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This is inspired by Evidence that the Solar System is expanding like the Universe?, which referenced an article by G. A. Krasinsky and V. A. Brumberg, "Secular Increase of Astronomical Unit from Analysis of the Major Planet Motions, and its Interpretation," pdf available here.

The article seeks to explain the increasing Earth-Sun separation (orbit-averaged separation, as in semi-major axis), and a large part of it is focused on the expansion of the universe. Now Ben Crowell gave a great explanation here as to why the effect of cosmic expansion only comes in with the third time-derivative of the scale factor, noting that the effect is "undetectably small."

So then I wonder: What is the rate at which the Earth-Sun distance is changing? Krasinsky and Brumberg cite something like $15\ \mathrm{cm/yr}$, but given the number of wrong calculations they do elsewhere, I don't entirely trust this claim. Who has done this analysis, and what do they find? Also, what methods are used? I would guess that measuring the precise distance to the Sun's surface would be challenging, due to all the activity it undergoes.


Note that discussion of cosmic expansion was background motivation for what sparked this question - I fully understand that any secular change in the Earth's orbit will be dominated by other effects.

Also note there is a related question on this site: How is distance between sun and earth calculated? However, that does not go into detail about the method. Moreover, it is not clear that the answers provide work for measuring small changes in the separation. For example is radar really that precise? Perhaps it is, in which case justification of that would serve as an answer here.

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  • $\begingroup$ Do you mean average distance (equivalently semi-major axis) or instantaneous distance? The Earth's orbit is not a perfect circle so the distance changes measurably on an annual basis. I forget the numbers, just look up "Earth orbital elements" if you want to see how much it varies. A long term secular change in the orbit, of the kind you probably mean, would manifest as changes in the semi-major axis and/or eccentricity. $\endgroup$ – Michael Brown Jul 20 '13 at 2:04
  • $\begingroup$ Re previous comment: I didn't even look at who posted this. Of course you know about eccentricity! Please don't be offended. Just the wording of the question was a little less than precise on that point, so maybe the comment would be useful to someone else someday. :) $\endgroup$ – Michael Brown Jul 20 '13 at 2:49
  • $\begingroup$ @MichaelBrown No problem ;) Eccentricity would be interesting to know about too, especially given that I suspect it can change on shorter timescales. $\endgroup$ – user10851 Jul 20 '13 at 2:57
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    $\begingroup$ Potentially relevant: what fraction of sun's mass gets ejected annually beyond earth's orbit. $\endgroup$ – Johannes Jul 20 '13 at 5:38
  • $\begingroup$ The Krasinsky & Brumberg paper has 59 citations on ADS. That should be a good starting point. $\endgroup$ – Pulsar Jul 20 '13 at 11:54
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According to E. V. Petjeva (2011), the measured rate of change of the Earth-Sun distance (astronomical unit) is (1.2 +/- 3.2) cm/yr, with the uncertainty value representing 3 standard deviations. In other words, any change is within the uncertainty of the measurement. She specifically addresses the Krasinsky and Brumberg value. E. M. Standish has also addressed this issue.

The measurements are by radar echos off other planets and radio signals from space craft. See the below references for details.

(Note that "astronomical unit" is not technically the same as Earth-Sun distance, see Standish reference for details, Krasinsky and the others are all measuring the "astronomical unit"; also "astronomical unit" was redefined to be a constant in 2012, see Nature reference for details)

http://syrte.obspm.fr/jsr/journees2011/pdf/pitjeva.pdf

http://adsabs.harvard.edu/abs/2005IAUCo.196..163S

http://www.nature.com/news/the-astronomical-unit-gets-fixed-1.11416

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    $\begingroup$ Yes! Thank you! I feared no one would ever post an answer. Also, welcome to the site! $\endgroup$ – user10851 Feb 26 '14 at 1:34
  • $\begingroup$ Read the article again, @DavePhD. It specifically states in table 3 that the rate of change of the Earth-Sun distance (specifically, the Earth-Sun semimajor axis length) is $\dot a/a = (1.35\pm 0.32)·10^{−14}/\text{century}$, or $\dot a = 20.2\pm4.8$ μm/yr. The astronomical unit was effectively divorced from the Earth-Sun semimajor axis length in the late 19th century and officially divorced from it in 1976. $\endgroup$ – David Hammen Aug 10 '16 at 7:24
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This is inspired by ... an article by G. A. Krasinsky and V. A. Brumberg, "Secular Increase of Astronomical Unit from Analysis of the Major Planet Motions, and its Interpretation."

