solar system - What was the length of year 1 million years back?

(Disclaimer: As I already pointed out in a comment to the question above, I never did a calculation with $H_0$ before and I might be utterly, horrible wrong with my interpretation.)

If you completely ignore the slowly changing orbit of earth and only take expansion of space into account and assume the Hubble-parameter to be pretty constant in the timeframe of 1 My, we can calculate the difference of the orbital period of earth using Keppler's third law [3]:

$T = 2pisqrt(a^3/GM)$

for

$a = 1.4959789*10^{11} m$ (semi-major axis of earth today) [1]
$G = 6.67*10^{-11} Nm^2/kg^2$ (gravitational constant)
$M = 1.988435*10^{30} kg$ (mass sun) [1]

We also assume: $H_0 = 2.3*10^{-18} s^-1$ [2] (Hubble parameter then and today in SI-units) which basically means "in every second a meter get $2.3*10^{-18} m$ longer".

Instead of taking the length of an (siderial) orbital period of earth from some source, let's calculate it manually first and take it as a reference.

$T_{today} = 2 pi sqrt((1.4959789*10^{11}m)^3/(6.67*10^{-11} Nm^2/kg^2 * 1.988435*10^{30} kg))$ = 365 days 8 hours 56 minutes 13.45 seconds

Pretty close and a good reference for more calculations.

Now, what was earth's semi-major axis 1 million years ago, only taking into account a constant $H_0$?

$x - (2.3*10^{-18} s^-1 * 1 My * x) = 1.4959789*10^{11} m$
Solving for $x$ leads to $x = 1.49598*10^{11} m$.

(Sorry for the lousy precision; I only have Wolfram Alpha at my hands right now.)

The old semi-major axis is a little smaller. Using Keppler's law again we can calculate the orbital period again:

$T_{old} = 2 pi sqrt((1.496*10^{11} m)^3/(6.67*10^{-11} Nm^2/kg^2 * 1.988435*10^{30} kg))$ = 365 days 8 hours 56 minutes 48.26 seconds

So, subtracting both times from another we can say that 1 My ago the year was indeed 34.81 seconds shorter.

However. This probably doesn't mean much; the orbit changes slightly over time anyway; the Hubble-parameter is not considered a constant any more, it changes slightly over time; and while this was an interesting question I don't trust my interpretation much and hope that someone else who's more qualified than me could enlighten the question better than I ever could.

(I hope I didn't botch anything somewhere. I need more coffee.)

[1] Source: Wolfram Alpha
[2] Source for Hubble-parameter in SI-units taken from the German Wikipedia: http://de.wikipedia.org/wiki/Hubble-Konstante#Definition
[3] http://en.wikipedia.org/wiki/Orbital_period#Small_body_orbiting_a_central_body