# Either the Sun Is Getting Smaller or Gravity Is Getting Weaker

**Paper discussion**: Solar system expansion and strong equivalence principle as seen by the NASA MESSENGER mission. Antonio Genova, Erwan Mazarico, Sander Goossens, Frank G. Lemoine, Gregory A. Neumann, David E. Smith & Maria T. Zuber. Nature Communications volume 9, Article number: 289.

Students of physics learn some interesting facts about the sun, spread over different lessons and different classes. When learning about orbital mechanics, students are taught that gravitational field of sun exerts a long-range force proportional to its mass and causes planets to orbit in an elliptical path around it. When encountering “modern physics” for the first time, students are taught the sun in its interior is “converting mass to energy,” maintaining its temperature and pressure by fusion of hydrogen into helium and producing energy that is eventually radiated from its surface and eventually felt by us on the Earth. An astute student may ask, if the sun is decreasing in mass to produce and radiate energy, is its gravitational influence on the planets getting weaker? For many years this has been handwaved away as a negligibly small effect, but now a probe orbiting the planet Mercury has made the first measurements of the sun’s loss of mass.

Before we discuss this recent finding, let’s estimate what kind of effect this would have. The sun’s luminosity is on the order of ##10^{26}## Watts. If we apply Einstein’s relation and divide this by ##c^2## this luminosity is consistent with a rate of mass loss of a million tons per second, which through energy conservation can be thought of as the rate that “mass is converted to energy” by fusion. For context, this is a reduction of about twice the mass of the largest ship ever built every second. The fusion-radiation process is not the only way that the sun loses mass; there is also the protonic outflux that forms the solar wind, as well as more coherent losses such as coronal mass ejections. The sun, however, is very big, and this is a few parts per quadrillion per year. From Kepler’s laws we know that with a given axis, the orbital period is proportional to the reciprocal square-root off the central mass, and even after billions of years we would not expect a noticeable difference in the orbit. Measuring something at the parts per trillion scale is extremely difficult, so it would have been a safe bet to assume that the gravitational effect of the sun’s shrinkage would never be observed.

Mercury is a well-known participant in the sun’s gravitational antics. Its over-eager perihelion precession of an extra 43 arc-seconds per century was a mystery at the end of the nineteenth century and was famously figured-out by Einstein with the general theory of relativity. From 2011 to 2015, Mercury had a visitor from Earth, an orbiting probe called “Mercury Surface, Space Environment, Geochemistry, and Ranging” which totally randomly and coincidentally spells out **MESSENGER**. Its goal was to make measurements of Mercury’s surface topography and gravitational field to determine its internal structure. During that time, it gave us some stunning images of our smaller sibling.

A number of papers were published by the MESSENGER collaboration about, as I mentioned, Mercury’s surface and interior, e.g. here and here and here. The probe was deliberately crashed into the surface of the planet on April 30, 2015, but unfortunately did so on the side that was not facing Earth, so we couldn’t watch. Since then, the various members of the collaboration have been busy analyzing all the data and learning the most they can. In their recent paper in Nature Communications (Nature’s journal for freely-readable papers), they used seven years of data to measure various parameters related to general relativity and its potential violations. This was done by carefully tracking the position of the probe with respect to Mercury, to see how well its orbit corresponded to what we know about gravity.

One of these violations is the Nordtvedt effect, which would exist if gravity gravitated differently from mass (meaning, if the contribution to the mass of a body due to its gravitational binding energy coupled differently to an external gravitational field). This would violate the equivalence principle, and objects would orbit differently depending on their internal composition. If this were real but small enough to have eluded previous measurements, it would have lead to a discrepancy in the relative position of Earth and Mercury by about three meters. MESSENGER data showed that the strength of this effect is consistent with zero, constraining it to less than one part in ten-thousand. While we already knew it was consistent with zero by tracking the distance to the moon (using the reflectors left there by Apollo astronauts), now we know it’s consistent with to zero with like three times the precision.

