Different disturbances have been identified which need to be diagnosed in the LTP. The ones that will be monitored by the DDS are: temperature fluctuations, magnetic fluctuations and incident fluxes of charged particles [6, 145]. A quantitative assessment of the actual contribution of each of these items to the system noise requires knowledge of the transfer functions relating the disturbance and the effect on the displacement of the test masses. Such transfer functions will be determined on the basis of in-flight experiments, which will consist in measuring (by means of thermometers and magnetometers) the effect of induced perturbations (by means of heaters and coils) on the system response.
Once the transfer functions are determined, we project the different noisy sources to the total measured acceleration noise in nominal conditions, i.e., in absence of induced perturbations. This will permit to identify critical systems and/or unforeseen effects in the systems. In the following sections we focus on the aspects related to the thermal effects in the LTP which set the requirements of the thermal diagnostic subsystem.
1.5.1
Thermal disturbances in the LTP
Equation (1.8) gives the total noise budget permitted for residual acceleration in the LTP. This total noise is made up of contributions from different perturbing physical phenomena, both of in-
strumental and environmental origin. One of these perturbing elements is temperature fluctuations, for which a stability requirement for the LTP experiment has been set to [86, 145, 84]
ST1/2(ω) ≤ 10−4 K Hz−1/2, 1 mHz ≤ ω/2π ≤ 30 mHz (1.10)
in the absolute temperature range of 10 to 30 .
Because temperature stability is important, high precision temperature thermometers will be placed in different strategic points over the LTP. Such measurements will be useful to identify the fraction of the total system noise which is due to temperature fluctuations only, and this will in turn provide important debugging information to assess the performance of the LTP. In addition, by applying controlled temperature signals and measuring their effect on the system read out, the transfer function between temperature fluctuations and displacement (real or faked11) in the test masses will be estimated. In this manner we want to assess whether or not the modeling of the thermal effects in the subsystems of the LTP are accurate.
In the following the temperature stability requirement expressed in Eq. (1.10) is analysed. The sources of temperature fluctuations that contribute to the total noise budget of Eq. (1.8) are iden- tified and quantified. Afterwards we discuss how the temperature stability requirement results in the definition of the temperature measurement subsystem performance.
Random temperature fluctuations in the LTP introduce noise in the system through different mechanisms. Proper characterisation of these effects will set the limits of temperature fluctuations compatible with the LTP requirements. As a rule of thumb, the total contribution of temperature fluctuation noise to the total acceleration noise —Eq. (1.8)— should not exceed 10%. Thus, it is required that Sa,T1/2(ω) ≤ 3 · 10−15 " 1 + ω/2π 3 mHz 2# m s−2 Hz−1/2 (1.11)
within the LTP MBW. This assumption is in fact somewhat conservative, as it has been estimated that more than twice this value is actually compliant with the overall LTP noise budget [155]. However, here Eq. (1.11) is adopted as reference to ensure that we are on the safe side.
The influence of the temperature in the whole LTP experiment has been separated into two subsystems: the GRS and the OMS.
1.5.1.1 Noise effects inside the GRS
Temperature differences between the walls of the electrode housing inside the GRS cause differential pressures on opposite faces of the test masses, which in turn result in net forces on them, and real motion of the test masses. Three different mechanisms have been identified: radiation pressure, radiometer effect and asymmetric outgassing [145, 84, 26].
