Modelling the Pioneer Anomaly as Modified Inertia.

Mike McCulloch,

Ocean modelling, Met Office, UK.

Edinburgh, 20th 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

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 accelerations2. The inertial mass is lower at long distances or low accelerations.

2

g

i i

GMmFa

m m r

a’

http://www.nineplanets.org/overview.html

1972/3

Anderson et al (1998)a’=8.7*10-10 ms-2

What should a modification of inertia look like?

To fit galaxy curves Milgrom (1983) derivedempirical correction for Newton’s 2nd law (MOND):

0( )ai i aF m a m a

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..

A possible model for inertia: Hawking & Unruh radiation

Hawking (1973) Unruh (1974).

2

aT

ck

Alokik KanwalAlokik Kanwal

3

8

cT

GMk

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.

2

aT

ck

m

W

T

2m

Wck

a

Can Milgrom’s break account for the Pioneer Anomaly?

m

W

T

Acceleration of Pioneer is still larger than the cut-off.So Pioneer should be unaffected…

10 22~10 /

m

Wcka m s

2

aT

ck

At this acceleration the Unruh spectrumAnd inertial mass might disappear.Feedback…minimum acceleration

mi α the totalenergy in thespectrum

λ (m)

DisallowedHigh acceleration

Low acceleration

Tiny acceleration

2c

H

=1.4*10-10 m/s2

a0=1.2*10-10 m/s2

H=2.3*10-18

A more gradual break in the response of the vacuum (new).

' 1 14 2m Wck

E E Ea

Since E’ = E when λ > Hubble distanceE’ is zero when λ approaches zero.

λ (m)

2c

H

High acceleration

Lower acceleration

Tiny acceleration

Pioneer

Only wavelengths of Unruh radiation that fit exactly into 2c/H are allowed.

' 12i i

Wckm m

a

mi α the totalenergy in thespectrum

Modelling Pioneer with & without Modified Inertia: Results

Outside ~15 AU, the Pioneer Anomaly is predictedwithout any adjustable parameters (although depends on choice of Λ)

Anomalous acceleration towards the sun (ms-2)

0

2

4

6

8

10

5 10 15 20 25 30

Distance from the sun (AUs)

An

om

alo

us

Ac

ce

lera

tio

n

(m/s

2)

Observed (Anderson et al., 1983) Predicted

' 12i i

Wckm m

a

Simulated Pioneer’s trajectory with & without the new term, v0=20,000 m/s.

OKNot OK

*10-10

Hawking (1973)’s expression for a black hole’s temperature

The theory predicts a maximum black hole mass of 1023 solar masses.

Also implies there is a minimum allowed acceleration for all bodies.

3

8

cT

GMk

A prediction: maximum black hole mass.

Using Wien’s law again:

3

8m

GMkW

c

As before, assume Hawking radiation above a limiting λ is not allowed:

3

2 8c GMkW

H c

M < 1053 kg

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 1023 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.