CAPE CANAVERAL — NASA decided Thursday that no repairs are needed for a deep gouge in Endeavour’s belly and the space shuttle is safe to fly home.

Mission Control notified the seven shuttle astronauts of the decision right before they went to sleep, putting an end to a week of engineering analyses and anxious uncertainty — both in orbit and on Earth.

“Please pass along our thanks for all the hard work,” radioed Endeavour’s commander, Scott Kelly.

Mission Control replied, “It’s great we finally have a decision and we can press forward.”

The astronauts had spent much of the day running through the never-before-attempted repair methods, just in case they were ordered up.

After meeting for five hours, mission managers opted Thursday night against any risky spacewalk repairs, after receiving the results of one final thermal test. The massive amount of data indicated Endeavour would suffer no serious structural damage during next week’s re-entry.

Their worry was not that Endeavour might be destroyed and its seven astronauts killed in a replay of the Columbia disaster — the gouge is too small to be catastrophic. They were concerned that the heat of re-entry could weaken the shuttle’s aluminum frame at the damaged spot and result in lengthy postflight repairs.

Endeavour’s bottom thermal shielding was pierced by a piece of debris that broke off the external fuel tank shortly after liftoff last week. The debris, either foam insulation, ice or a combination of both, weighed just one-third of an ounce but packed enough punch to carve out a 3 1/2-inch-long, 2-inch-wide gouge and dig all the way through the thermal tiles. Left completely exposed was a narrow 1-inch strip of the overlying felt fabric, the last barrier before the shuttle’s aluminum structure.

The only way to fix the gouge would have been to send a pair of spacewalking astronauts out with black paint and caulk-like goo, and maneuver them beneath the shuttle on the end of a 100-foot robotic arm and extension boom, with few if any close-up camera views of the work.

The spacewalk would have had added risk, so much so that mission managers did not want to attempt it unless absolutely necessary. Wednesday’s spacewalk, cut short by an astronaut’s ripped glove, showed how hazardous even a relatively routine spacewalk can be.

Earlier, astronaut Alvin Drew said from Endeavour that he was comfortable with the prospect of flying back to Earth in a gouged ship. Engineers seem confident, he said, “and I trust their confidence that we can get home safely even with the divot that we have in the belly.”

“Spaceflight is risky,” noted astronaut Barbara Morgan, the backup teacher for Challenger’s doomed mission, “but we have all confidence that we’re going to be able to do the right thing.”

But a Nobel Prize-winning physicist who served on the Columbia investigation board four years ago, Stanford University’s Douglas Osheroff, questioned NASA’s hesitancy to perform the repairs since they “can only increase their chances of making it down.”

I don’t see why NASA is going to invent a fix and not use it,” Osheroff said. He added: “This attitude of, ’It looks like it’s OK, let’s not do anything about it,’ it seems like the Columbia NASA.”

In a poignant reminder of NASA’s other shuttle accident, the 1986 Challenger launch explosion, Morgan — Christa McAuliffe’s backup — answered questions from youngsters gathered at the Challenger Center for Space Science Education in Alexandria, Va.

The moderator was June Scobee Rodgers, the widow of Challenger’s commander and the founding chairman of the Challenger center’s board. “Barb, we have been standing by waiting for your signal from space for 21 years,” she said.

One girl asked if Morgan had a special teacher or mentor when she was young.

“Some of my mentors that have meant more than anything to me are seven very special people who I believe are mentors to you, too, and that was the Challenger crew,” Morgan replied.

Morgan closed the teaching session by holding up an emblem of the Challenger crew’s mission patch.