1. Modelling the Pioneer Anomaly as Modified Inertia. Mike McCulloch, Ocean modelling, Met Office, UK. Edinburgh, 20 th April 2006 Why modify inertia? Look at a possible cause of inertia Show how this cause could fail at low accelerations Derive the implied expression for inertial mass Show it forecasts the Pioneer anomaly when r > 15 AU Discuss problems with orbital motion

2. Is the Pioneer Anomaly a gravity or inertia problem? If the a’ is new physics it could mean that: 1.G is stronger than expected at long distances or low accelerations 2.The inertial mass is lower at long distances or low accelerations. a’ http://www.nineplanets.org/overview.html 1972/3 Anderson et al (1998) a’=8.7*10 -10 ms -2

3. What should a modification of inertia look like? To fit galaxy curves Milgrom (1983) derived empirical correction for Newton’s 2 nd law (MOND): when accelerations are familiar then μ =1 when accelerations ~ 1.2*10 -10 ms -2 then μ α a/1.2*10 -10 Aim: find a theory that produces a function like μ..

4. A possible model for inertia: Hawking & Unruh radiation Hawking (1973) Unruh (1974). Alokik Kanwal

5. A break in the response of the vacuum. Haisch, Rueda & Puthoff (1994): Unruh radiation could cause inertial drag. Milgrom (1999): At very low accelerations, these wavelengths might be too large to fit into the Hubble distance. There should then be a break in the response of the vacuum.

6. Can Milgrom’s break account for the Pioneer Anomaly? Acceleration of Pioneer is still larger than the cut-off. So Pioneer should be unaffected… At this acceleration the Unruh spectrum And inertial mass might disappear. Feedback…minimum acceleration m i α the total energy in the spectrum λ (m) Disallowed High acceleration Low acceleration Tiny acceleration =1.4*10 -10 m/s 2 a 0 =1.2*10 -10 m/s 2 H=2.3*10 -18

7. A more gradual break in the response of the vacuum (new). Since E’ = E when λ > Hubble distance E’ is zero when λ approaches zero. λ (m) High acceleration Lower acceleration Tiny acceleration Pioneer Only wavelengths of Unruh radiation that fit exactly into 2c/H are allowed. m i α the total energy in the spectrum

8. Modelling Pioneer with & without Modified Inertia: Results Outside ~15 AU, the Pioneer Anomaly is predicted without any adjustable parameters (although depends on choice of Λ) Simulated Pioneer’s trajectory with & without the new term, v 0 =20,000 m/s. OK Not OK *10 -10

9. Hawking (1973)’s expression for a black hole’s temperature The theory predicts a maximum black hole mass of 10 23 solar masses. Also implies there is a minimum allowed acceleration for all bodies. A prediction: maximum black hole mass. Using Wien’s law again: As before, assume Hawking radiation above a limiting λ is not allowed: M < 10 53 kg

10. Conclusions Assuming that inertia is caused by Unruh radiation, and is quantised, it is possible to predict the Pioneer Anomaly (for radii > 15 AU) without any adjustable parameters. These ideas could provide a physical reason for MOND? Theory also predicts a maximum mass of 10 23 solar masses for black holes. Theory predicts all bodies have a minimum possible acceleration. However: The anomalous acceleration close the sun, and in galaxies, is overestimated by a factor of about five. A possible reason is that Unruh’s equation is only valid for linear acceleration. An Unruh equation valid for circular motion would be very helpful.