It is well known that a detector, coupled linearly to a quantum field and accelerating through the inertial vacuum with a constant acceleration g, will behave as though it is immersed in a radiation field with temperature T = (g / 2 π). We study a generalization of this result for detectors moving with a time-dependent acceleration g (τ) along a given direction. After defining the rate of excitation of the detector appropriately, we evaluate this rate for time-dependent acceleration, g (τ), to linear order in the parameter η = over(g, ̇) / g2. In this case, we have three length scales in the problem: g- 1, (over(g, ̇) / g)- 1 and ω- 1 where ω is the energy difference between the two levels of the detector at which the spectrum is probed. We show that: (a) When ω- 1 ≪ g- 1 ≪ (over(g, ̇) / g)- 1, the rate of transition of the detector corresponds to a slowly varying temperature T (τ) = g (τ) / 2 π, as one would have expected. (b) However, when g- 1 ≪ ω- 1 ≪ (over(g, ̇) / g)- 1, we find that the spectrum is modified even at the orderO (η) . This is counter-intuitive because, in this case, the relevant frequency does not probe the rate of change of the acceleration since (over(g, ̇) / g) ≪ ω and we certainly do not have deviation from the thermal spectrum when over(g, ̇) = 0. This result shows that there is a subtle discontinuity in the behavior of detectors with over(g, ̇) = 0 and over(g, ̇) / g2 being arbitrarily small. We corroborate this result by evaluating the detector response for a particular trajectory which admits an analytic expression for the poles of the Wightman function. © 2010 Elsevier B.V. All rights reserved.