The paper presents a description of the new Tesseract null fluxgate magneometer's mechanical design, magnetic uniformity across the racetrack cores, and thermal stability tests. Relatively minor comments, questions and suggestions included in attached PDF....

The authors present a vector compensated fluxgate magnetometer based on six racetracks inside a three axes coil system. The design is unique and the demonstrated performance is excellent. Thus, the paper is worth to publish.

Features of magnetic field experiments on constellation mission are discussed in the introduction. Please add a brief statement, why particularly these applications are used as reference for the new developed magnetometer.  An argument could be, that these missions are representative for almost all space born magnetometers; wide range is required (low field at apogee, high field at perigee), exposed to radiation, temperature changes due to eclipse crossings ..

The comparison is made disordered, some parameters are listed for the one, others for the second type of magnetometers. This should be harmonised, may be supported in a table format.  Noise, mass, scale value and axis stability vs. temperature shall be given for all of them.

The paper attention turns to stability of the sensors and to the advantages of the vector compensated system.  All constellation missions mentioned in this paper (Themis, MMS, SWARM) are since many years in space. Long period data for offset and axis stability should be available from inflight calibration. Contact magnetometer PI’s for these data (if not published) and include the inflight measured drifts into your comparison.

The presented sensor design is impressive. In contrast to the straight forward OERSTEDT/SWARM design (feedback system over three single ringcores) and the more compact THEMIS design (feedback system over crossed ringcores) the cores (racetracks) are accommodated symmetric and identical for all three components. It is made similar to the very innovative Xavier Lalanne design from the 1990^th.  He placed six ringcores at the six planes of a cube. Please refer to it.

Chapter 3 has to be rewritten. Promoting the presented sensor is ok, however, the comparison with a user defined ringcore sensor, which should imply that the presented sensor is much better than ringcore sensors in general is not acceptable. The comparison has to be made with the vector compensated ringcore sensors you have studied in the introduction.

Quantities are mixed up. It shall be clearly distinguished between stability of offsets, scale values and orthogonality. The vector compensation stabilises the orientation of the magnetic axis while the offset stability depends on core properties only. Thus for scale value and axes stability it is fully unimportant which type and geometry of magnetic material is used as core.

The analysis of the uniformity of the feedback coils has been intensively discussed. No question, high homogeneity is better than low homogeneity, however in case you want to underline the importance of the uniformity, you have to quantify it. What is the impact on offset, scale value, linearity and orthogonality behaviour really? Particularly racetracks with a significant length/diameter ratio might disturb the uniformity you have hardly achieved by the sophisticated feedback coil design.

The discussion of thermal expansion of the feedback system is not as simple. A high scale value stability of <10ppm/K, achieved by a combination of materials with different expansion coefficients must not be better than a scale value stability of 20ppm/K, if this one is linear over the whole temperature range and has a lower hysteresis. Thus linearity, the reaction on fast temperature changes (e.g. during eclipse) and the hysteresis are criteria which have to be discussed and compared, not the number itself.

The authors present a nice sensor with an excellent performance. The demonstration of its parameter doesn’t need to be underlined by an artificial comparison with an arbitrary sensor. Only drawback is the mass, may be a little bit too heavy for constellation missions or small satellites. However, with respect to the mass of boom and harness, a few 100g should be acceptable for the most important part of a magnetic field experiment, the sensor.

"General comments"

The paper “Tesseract - A High-Stability, Low-Noise Fluxgate Sensor Designed for Constellation Applications” by Green et al. is a useful contribution to еру design of the fluxgate magnetometers for space applications, particularly those for multi-point magnetic field measurements. The scientific and technical questions addressed in the paper are within the scope of GI.

The paper proposes further development of some ideas intended to improve the accuracy of fluxgate magnetometers including but not limited to such approaches: a) selection of the sensor components with similar coefficients of linear thermal expansion; b) the use of a three-axial system of magnetic feedback coils for creating near-zero field inside ferromagnetic cores of the sensor; c) using a solid homogeneous base as a support for windings and sensors for minimizing mechanical deformations, that is expected to provide excellent stability of the sensor magnetic axes; d) selection a compact configuration of two cores per axis for further reducing cross-talk between orthogonal components.

