One possible indication of this is by using Milgrom’s MOND’s a = (GM a_o/ r^2)^1/2 and modifying it for our solar system, but to drop Newton’s assumption of a constant universal G, and give it a variable value instead. For example, if we ‘assume’ a variable G at the rate of 1G per 1AU with distance from the Sun (at present unsubstantiated empirically), we get an approximation of the Pioneer Anomaly, as follows:
-a = (GM a_o/ r^2)^1/2, which becomes modified with 1G per 1AU as:
-a = [G(AUn)M a_o/ r(AUr)]^1/2, where AUn is the number of AU distance, and AUr is the distance r for one AU, so with numbers, for Earth’s orbital:
-a = [(6.67E-11)(1)(1.98E+30)(1.2E-10) / (1.5E+11)(1.5E+11)]^1/2, gives us a value of:
-a = (15.8479E+9 / 2.25E+22)^1/2 = (70.435E-14)^1/2
-a = 8.3934E-7 m/s^2, which is three orders of magnitude greater than Pioneer’s –a = ~8E-10 m/s^2, too far out of ball park.
The same calculation for any distance in AU will yield the same result, i.e., at Saturn’s 9.5AU, where r = 1.429E+12 m, gets nearly same result, viz. –a = 8.38E-7 m/s^2
However, what Milgrom calculated for the outer galaxy flat rotation curves may not be the same as what is operable within the limits of our solar system, so that a ‘gentler’ MOND effect may be the case here, which can be calculated as follows, solving for a_os within our solar system:
-a = = [G(AUn)M a_o/ r(AUr)]^1/2, and plugging in known values for Pioneer Anomaly:
-8E-10 m/s^2 = [(6.67E-11)(1)(1.98E+30)(a_o) / 2.25E+22]^1/2, and solving for our solar system’s a_os we get:
-8E-10 m/s^2 = [13.2066E+19)(a_os) / 2.25E+22 ]^1/2
a_os = 1.0908E-16 m/s^2
, for our solar system, which is a far lower, gentler value for our solar system then what was computed for the outer galaxy curves, viz. a_o = 1.2 E-10 m/s^2.