The most interesting result in the paper, in my opinion, was the measured decrease in the sun’s gravitational parameter. As readers may be aware, the equations of orbital mechanics all have the product of the mass and Newton’s gravitational constant, together GM. Each parameter cannot be separately analyzed by examining orbits. This is why Cavendish measured the deflection of a pendulum between giant lead spheres to separately measure G and Earth’s mass in 1798. By very precisely tracking the distance to MESSENGER from Earth, the collaboration was able to measure the change in the Earth-Mercury distance over the several years of observation. This amounted to about two meters, and this was the first nonzero measurement of the changing gravitational parameter of the sun (keeping in mind the distance between them is at least 77,000,000,000 meters). They found that the sun’s gravity was getting proportionally weaker by ##6.13 \pm 1.47 \times 10^{-14}## per year (61 parts per quadrillion), and thus the solar system is slowly expanding as well.

There are two potential causes for this decreasing gravity: the M of GM is changing and the mass of the sun is decreasing as it fuses, radiates, and spews, or the G of GM is changing and the whole universe’s gravity is getting weaker. To separate these two, we can consider how the measured value compares to our expectations. Solar physicists can make a more precise estimate than my “luminosity over c-squared” calculation, and the expected mass loss rate due to fusion is about 68 parts per quadrillion per year, and the mass loss due to the solar wind was estimated at 10 parts per quadrillion per year (this was harder to pin down due to the fact that the observation window didn’t span a full sunspot cycle). Together, the predicted mass loss rate is consistent with the measured decreasing gravitational parameter given its uncertainty.

The curious skeptic might ask how we know that the laws of physics are unchanging over time. To answer that, we can examine what the effect would be *if* they varied over time, and go look for those effects. One possibility is that the strength of gravity changes over time, and looking for a time-varying Newton’s constant would verify or constrain that. Given our confidence in the physics of the sun and the measurements from MESSENGER, they were able to use the data to constrain this to below 40 parts per quadrillion per year and consistent with zero. There have been previous attempts to look for a changing gravitational constant, also consistent with zero, but less constraining. Before this study, the best estimate was based on long-term observation of binary neutron stars, constraining it to about 600 parts per quadrillion per year. So, we still have no evidence that gravity is changing (my intuition says it isn’t), and now we know that if it is, it decrease proportionally like 400 times slower than the universe is expanding.

So, now we have empirical evidence that the sun is decreasing in mass (even though it is very slowly increasing in volume as well), and that it leads to a change in the orbital axis of Mercury consistent with a few meters over a few years.

Postdoctoral Research Associate at Massachusetts Institute of Technology (MIT)

Very interesting. There's a minor LaTeX error after "77,000,000,000".

And last but not least a good reference to quote on the FAQ "Do constants change over time or space?"

To one point: The Solar System — the orbital radii of the planets and other bodies — could be getting larger as the Sun's mass slowly shrinks. However, it could also be getting

smaller, due to friction from the solar wind. (Also radiation pressure, in the case of tiny bodies, especially dust.) The getting-larger effect should apply equally to all bodies, and the increase should be a simple proportionality with distance. But the getting-smaller effect(s) should be greater at smaller solar distances and for less dense bodies. Which effect is dominant, and for which bodies?Really enjoyed this Insight!

There is also one at "10[sup]2[/sup]6 Watts"

> Together, the predicted mass loss rate is consistent with the measured decreasing gravitational parameter given its uncertainty.

So… we expect a mass loss, and we see the effect of the mass loss in the expected amount. Nothing surprising, unlike the topic would suggest.

Some would hope the Earth's gravity is getting weaker. It would certainly make people feel better after the end of year holiday dinners. :biggrin:

Well isn't the momentum of the solar wind directed, for the most part, directly away from the sun? So it would contribute to an expansion overall, yes? Perhaps when the charged particles interact with magnetic fields of (some) planetary bodies and they're deflected? But still seems like an "outward" directed force.

A uniform outwards force does not increase orbits, it gives them a different period. It acts like a slightly lower (but constant) solar mass. What is left is some sort of friction, which decreases the orbital radius of objects over time.

A uniform outwards force does not increase orbits, it gives them a different period. It acts like a slightly lower (but constant) solar mass. What is left is some sort of friction, which decreases the orbital radius of objects over time.

Isn't sixty parts per quadrillion decrease in gravity per year smaller than the uncertainty in the gravitational constant given by CODATA?

Yes. So what?