The last method - the pairing of identical cores on each axis – is also useful, in my opinion, for reducing noise levels. Besides, such a configuration of the cores makes it possible to measure the three components of the magnetic field at the same point. See, for example, patents “EP0275767A1. Triaxial homo-centered magnetometer” (https://patents.google.com/patent/EP0275767A1/en) and “FR2740556A1. Single symmetry center magnetic core for multi-axial magnetometer” (https://patents.google.com/patent/FR2740556A1/en?oq=FR2740556A1). As far as I know, the idea of the triaxial homo-centered magnetometer was implemented in the variometer VM391 developed for geomagnetic observatories (http://www.ipgp.jussieu.fr/~chulliat/papers/Chulliat_etal_2009b.pdf).

The paper describes the new construction of the sensor in detail with excellent graphical materials. The computer simulation and experimental results mainly concern characteristics of the three axial feedback windings without magnetic cores inside. The temperature and long-term stability of the complete flight-ready magnetometer are going to be characterized in the next stages of work.

The results obtained at the current stage of research are very promising.

However, some details of the exploited procedure for sensor calibration during temperature tests are not completely clear, in my opinion. This would cause some difficulties in reproducing similar experiments by other researchers.

The authors based their research on an analysis of a large number of related studies and this is reflected in the appropriate list of references.

The title and abstract clearly and completely represent the contents of the paper. The overall presentation of the research results is well structured and clear. Abbreviations, symbols, and units are correctly defined and used.

The equations (2), (3), and (4) for estimating sensor orthogonality have to be explained in detail or appropriate reference should be added.

"Specific comments"

In my opinion, more details about the method used for measuring sensitivity and orthogonality of the feedback coils (subsection 3.2) have to be provided.

The references (Brauer et al., 1999; Miles et al., 2017) exploit some other approaches in comparison with that described in the manuscript. Brauer et al. (1999) used so-called “thin shell” calibration - the calibrating signals “were randomly distributed over shells of fixed field magnitudes”. The data processing was also different – the overdetermined system of linear equations was solved for a parameters matrix by singular value decomposition.

Miles et al. (2017) did not estimate orthogonality. In both references, a fluxgate magnetometer as a whole unit was calibrated, whereas the manuscript estimates the temperature characteristics of the feedback coils only, without magnetic cores inside.

page 12, lines 347-349

“To characterize the thermal stability of the Tesseract sensor’s design, we temporarily configured it as an air-core search coil magnetometer to directly access the attributes of the sensor base and feedback windings without any dependence on cores or electronics.”

Were the feedback coils used in the air-core search coil magnetometer to form feedback signals or to serve as sense windings? What was a sense winding in the first case?

In the second case (the feedback winding is used as a sense one) the temperature dependence of the sensitivity or gain of such air-core search coil magnetometer was actually tested. The temperature stability of the fluxgate magnetometer’s scale factor depends on the stability of the coil constant of the feedback winding. Is it assumed that the gain of the air-core search coil magnetometer based on the feedback winding depends on the temperature in a similar way as the coil constant of the feedback winding does?

pages 12-13, lines 350-351:

“The polystyrene box is then placed within the two-meter Merritt coil system and fixed to the table so that Tesseract’s axes are aligned with the coil systems axes as shown in Figure 9a”

The mutual orientation of the axes of the calibrating system and the device under test is not clear in Figure 9a.

How accurately were aligned the magnetometer feedback coil axes with that of the Merritt coil system and what method was used to achieve this?

pages 13-14, lines 370-378.

The way Equations (2), (3), and (4)  for estimating orthogonality angles were derived is not clear. Why are these equations different for the XY pair and the XZ, and YZ pairs? How was the total magnitude (A) of the applied field calculated or measured?

"Technical corrections"

Which component of the magnetic field generated by the feedback coil is presented in the color map in Figure 4a? Bx? It would be useful to clarify.

The length of the racetrack sensor is equal to 31.45 mm (Subsection 2.1, p. 5, line 132 ), but the Racetrack boundaries are equal to +/-14.5 mm in Figures 6, 7, and +/-15 mm in Figure 8.

page 9, Figure 6

The last part of the caption of Figure 6: “...Configuration (b) was optimized for best homogeneity while sensor while (c) was chosen for good homogeneity with very low power consumption.”

Should it be “...Configuration (b) was optimized for best homogeneity within the sensor while (c) was chosen for good homogeneity with very low power consumption.” or  “...Configuration (b) was optimized for best homogeneity while sensor (c) was chosen for good homogeneity with very low power consumption.”?