Pitjeva and Pitjev (1) provide a simple explanation of the very large secular change in the astronomical unit found by Krasinsky and Brumberg:

"In the paper by Krasinsky and Brumberg the au change was determined simultaneously with all other parameters, specifically, with the orbital elements of planets and the value of the au astronomical unit itself. However, at present it is impossible to determine simultaneously two parameters: the value of the astronomical unit, and its change. In this case, the correlation between au and its change $\dot{au}$ reaches 98.1 %, and leads to incorrect values of both of these parameters".


What is the rate at which the Earth-Sun distance is changing?

The first article cited in DavePhD's answer by E.V. Pitjeva is based on the refereed article by Pitjeva and Pitjev (1). Both articles provide rates of change of the Earth-Sun distance (specifically, the Earth-Sun semimajor axis length $a$) and of the 1976 definition of the astronomical unit $au$ as $$\begin{aligned} \frac{\dot a} a &= (1.35\pm0.32)\cdot10^{-14}/\text{century} \\ \frac{\dot{au}}{au} &= (8\pm21)\cdot10^{-12}/\text{century} \end{aligned}$$ (Note: the latter is based on the published value of $\dot{au} = 1.2\pm 3.2$ cm/yr.)

The reason for the nearly three order of magnitude difference between these two figures is that the astronomical unit is not the distance between the Sun and the Earth. While that is how the astronomical unit was originally defined, the two concepts have been effectively divorced from one another since the end of the 19th century, when Simon Newcomb published his Tables of the Motion of the Earth on its Axis and Around the Sun. The divorce was made official in 1976 when the International Astronomical Unit redefined the astronomical unit to be the unit of length that made the Gaussian gravitational constant k have a numerical value of 0.017202098950000 when expressed in the astronomical system of units (the unit of length is one astronomical unit, the unit of mass is one solar mass, and the unit of time is 86400 seconds (one day)).


Who has done this analysis, and what do they find? Also, what methods are used?

There are three key groups:

  • The Institute of Applied Astronomy of the Russian Academy of Sciences, which produces the Ephemerides of Planets and the Moon series (EPMxxxx) of ephemerides (1);
  • The Jet Propulsion Laboratory of NASA, which produces the Development Ephemeris series (DExxx) of ephemerides (2), and also ephemerides for small solar system bodies; and
  • The Institute of Celestial Mechanics and of the Calculation of Ephemerides of the Paris Observatory (L'institut de mécanique céleste et de calcul des éphémérides, IMCCE), which produces the Integration Numerique Planetaire de l’Observatoire de Paris series (INPOPxx) of ephemerides (3).

All three numerically solve the equations of motion for the solar system using a first order post Newtonian expansion given a set of states at some epoch time. This integration of course will not match the several hundred thousands of observations that have been collected over time. All three use highly-specialized regression techniques to update the epoch states so as to somehow minimize the errors between estimates and observations. All three carefully address highly correlated state elements, something that Krasinsky and Brumberg did not do. All three share observational data, sometimes cooperate (joint papers, IAU committees, ...), and sometimes compete ("our technique is better than yours (at least for now)").


For example is radar really that precise?

Regarding radar, the distance to the Sun was never measured directly via radar. Unless massively protected with filters, pointing a telescope of any sort directly at the Sun is generally a bad idea. If massively protected with filters, a radio antenna would not see the weak radar return. Those 1960s radar measurements were of Mercury, Venus, and Mars. There's no compelling reason to ping those planets now that humanity has sent artificial satellites in orbit about those planets. Sending an artificial satellite into orbit about a planet (as opposed to flying by it) provides significantly higher quality measurements than do radar pings.


References:

  1. E. V. Pitjeva and N. P. Pitjev, "Changes in the Sun’s mass and gravitational constant estimated using modern observations of planets and spacecraft," Solar System Research 46.1 (2012): 78-87.

  2. E. Myles and Standish and James G. Williams, "Orbital ephemerides of the Sun, Moon, and planets," *Explanatory Supplement to the Astronomical Almanac (2012): 305-346.

  3. A. Fienga, et al. "INPOP: evolution, applications, and perspectives," Proceedings of the International Astronomical Union 10.H16 (2012): 217-218.

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