The Juno flyby

On October the 9th, 2013 the Juno spacecraft en route to Jupiter will fly by the Earth for a gravity assist. Rather like Marty McFly in Back to the Future, who hitched a skateboard ride by holding on to a truck, Juno will hitch a ride for a short time behind the Earth in its orbit. The Earth's gravity will pull it along and speed it up. This is how NASA manage to get spacecraft to Jupiter without the need to launch lots of heavy fuel.

The particularly interesting aspect of this flyby for me is that the spacecraft will approach Earth at a declination of -14 degrees (near the equator) and leave it at +39 degrees (a bit closer to the spin axis). As a result of this, MiHsC predicts a slight jump in speed, beyond the usual one from the gravity assist. This is because as the spacecraft comes in near the equator the acceleration vector of the craft and the acceleration vector of all the masses in the spinning Earth (which point at the spin axis) are pointing at or away from each other, so the mutual accelerations are large, the Unruh waves seen are short and a lower proportion of them are disallowed by the Hubble-scale Casimir effect, so MiHsC doesn't reduce the inertial mass of the craft much. However, when the spacecraft leaves at an angle closer to the Pole (the spin axis) the acceleration vectors of the masses in the spinning Earth are still pointing at the Earth's spin axis and so are now not pointing at the craft, so the mutual Earth-craft accelerations are lower, the Unruh waves lengthen and a greater proportion are disallowed by the Hubble-scale Casimir effect and MiHsC decreases the inertial mass of the spacecraft. To conserve momentum (mv) the craft has to speed up slightly.

Anomalous speed ups like this have been seen in previous flybys (they were first noticed by Antreasian and Guinn, 1998 and Anderson et al., 2008, see also the ISSI Flyby Workshops, 2009, 2010) and MiHsC, as described above, predicts them fairly well: an agreement that is not perfect (follow the link below to see my paper), but is encouraging given there are no adjustable parameters in MiHsC. Of course, you could to some extent use past flybys to predict the next anomaly: the upcoming Juno one will be similar geometrically to the first Galileo flyby which had an anomaly of 4.2 mm/s, but it's not identical and it's surely better to have a theory.. The paper I published on this is freely available from MNRAS here. The specific formula for the predicted anomalous velocity jump (flyby anomaly, or dv) (Eq. 8 in the above paper) is:

dv = 2.8*10^-7*(v2*cos(dec1)-v1*cos(dec2))/(cos(dec1)*cos(dec2))

where the first factor is not adjustable, and depends only on observed and fixed parameters such as the rotation rate of the Earth, the speed of light, the Hubble diameter and the variation of density towards the Earth's core. Dec1 is the incoming declination of the craft (like its latitude in the sky), dec2 is its outgoing declination and v1 and v2 are the initial and final geocentric velocities (in the paper I wrongly used heliocentric ones).

In deriving this formula I made certain geometric assumptions, that seemed reasonable at the time and probably have only a small effect, but with hindsight could be improved on and I'll try to correct those before October (it's not trivial), but into this formula I can put the values for the upcoming Juno flyby that I have obtained from the indispensible JPL HORIZONS website: dec1 = -14 degrees, dec2 = 39 degrees, v1 ~ 10,500 m/s and v2 ~ 10,500 m/s. Therefore the flyby anomaly predicted by MiHsC for Juno on October 9th is: dv ~ +0.75 mm/s ( a speed up) (using heliocentric velocities, of 35000 m/s, as in my previous paper, the prediction is 2.9 mm/s).

Note: the observed Juno flyby appears to be zero, although nothing has been publish to confirm this yet. This certainly puts the cat among the pigeons since it contradicts the pattern seen in the others..

References

Anderson, J.D., J.K. Campbell, J.E. Ekelund, J. Ellis, J.F. Jordan, 2008. Phys. Rev. Lett., 100, 091102.
Antreasian, P.G. and J.R. Guinn, 1998. Paper no. 98-4287 presented at the AIAA/AAS Astrodynamics specialist conference in Boston, USA. http://www.issibern.ch/teams/Pioneer/pa-literature.htm
McCulloch, M.E., 2008. MNRAS, 389, L57-60. PDF
Lammerzahl, C. et al., 2009-2010. Flyby workshop: http://www.issi.unibe.ch/teams/investflyby/index.html