Question

（08寧夏卷）23.（15分）天文學(xué)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(tǒng)在銀河系中很普遍。利用雙星系統(tǒng)中兩顆恒星的運動特征可推算出它們的總質(zhì)量。已知某雙星系統(tǒng)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(tǒng)的總質(zhì)量。（引力常量為G）

Answer 1

解析：設(shè)兩顆恒星的質(zhì)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(jù)題意有

                            w₁=w₂                                                                                                                                                                                  ①

                                                        r₁+r₂=r                                                                                        ②

根據(jù)萬有引力定律和牛頓定律，有

                            G                                                                      ③

G                                                                      ④

聯(lián)立以上各式解得

                                                                            ⑤

根據(jù)解速度與周期的關(guān)系知

                                                                                                       ⑥

聯(lián)立③⑤⑥式解得

                                                                                                ⑦