Question

（08寧夏卷）23.（15分）天文學家將相距較近、僅在彼此的引力作用下運行的兩顆恒星稱為雙星。雙星系統在銀河系中很普遍。利用雙星系統中兩顆恒星的運動特征可推算出它們的總質量。已知某雙星系統中兩顆恒星圍繞它們連線上的某一固定點分別做勻速圓周運動，周期均為T，兩顆恒星之間的距離為r，試推算這個雙星系統的總質量。（引力常量為G）

Answer 1

解析：設兩顆恒星的質量分別為m₁、m₂，做圓周運動的半徑分別為r₁、r₂，角速度分別為w₁,w₂。根據題意有

                            w₁=w₂                                                                                                                                                                                  ①

                                                        r₁+r₂=r                                                                                        ②

根據萬有引力定律和牛頓定律，有

                            G                                                                      ③

G                                                                      ④

聯立以上各式解得

                                                                            ⑤

根據解速度與周期的關系知

                                                                                                       ⑥

聯立③⑤⑥式解得

                                                                                                ⑦