy the Earth? Does it mean the volume increases along with this increased mass or does the volume stay constant and only the mass increases?

If it means to literally destroy the Earth, the Earth's orbit has to change essentially until perigee reaches the Sun's surface (or at least until the Sun's atmosphere, solar flares, etc, start slowing the Earth down). The only way this question would make sense is if the Earth kept it's current velocity while increasing the Sun's mass.

If it means to destroy life on Earth, then you first have to figure out how high the temperature has to get in order for the Earth to be uninhabitable - and is that uninhabitable by life or uninhabitable by human life?

Saying an average temperature of 373 degrees Kelvin (100 deg C) would at least be reasonable, except I'm not positive that would really be accurate (you'd have to research life existing near fissures on the ocean floor, etc since I'm not sure what their environment is - just that it's pretty extreme). Since intensity of light (and the energy from the Sun) is inversely proportional to the square of the distance, you could calculate how close the Earth's perigee would have to get to the Sun to boil all life periodically (average temperature on Earth is about 288 deg K, or about 15 deg C).

If you're talking about human life, then it would take a much smaller temperature difference to upset the ecosystem and pretty much change life on Earth as we know it even if it doesn't destroy all life.

Or, is the question assuming the density of the Sun stays as it is and the increase in mass also results in an increase in volume, thereby reducing the distance between the Earth and Sun by that means? And, once again, does it mean to literally destroy the Earth or to destroy the Earth as a life sustaining planet?

Or, the question is asking about the Roche limit, as Dave suggested.

 

[DaveC426913]

If it means to literally destroy the Earth, the Earth's orbit has to change essentially until perigee reaches the Sun's surface (or at least until the Sun's atmosphere, solar flares, etc, start slowing the Earth down).

No. As I said, the Earth will break up once inside the Sun's Roche limit, which is several solar radii above the sun.