Radiation pressure A body at any (absolute) temperature T emits thermal radiation. This exerts pressure on any surface the radiation hits —see Figure 1.15. According to electromagnetic theory, such pressure is given by
pe.m.= 4 3 σ cT 4 (1.12)
where σ=5.67·10−8 W m−2 K−4 is the Stefan-Boltzmann constant, and c is the speed of light. Consequently, if there are temperature fluctuations around the test mass, a noisy net force will appear on it —see Figure 1.15. The pressure gradient through x is easily inferred from Eq. (1.12) as: dpe.m. dx = dpe.m. dT dT dx = 16σ 3c T 3dT dx , (1.13)
11For instance local temperature fluctuations in different components of the OMS cause errors in the measurement
hence
∆pe.m. = 16σ
3c T
3∆T (1.14)
where ∆p and ∆T are the differences of pressure and temperature between the sides of the test masses, respectively. Associated acceleration noise is easily obtained multiplying Eq. (1.14) by the TM surface area, `2
TM, and dividing by its mass12, mTM, i.e.,
ae.m.= 16`2 TMσ 3mTMc T3∆T . (1.15) electrode housing T1 T2 p 1 p2 test mass
Figure 1.15: Effect of the different pressure on opposite faces of a test mass due to differences of temperature in the walls of the EH.
Radiometer effect [96] This effect happens in rarefied gas atmospheres. In low pressure at- mospheres, where the gas particles have a mean free path well in excess of the dimensions of the containing vessel, equilibrium conditions do not happen when pressure is uniform, but rather when the ratios of pressure to square root of temperature equal each other. Let A and B be the two sides
1 T p T p 2 2 1 S A B
Figure 1.16: Receptacle with different temperature and pressures.
of a receptacle with temperatures T1and T2, separated by a wall with a hole S —see Figure 1.16. The equilibrium condition between both sides occurs when the number of molecules per time unit passing from A to B and from B to A is the same, i.e.,
n1c1= n2c2 (1.16)
where n1 and n2 are the number of particles per unit volume of both receptacle sides, and c1 and c2 are the average speeds of the particles. They can be expressed as
ni = pi kBTi , (1.17a) ci = r 8RTi πM (1.17b)
where kB = 1.38 · 10−23J K−1 is Boltzmann’s constant, R = 8.314 J K−1mol−1 is the ideal gas constant and M is the molar mass of the gas.
Substituting Eqs. (1.17a) and (1.17b) into Eq. (1.16) the equilibrium condition is obtained: p1 √ T1 = √p2 T2 . (1.18)
Equation (1.18) can be expressed in the form of
p(T ) = p0 r T
T0
(1.19)
and its gradient through x is easily derived from Eq. (1.19), i.e.,
dp(T ) dx = dp(T ) dT dT dx = 1 2p(T ) 1 T dT dx . (1.20) Re-arranging we obtain ∆pr.m. p(T ) = 1 2 ∆T T (1.21) and considering ∆pr.m.=(mTMar.m.)/`2 TM we arrive to 13 ar.m.= 1 2 p(T )`2TM mTM ∆T T . (1.22)
Outgassing Outgassing is one of the causes of the presence of gas within the walls of the GRS. In the present context, outgassing problems actually derive from temporal fluctuations in rate, which once more result in pressure fluctuations, thence in noise. Partial evidence has been gathered that outgassing might be in practise a small effect in the LTP [25, 26]. Moreover, a baking process to the GRS will be done in order to reduce the potential effects of this issue. Therefore this effect will be omitted in further analysis.
Total temperature fluctuation noise in the GRS Radiometer and radiation pressure accel- eration fluctuations are of course totally correlated since they have the same noisy source, ∆T . If we neglect outgassing then Eqs. (1.15) and (1.22) are linearly added, and hence the spectral densities of acceleration and temperature in the GRS are related by
Sa,GRS1/2 (ω) = 16` 2 TMσ 3mTMc T3+ p` 2 TM 2mTM T−1 S∆T , GRS1/2 (ω) (1.23)
Nominal conditions in the GRS are as follows, `TM=46·10−3m, mTM= 1.96 kg, T =293 K, p=10−5Pa which yield S∆T , GRS1/2 (ω) = [2.2 · 1010 K (m s−2)−1] · Sa, TF GRS1/2 (ω) (1.24) with 60% coming from the radiation pressure term, and 40% from the radiometer effect term in Eq. (1.23). Equation (1.24) gives 70 µK Hz−1/2as the permitted fluctuations in the worst case that all the thermal acceleration budget, Eq. (1.11), is allocated to temperature fluctuations in the GRS.
