The dustjacket has a nice, fairly long summary of the story that I have copied here, for your own personal use:

Here is an exciting tale of action and suspense, set in the world of tomorrow. As the story opens, two brothers, Jim and Ken Barry, are traveling by space ship from Earth to a new city on the moon. Aboard the space ship they discover a stowaway, young Dig Allen. Almost at once the three youths are plunged into an adventure that threatens to spell destruction for them all.

Dig has set out to find his father, a space scientist who mysteriously disappeared months before. The Barry brothers agree to help Dig. Their search takes them to the "Graveyard of Space" and to Mars. There they meet Old Dorkas, the one person who is able to decipher the last message received from the missing man. Finally, the boys are forced to set out along in an unauthorized space ship. They soon find themselves marooned on a weird, forgotten world at the outer edges of the solar system -- where unknown to them, their greatest adventure is about to begin!

"The Forgotten Star" is, of course, the first book in the series. The book acts as kind of an introduction to the series. In this first book, all of the major, reappearing characters of the series (Jim Barry, Ken Barry, Dig Allen, Boyd Allen, Old Dorkas, and Sergeant Brool), are introduced. The state of the civilization -- both socially and scientifically -- is set forth, and a good feel can be had for the general state of things.

The general plot for the book is quite straightforward -- Jim, Dig and Ken all go to the satellite Eros to rescue Dig's father. Once on the satellite, they find out that Eros is not a natural asteroid at all, but instead had been hollowed out and converted into a huge, sprawling spaceship -- complete with natives. After rescuing his father and being made a Space Explorer, Dig's father moves his team of scientists to Eros and proceeds with unraveling the satellite's secrets.

This book definitely gives the most detailed account of the artificial satellite Eros that is found in the series. No other book hints at Eros' true size or immensity -- or science. Eros houses several devices that are definitely scientific mysteries. Its basement, for example, houses hundreds of gigantic machines (which later books call nuclatomizers) that can disassemble atoms and then recombine them to form whatever they wish. The steel elevators are quite odd, too -- somehow they can pressurize to normal Earth atmospheric pressure without the use of air vents or any other openings through which air might be let in.

The satellite Eros was, according to Dig's father, a huge spaceship, capable of going "at least as fast as light". It seems that the Asterians, as they were called, constructed it to journey to some far-off location. However, the journey took such an extraordinarily long time that the Asterian population housed inside the asteroid crumbled and reverted to a primitive state, living as savages off of a forest housed deep inside the asteroid. Why the civilization crumbled is not hinted at in this book, but one can take a good educated guess as things.

The title of the book does not, in fact, refer to the satellite Eros itself, but instead refers to the home of the Asterians. It seems that, although many "star charts" were provided on Eros, none of them could be reconciled to the currently known stars. Thus, said the book, the Asterians were from some "long forgotten star", which, I suppose, they were.

3 stars out of 5. This is an enjoyable book which introduces both the characters and the world of Dig Allen. Normally the first volume of a series is the most thought-out and imaginative, but this time that is not the case. Like the rest of the books in the series, this title wins high marks for originality and creativity: back at the time of Sputnik and before Star Trek and Star wars the author had created a world of interplanetary commerce, nuclear rockets, lunar settlement and the Graveyard of Space. The asteroid Eros, which was really an interstellar spaceship in disguise, was a brilliant idea which was made better by having its builders still housed inside unaware that they were on a spaceship.

However, the book could have been better. The characters act somewhat implausibly, and the plot is not as well-developed as the plot in later volumes. The book lacks the coherent excitement present in the other volumes, and the Asterians are hard to take. Overall, it is a good book to read to become familiar with the beginnings of the characters, but the later volumes are better.