Why does the Earth move away from the Sun?

Question

From the (Wikipedia's) definition of the astronomical unit $AU$, we have that it is defined as:

$AU=\sqrt[3]{\dfrac{GMD^2}{k^2}}$

Where $k\approx0.01720209895$ is Gauss' gravitational constant, $G$ is Newton's gravitational constant, $M$ is the mass of the Sun and $D$ is the time period of one day. Therefore, according to several observations, pointing to an increasing astronomical unit at a tremendous pace (around $15$ cm a year), this means that the Sun is gaining approximately $6\times10^8$ kg per year (taking into account Sun's mass year losses).

Questions: (1) first of all, why does the distance from the Sun increase with Sun's mass augment? Shouldn't the gravitational attraction be stronger and attract the Earth more rapidly, instead of getting it away? And (2) how is this increment in the $AU$ explained? $6\times10^8$ kg per year seems unphysical.

Answer 1

The source of the confusion is contained in Gauss's gravitational constant $k$. It's defined to be such that the equation you gave is true. So the definition is really giving nothing more than a conversion between $AU$ and $k$. You can't use this equation to calculate how far away earth will be when the mass of the sun changes or what the mass of the sun is given how far away the earth is; both $k$ and $AU$ would just be updated accordingly so that they both still have the relation given by your definition.

In actuality, $AU$ is now defined to be a specific distance that is not directly tied to the distance of the earth from the sun, even though of course it is approximately equal to it, and $k$ has been recommended to be discontinued in its use.