13In [26] this effect is modelled as (1/4)p(T )`2 TMm
−1 TM∆T /T .
1.5.1.2 Noise effects inside the OMS
The optical metrology system is affected by temperature fluctuations basically through three distinct effects [145, 84, 97]:
the index of refraction of optical components depends on the temperature,
temperature changes cause dilatation (and contractions) of optical elements, which in turn cause light’s optical path to change accordingly and,
the temperature effect on the analog electronics, in particular a change of capacitances in the photodiodes which converts into a phase change of the measured signal.
It is not difficult to characterise how individual components are influenced by the above effects. For instance a set of on-ground tests in the optical window (OW) and the optical bench (OB) of the OMS have been done in order to estimate the relationship between temperature and interfer- ometer performance when this thermally disturbed. In the case of the OW it has been estimated ∼6.5 nm K−1, which allows a maximum temperature fluctuations in the OW of ∼10−4 K Hz−1/2 not to exceed the noise budget allocated for the interferometer [98, 58].
However, the assessment of the behaviour of the fully integrated optical metrology is a com- plicated task. Significant progress has been made since the early design proposals, and improved materials and designs more immune to temperature fluctuations are now available. Altogether, it appears that
ST ,OMS1/2 (ω) . 10−4K Hz−1/2 (1.25)
in the LTP MBW is a requirement which should guarantee the performance of the OMS against temperature fluctuations in flight. Again, the noise level in Eq. (1.25) has been estimated for about 10% of the total LTP acceleration noise.
1.5.2
Temperature measurement subsystem sensitivity requirement
Estimates so far indicate that both GRS and OMS noise must be in the order of 10−4K Hz−1/2, as shown in the requirement in Eq. (1.10). Noise in the GRS should be, in principle, uncorrelated with the noise in the OMS since they are of different nature: temperature gradient fluctuations across the test masses cause noise in the GRS, local temperature fluctuations affect the OMS read out. Then, the zero-correlation hypothesis implies that both kinds of noise add quadratically:
S∆T ,T1/2 (ω) = [S∆T , GRS(ω) + ST , OMS(ω)] 1/2
6 10−4 K Hz−1/2 (1.26)
Equation (1.26) sets the maximum temperature fluctuations permitted in the LTP to respect the requirement given in Eq. (1.11). We are thus reassured that Eq. (1.10) is a sensible requirement for the temperature fluctuations which can be tolerated in the LTP, and this will be designed to ensure such temperature stability. Nevertheless, it is necessary to know if the stability is actually met during the mission. For this reason it is required to monitor the temperature across different places in the LTP by means of temperature sensors able to detect small temperature fluctuations: 10−4K Hz−1/2 in the frequency range of 1 mHz to 30 mHz. The temperature measurement subsys- tem (TMS) should be one order of magnitude less noisy than the maximum noise level that has to be measured in the temperature range from 10 to 30 [83, 119], or
S1/2T , TMS(ω) ≤ 10−5K Hz−1/2, 1 mHz ≤ ω/2π ≤ 30 mHz,
10oC ≤ T ≤ 30oC (1.27)
which, in fact, becomes a mission top level requirement [146]. There are mainly two reasons which support this decision: (i) Eq. (1.10) defines the maximum acceptable level of temperature fluc- tuations in the LTP. This, of course, must be satisfied by proper satellite design. Hence, actual
fluctuations will be, in principle, less than that. Requirement (1.27) sets a 10% minimum discrim- ination capability for the measuring system, a standard approach which is compatible with better performance and (ii) LISA is more demanding than LPF as regards thermal stability (by one order of magnitude [145, 14, 126]). If we require (1.27) for LTP then we are in a position where analysis of thermal sources of noise of relevance for LISA can be identified and tagged for improvement. This prospect is in line with the very concept of LPF as a precursor mission.