Cryogenic inertial confinement fusion targets at the National Ignition Facility and the Laser Megajoule will be protected from thermal infrared radiation by a cold shroud. As the shroud is removed just before the laser pulse, infrared radiation will heat and possibly degrade the symmetry of the solid hydrogen fuel layer. A lumped component mathematical model has been constructed to calculate how long an indirect drive target can be exposed to thermal radiation before the fuel layer degrades. The allowed exposure time sets the maximum shroud removal time and therefore has important implications for the design of the cryogenic shroud systems. The model predicts that the maximum exposure time is approximately 0.18 s for plastic capsules inside hohlraums with transparent laser entrance holes. By covering the laser entrance holes with a partially reflective coating, the exposure time can be increased to approximately 1 s. The exposure time can be increased to about 2 s by using beryllium capsules. Several other design concepts could increase the exposure time even further. Lengthening of the allowed exposure time to 1 s or longer could allow a significant cost savings for the shroud system.