Expert: Philip Stahl - 6/15/2008

Question
How has the Earth's orbital period changed over time? I assumed it has changed due to the Sun losing mass each day and perhaps other factors. How much is it changing per year or other unit of time?

Answer
Hello,

The answer here is a minscule amount, so minsucule it would be totally unnoticeable over human generations.

Using a summary form of the basic fusion reaction in the Sun:

H1  + H1  + H1 + H1  ->  He 4 + Energy

one can apply Einstein's mass-energy eqn. (E = mc^2) and find that the Sun loses approximately 4.2 x 10^9 kg of mass each second (that is, 4.2 billion kilograms). This sounds like an enormous amount of mass, but not in relation to the Sun's total mass of 2 x 10^30 kg!

Taking that fraction, one finds:

f =  (4.2 x 10^9 kg)/  (2 x 10^30 kg) = 2.1 x 10^-21


Even over one billion years (e.g. considering the mass associated with the loss per second), the fraction lost would be barely:

6.5  x 10^-5

or 6.5 parts in 100,000

What does this translate into in terms of actual orbital changes?

Bearing in mind that the force of gravitational attraction

F =  GMm/ r^2

where r is the distance between Sun and Earth, M is the mass of the Sun and m the mass of the Earth, we see that if M decreases then r must increase and F must decrease since:

r =   [GMm/ F]^1/2

Specifically, if one does the calculations one finds that over a period of 100 years, r will increase by 1 meter (3.3 ft.) and thus the orbital period will increase by about:  delta T = 0.004 sec.

In other words, about 4 milliseconds. That is 4 milliseconds over one hundred years!

Not enough to even bother about!