13 June 2014

`research`

`hpf`

The Habitable Zone Planet Finder (HPF) - being an infrared spectrograph - must be kept from being saturated by infrared radiation emitted from the surroundings. This can be done by keeping the instrument extremely cold, or at 180K, and must be kept at that fixed temperature to milli-Kelvin precision as any variations will increase measurement errors. How do we achieve this? Take a look at the figure below.

We need to consider four things:

**Cooling agent**- We use liquid nitrogen (LN2) to cool the instrument. The LN2 tank is at a fixed temperature of 77K at atmospheric pressure.**Conductive paths to the vacuum chamber.**- To cool down the radiation shield we connect it with the LN2 tank with highly thermally conductive*copper thermal straps*. The straps need to be sized properly to not draw*too much*heat (resulting in a too cold chamber) nor*too little*heat (chamber too warm) from the vacuum chamber over long periods of time. The copper straps are sized to cool the vacuum chamber down to temperatures slightly cooler (around 160K - 170K) than the 180K end temperature goal.**Heater Panels**- These panels heat the overcooled radiation shield to the 180K temperature goal, and at the same time give us the milli-Kelvin precision required.**Thermal Insulation**- Lastly, all of the above components are kept under a high vacuum - the radiation shield, the heater panels, the thermal straps, and the LN2 tank - and are all covered with Multi-Layer Insulation blankets (MLI - commonly used for space probes!), to provide effective thermal insulation from the outside world.

The radiation shield is on one hand being heated up by the radiation from *the surroundings* (at room temperature) and the *heater panels*, and being cooled down from the *copper thermal straps* on the other.
At equilibrium we can write:
\begin{align}
H_{\mathrm{rad}} + H_{\mathrm{Heaters}} = H_{\mathrm{Cu}},
\end{align}
where \( H_{\mathrm{rad}}, H_{\mathrm{Heaters}}, H_{\mathrm{Cu}} \), denote *the heat current* from *the net incoming radiation, the heaters*, and *the copper straps*, respectively.
Each of these factors are discussed below.

First off, lets consider the incoming radiation. All objects, regardless of their temperature, emit energy in the form of electromagnetic radiation: the warmth of the Sun and glowing coals in a fireplace is just infrared radiation emitted from these objects.

Then to the math.
The net heat current absorbed by an object with surface area \( A \) and emissivity \( e_{\mathrm{eff}} \) (a dimensionless number between 0 and 1 - larger for darker surfaces) sitting in a room at absolute temperature \( T_{\mathrm{room}} \), can be expressed by the *Stefan-Boltzmann* law:
\begin{align}
H_{\mathrm{rad}} = A e_{\mathrm{eff}} \sigma (T_{\mathrm{room}}^4 - T_{\mathrm{HPF}}^{4})
\end{align}
where \(T_{\mathrm{HPF}}\) is the temperature our object: the HPF instrument.

The emissivity of MLI blankets is very low (for good blankets \( 0.005 \lesssim e \lesssim 0.1 \); see here) and therefore offer very good radiative thermal insulation. By covering HPF in MLI blankets, the emissivity of the blankets governs \( e_{\mathrm{eff}} \) - the effective emissivity of the instrument. However, the actual value of \( e_{\mathrm{eff}} \) is highly dependent on the overall quality of the MLI blankets and the surface finish of the radiation shield, etc. and is very difficult to calculate the value exactly. Our best bet is then to empirically derive the effective emissivity from the APOGEE instrument - built for the Sloan Digital Sky Survey. This gives us a value of \( e_{\mathrm{eff}} \sim 0.0087 \). We will defer further MLI-blanket discussion - how we prepare and size the blankets for HPF - material for a whole blog post in itself!

When a quantity of heat \( dQ \) is transferred through a conductive material in time \( dt \), the rate of heat flow is given by \( H = \frac{dQ}{dt} \).
More specifically, we can relate the heat current to other properties of our copper thermal straps with the following equation:
\begin{align}
H_{\mathrm{Cu}} = \frac{dQ}{dt} = k A \frac{T_{H} - T_{C}}{L}
\end{align}
where \( k \) is the *thermal conductivity* of the material - *copper* in our case - and \( T_C = 77 K \) is the LN2 temperature, and \( T_H = 180 K \) is the cryostat temperature, and \( L \) and \( A \) are the length and cross-sectional area of our thermal strap, respectively.

There is one issue however: the thermal conductivity of copper varies with temperature, so \( k \) is not a constant in our operating temperature range (see figure below). By integrating over the temperature range: \begin{align} H_{\mathrm{Cu}} = \frac{dQ}{dt} = \frac{A}{L} \int_{T_C}^{T_H} k(T) dT, \end{align} we can account for this secondary effect - ignoring it, we would underestimate \( H_{\mathrm{Cu}} \).

The images below show a few photos from the copper straps preparation; we will need 16 straps in total.

Like mentioned above, the heater panels heat up the overcooled radiation shield to the 180K temperature goal. Each panel has 4 thermal resistors (\( 150 \Omega \) each), which heat up in proportion to the current going through them. As we now know the heat current from (I) the net incoming radiation, and (II) the copper straps, we can calculate the heat current needed from the heater panels to keep the system at equilibrium: \begin{align} H_{\mathrm{Heaters}} = H_{\mathrm{Cu}} - H_{\mathrm{rad}}, \end{align} which can be used to calculate the current needed per panel and per thermal resistor. The exact current running through per resistor is controlled by a thermal feedback control system which monitors any external temperature fluctuations at the observatory - we can't control the weather!. The system then compensates for these changes by controlling the amount of current going through the resistors, warming them up as needed, keeping the instrument stable at 180K with the milli-Kelvin precision needed.

Guðmundur Kári Stefánsson

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