How fast is the Earth-Sun distance changing
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
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:
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.
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.
A. Fienga, et al. "INPOP: evolution, applications, and perspectives," Proceedings of the International Astronomical Union 10.H16 (2012): 217-218.