For instruments operating in the soft (meaning relatively low-energy) gamma-ray and hard X-ray energy ranges, instrumental background is very important. Instrumental background refers to events that trigger the detector that are not caused by incoming gamma rays, but most commonly as a result of charged particles from cosmic rays, from the Sun, or from the Earth’s radiation belts. As a rule of thumb, only about one percent of events are due to the gamma rays one wants to measure in the soft gamma range.
Since INTEGRAL’s instruments work using the coded aperture principle, which relies on measuring the change in count rate when the satellite is pointed in different directions, a low rate of background events is desirable, but it was decided1 that a background rate that is moderate but varying slowly over time would be preferable to one that is lower but varying quickly over time.
These aspects were discussed in older pre-launch articles such as Winkler (1994)
INTEGRAL (with a total launch mass of 3.6 tons) will be launched early 2001 into a geosynchronous Highly Eccentric Orbit (HEO) with high perigee in order to provide long uninterrupted observation periods at nearly constant background and away from trapped radiation. The launcher will be a Russian PROTON D1-e to launch the spacecraft into a 72 hour orbit with 51.6° inclination, height of perigee of 48,000 km, and 115,000 km apogee. The maximum lift capacity of the launcher for this orbit is 4180 kg.
Alternatively, an ARIANE 5 could be used for a 24 hour orbit, 65° inclination and 4000 km (perigee) X 68,000 km (apogee). During Phase A a TITAN III plus TOS (or equivalent) has also been studied and shown to be a feasible launcher for the same spacecraft for a 48 hour orbit, 28.5° inclination, 4000 km perigee, 117,000 km apogee.
A high orbit has these desired properties in contrast to a low orbit that would be affected by passages close to the South Atlantic Anomaly, which leads to a loss of a substantial fraction of observing time as discussed in Gehrels (1992). For a high elliptical orbit, it is at least the case for the large fraction of the orbit spent at high altitude, after the satellite has left the radiation belts. Inside the belts, the scientific instruments are switched off. However the onboard radiation monitor remains on, you can see its measurements in Hajdas et al. (2003). INTEGRAL spends only a short time (~ 1.5 h) exposed to the protons in the belts which do negligible damage to the detectors (at least those of spectrometer SPI, Lonjou et al., 2005) and leaves the electron belt at altitudes over 60,000 km–70,000 km. A circular orbit would not have been high enough with these launchers.
The particle radiation has two distinct effects. One is the creation of instrumental background by hitting the detector itself or by inducing radioactivity in the spacecraft materials. This limits the sensitivity of the instruments. For good sensitivity, the background should be low, but it should also be known as precisely as possible because with some exceptions you cannot tell if an individual event in the detector was caused by gamma photon from an astronomical source or not. Only via statistics from long measurements can this ambiguity be resolved.
The second effect is the damage to the detectors. Energetic particles create defects in the crystal structure of the detector materials which degrade the electronic properties and reduce the detector performance over time. The spectrometer SPI has been designed with a capability to anneal the detectors by heating them to about 100 °C for about a day every six months which restores the detector performance to be almost as good as new. The other instruments do not have this capability. But anyway, at least for SPI, the damage is caused by cosmic ray protons and not by the belts. In a Low Earth Orbit, the Earth’s magnetic field would shield the spacecraft from the lower energy cosmic ray protons.
1personal knowledge primarily from in-person discussions at INTEGRAL team meetings.
There are other constraints. For example, the orbit should not be too high (> 160,000 km) because at those distances the data rate of the radio link to the ground station would be unacceptably low. Another limit is the performance of the launch vehicle which would have been sufficient to achieve a perigee height of 40,000 km, so a less elliptical orbit than the one chosen. However, this would have required the upper stage of the Proton to coast for half an orbit to this height, then ignite the engine again to change the orbit to the final one and this was judged as too risky.
As you saw from the table in Sec. 2 of the article you linked, there are more constraints, but I think the ones I mentioned are the most important ones.
P.S.: While this is not directly related to your question, you might be interested in reading about how INTEGRAL’s orbit was changed in Jan.–Feb. of 2015 for the first time in 12 years after launch to prepare for a controlled deorbit maneuvre in 2029 [1, 2]. “[…] INTEGRAL lowered its orbit from a 4310 min period to approx 3840 min period […]”.