Spacecraft calorimetry as a test of the dark matter scattering model for flyby anomalies

In several recent papers we have explored the possibility that dark matter scattering may be responsible for the anomalous geocentric frame orbital energy changes that are observed during earth flybys of various spacecraft, as reported by Anderson et al. [1]. Some flybys show energy decreases, and others energy increases, with the largest anomalous velocity changes of order 1 part in 10 6 . While the possibility that these anomalies are artifacts of the orbital fitting method used in [1] is being actively studied, there is also a chance that they may represent new physics. In [2] we explored, through order of magnitude estimates, the possibility that the flyby anomalies result from the scattering of spacecraft nucleons from dark matter particles in orbit around the earth, with the observed velocity decreases arising from elastic scattering, and the observed velocity increases arising from exothermic inelastic scattering, which can impart an energy impulse to a spacecraft nucleon. In [3] we constructed a concrete model, based on two populations of dark matter particles, one of which scatters on nucleons elastically, and the other of which scatters inelastically, each with a shell-like distribution of orbits generated by the precession of a tilted circular orbit around the earth's rotation axis. We showed in [3] that this model can give a good fit to the flyby data, with shell radii in the 30,000-35,000 km range.

In the present paper we follow up on the brief observation in [2] that if there is a spacecraft velocity change as a result of dark matter scattering, there must be a corresponding temperature increase arising from fluctuations in the scattering recoil direction. In Sec. II we develop formulas for giving a quantitative treatment of this effect. In Sec. III we give order of magnitude quick estimates, and in Sec. IV we give numerical results based on the model of [3]. In Sec. V we suggest that a thermally shielded, spacecraft based calorimetry experiment could potentially give crucial information on the dark matter scattering model for the flyby anomalies.

In [2] we considered the velocity change when a spacecraft nucleon of mass m 1 ≃ 1GeV and initial velocity u 1 scatters from a dark matter particle of mass m 2 and initial velocity u 2 , into an outgoing nucleon of mass m 1 and velocity v 1 , and an outgoing secondary dark matter particle of mass m ′ 2 = m 2 -∆m and velocity v 2 . (In the elastic scattering case, one has m ′ 2 = m 2 and ∆m = 0.) Under the assumption that both initial particles are nonrelativistic, so that | u 1 | « c, | u 2 | « c, a straightforward calculation shows that the outgoing nucleon velocity is given by

Here w > 0 is given1 by taking the square root of

and vout is a kinematically free unit vector. Denoting by θ the angle between vout and the entrance channel center of mass nucleon velocity

and assuming that the center of mass scattering amplitude is a function f (θ) only of this polar angle, the average over scattering angles of the outgoing nucleon velocity is given by

with cos θ given by

Subtracting u 1 from Eq. ( 3) gives the formula for the average velocity change used in [2] and [3] to calculate the flyby velocity change,

However, in addition to contributing to an average change in the outgoing nucleon velocity, dark matter scattering will give rise to fluctuations in this velocity, which have a mean square magnitude given by

This fluctuating velocity leads to an average temperature increase of the nucleon, per single scattering, of

with k B the Boltzmann constant. In analogy with the treatment of the velocity change δ v 1 in [2],

to calculate dT /dt, the time rate of change of temperature of the spacecraft resulting from dark matter scatters, one multiplies the temperature change in a single scatter δT by the number of scatters per unit time. This latter is given by the flux | u 1 -u 2 |, times the scattering cross section σ, times the dark matter spatial and velocity distribution ρ x, u 2 . Integrating out the dark matter velocity, one thus gets for dT /dt at the point x(t) on the spacecraft trajectory with

Integrating from t i to t f we get for the temperature change resulting from dark matter collisions over the corresponding interval of the spacecraft trajectory ,

In the elastic scattering case, with ∆m = 0, m ′ 2 = m 2 , the formula of Eq. ( 2) simplifies to

In the inelastic case, assuming that ∆m/m 2 and m ′ 2 /m 2 are both of order unity, Eq. ( 2) is well approximated by

Since u 1 and u 2 are typically of order 10 km s -1 , the temperature change in the inelastic case, per unit scattering cross section times angular factors, is larger than that in the elastic case by a factor

Before going on to detailed modeling calculations using Eq. ( 9), we first give quick estimates using Eqs. ( 7), (10), and (11), making the approximations that the dark matter mass m 2 is much smaller than the nucleon mass m 1 , and that cos θ in Eq. ( 7) is much smaller t

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