2018年IAO理论低年组第2题-火星大冲

英文题目

Great oppositions of Mars. On the morning of July 27, 2018, Mars was in a great oposition (see attached ephemeris), The great oppositons of Mars (the situations when this planet is seen most brightly in comparison with the visibility during other oppositions) are repeated every 15 or 17 years. So the previous opposition (which is ales called the "greatest", since Marswas so bright only once in many centuries) was on August 28, 2003, and the next one will occur in 2035.

2.1. Determine the constellation, in which Mars was during this year opposition.

2.2. Considering the orbits of the Earth and Mars as circular, and based on the date of opposition 2003, calculate the dates the oppositions in 2018 and 2035 would occur.

2.3 Explain, why the difference appeared between the calculated and real dates of the opposition 2018. (Draw a figure that clearly demonstrates your explanation.)

2.4. In which of the oppositions, 2018 or 2035 is Mars brightest? (The answer must be justified by drawing a figure).

中文题目

火星大冲 2018年7月27日上午，火星到达大冲位置（见附录星历表【注：暂无】）。火星大冲（大冲时行星的视亮度在所有冲日事件中是最亮的）每15或17年发生一次。所以上一次大冲（也被称为“最”大冲，因为这一次的火星是许多世纪以来最亮的）发生在2003年8月28日，下一次大冲将发生在2035年。

2.1. 写出今年火星大冲时，火星位于哪个星座。

2.2. 假设地球和火星的轨道为圆轨道，基于题目中给出的2003年火星大冲的发生日期，计算2018年和2035年发生火星大冲的日期。

2.3. 试解释计算得到的2018年火星大冲日期与实际日期间存在差别的原因。（请画草图说明）

2.4. 2018年与2035年的火星大冲，哪一次更亮？（请画图证明你的答案）

解答

英文解答

2.1. During the opposition, the planet is at a point approximately (up to an angular distance from the ecliptic at this moment) opposite to the Sun, that is, located where the Sun was six monthe ago. The Sun is in the constellation Capricorn on January 27.

2.2. The oppositions are repeatde every synodic period. The synodic of Mars is:

$$T_S = 1/(1/T_E - 1/T_M),$$

$$T_S = 1/(1/365.256 - 1/686.98)\quad 779.93\ days,$$

Approximately 15 years passed from 2003 to 2018, and there are 15·T_E / T_S ≈ 7.02 synodic periods of Mars during this period. So, the opposition 2018 was to occur after 7 synodic periods earlier than August 28.

The easiest way to calculate the date is as follows. For 7 synodic periods of Mars 7×779.93 ≈ 5459 days pass. From August 28 2003, to August 28, 2018, 365×15+4 = 5479 days pass. Thus, it turns out that in 2018 the opposition of Mars should be 20 days earlier than August 28, that is, August 8.

Similarly, we calculate the date of opposition 2035. 32·T_E/T_S ≈ 14.99. 15 synodic periods:15× 79.93 ≈ 11699 days. From August 26, 2003, to August 28, 2035, 365×32+8 = 11688 days pass. Thus, it turns out that in 2035 the opposition of Mars should be 11 days later than August 28, that is, Septmeber 8.

2.3. The difference appears from face that in our calculations we take the average angular velocity of Mars in orbit, and in previous close to great oppositions, Mars is near the perihelion of its orbit, and its speed is larger. As a result, the difference in the angular velocities of the Earth and Mars is smaller. Therefore, to change the position of Mars relative to the Earth takes more time. That is why, the farther the opposition from the perihelion point of Mars orbit (which is at the ecliptic longitude that the Earth passes around August 28), the greater the error. Calculations in the approximation of circular orbits give about 1.5 times the underestimated interval between the day of the calculated opposition and the date on which the "greatest oppositions" take place (that is, August 28).

2.4. The position of Mars during the opposition of 2035 is closer to the positiom of the perihelion of the orbit than in 2018, therefore in the opposition of 2035 Mars will be brighter.

中文解答

2.1. 摩羯座。冲日时，行星大约正好在太阳对侧（此时的行星位置距离黄道只有一些角距离）。所以，火星在太阳半年前所在的位置。1月27日太阳位于摩羯座。

2.2. 行星与地球的每个会合周期会发生一次冲日。火星的会合为：

$$T_S = 1/(1/T_E - 1/T_M),$$

$$T_S = 1/(1/365.256 - 1/686.98)\quad 779.93\ days,$$

其中T_S代表火星的会合周期，T_E代表地球的公转周期，T_M代表火星的公转周期。

从2003年到2018年经过了大概15年，这大约相当于15×T_E / T_S ≈ 7.02个火星与地球的会合周期。所以2018年的火星冲日应发生在2003年火星冲日的7个会合周期之后，比8月28日更早的日子。

以下是最简单的计算方法。火星的7个会合周期大约经过了7×779.93 ≈ 5459 天。从2003年8月28日到2018年8月28日，经过了365×15+4 = 5479 天。所以2018年的火星冲日发生的日期应该比8月28日早20天，也就是在8月8日。

同理，从2003年到2035年经过了大概32年，这大约相当于32×T_E / T_S ≈ 14.99个火星会合周期。火星的15个会合周期大约经过了15×779.93 ≈ 11699 天。从2003年8月28日到2035年8月28日经过了365×32+8 = 11688 天。所以2035年的火星冲日发生的日期应该比8月28日晚11天，也就是在9月8日。

2.3. 造成冲日日期计算与实际上有差异的原因是在我们的计算中，假设了火星的公转角速度为一个恒定的平均角速度，但实际上在上一次2003年火星冲日的时候，火星更接近它的近日点，所以轨道角速度更大。因此，那时候地球和火星的轨道角速度差更小。火星相对于地球位置的改变需要花更多时间。这就是当大冲发生时火星（此时火星的黄经大约即地球在8月28日左右所在的黄经位置）离其轨道近日点越远，计算与实际大冲日期间的误差就越大。如果取近似圆轨道来计算2018年的大冲日期，所得的日期与2003年“最”大冲的8月28日间的日期差会被低估1.5倍。（译注：圆轨道计算该日期差为20天，而实际日期差为约31天）

2.4. 2035年火星大冲时火星的在轨道上位置比2018年火星大冲时更靠近近日点。所以2035年的火星大冲更亮。