What astronomers measure is not the gravitational constant, but the product of the gravitational constant and the mass of objects, GM. This product can be measured much more precisely than either G or M.

Yes. So in principle, the title of this thread is misleading: The product is decreasing, but (a) it could be either or (b) it could be both or (c) one could be increasing slightly but the other decreasing more.

I'm curious if they took into account the momentum of the EM field (or photons) created by the Sun exerted on the planets (and the black body radiation of the planets and cosmic background blackbody radiation). By intuition, even though Mercury is a small planet, it's very close to the Sun and might be the most affected to it.

This is covered by @mfb's post #8 here (even though that was stated in the context of the solar wind).

It is very difficult to draw any meaningful conclusions about individual factor variance based on the product of individual factors across cosmological time scales.

Does this mean that if the Sun was, say, so shiny that the outward force more than counteract the gravitational force between the Earth and the Sun, the Earth would still have its orbit intact?

I do not understand how the Earth could stay with the same orbit if the photons hitting them from the Sun were so much more numerous than they are now that the outward force would be so great that there should not be any stable orbit. Am I missing something?

No, of course not. These particles are not in an orbit at all. That only affects particles smaller than ~500 nm, however.

Has anyone calculated the loss of energy from the gravitational waves emitted by the Sun and Jupiter, or the Sun and the other gas giants?

For the Sun and Jupiter,

m1 = 1.989 × 10^30 kg (Sun) ~ 2 × 10^30 kg

m2 = 1.898 × 10^27 kg (Jupiter) = 0.0009543 m1 ~ 2 × 10^27 kg

R = 778.5 million km = 7.785 x 10^11 m ~ 8 x 10^11 m

The best reference I could find with a quick Google search was:

http://www.physics.usu.edu/Wheeler/GenRel2013/Notes/GravitationalWaves.pdf

but I'm not familiar enough with the formulas to be confident that I'm applying them correctly.

About 200 W for Earth/Sun, a bit more for Jupiter. Wikipedia has formulas where you can just plug in masses, distances and eccentricities (use e=0 for a good approximation).

Some of you likely already know some of the following, but it was a very interesting point, about the Sun, when I heard it.

Jupiter is massive, and so is the Sun. Given the mass of Jupiter it has great total gravitational force. That force also is exerted noticeably on the Sun itself.

Due to the gravitational force of Jupiter it actually affects the Sun's position in our solar system, such that the Sun is not in a constant epicenter where all the planets form their orbits around that constant point. Rather, Jupiter's gravity pulls on the Sun such that the Sun also has an "orbit", and it's orbit is around an imaginary point that the sun orbits around. So, the Sun itself is also pulled towards Jupiter as Jupiter is pulled towards the sun, and their respective gravity creates both Jupiters's orbit as well as giving the Sun an "orbit" that is around the "center" of the solar system.

Very cool. And it demonstrates the massive gravitational power of Jupiter.

Question:

As the Sun's mass lessens, and with that it's gravitation force lessens, then at what point in the future will Jupiter's gravity have a greater effect on the planets between it and the Sun? IOW, at some future point, will Jupiter's gravitational effect become stronger on the Earth as the Sun's gravitational effect becomes lesser? And, when, or if, that happens, will the Earth's orbit change? Will the effect of Jupiter's gravitational pull slow the Earth's orbit around the sun? Or, can it create an effect of a slow orbit during certain Earth to Jupiter orbiting positions, and then speed up the orbit during different relative positions?

This of course is predicated on the idea that the Sun's mass is lessened due mainly to fusion and mass coronal ejections, while Jupiter's mass remains constant over this same period.

This is true of all the planets, not just Jupiter. Jupiter's effect is by far the largest, but I don't think it's the only detectable one. The strictly correct statement is that all of the objects in the Solar system orbit its barycenter, which can be thought of as the "center of mass" of the system as a whole, and this barycenter does not coincide with the center of any of the objects, including the Sun.

Never. The rate of mass loss of the Sun is much, much too small. The sun is over 1100 times as massive as Jupiter, and the mass loss rate is roughly 6 parts in ##10^{14}## per year, so even after 5 billion years more, when the Sun is expected to become a red giant, the Sun will have lost only about 1 part in 100,000 of its current mass, so it will still be over 1100 times as massive as Jupiter.