Ke Kānāwai ʻEkolu o Kepler: Ke ʻimi nei i ka Neʻe ʻana o nā Hōkū a me nā Kino Lani
He mea nui nā kānāwai o Kepler no ka hoʻomaopopo ʻana i ka neʻe ʻana o nā hōkūhele a me nā kino lani ʻē aʻe i loko o ka ʻōnaehana lā. Ua hoʻokumu ʻo Johannes Kepler, he kilo hōkū Kelemania, i ʻekolu mau kānāwai e wehewehe ana i ka neʻe ʻana o nā hōkūhele a puni ka Lā. E nānā kēia ʻatikala i ke Kānāwai ʻEkolu o Kepler, i ʻike ʻia hoʻi ʻo ke Kānāwai Harmonic, kona mau ʻano pili, a me kona hoʻohana ʻana i ka kilo hōkū hou.
Hoʻolauna i nā Kānāwai o Kepler
Ma mua o ke kūkākūkā ʻana i ke Kānāwai 3 o Kepler, he mea nui e hoʻomaopopo i ke ʻano o nā kānāwai ʻekolu o Kepler:
1. Ke Kānāwai Mua o Kepler (Kānāwai o nā Ellipses): Neʻe nā hōkūhele ma nā ʻōpuni elliptical me ka Lā ma kekahi kiko o ka ellipse.
2. Ke Kānāwai ʻElua o Kepler (Kānāwai o nā Velocities Area): ʻO kahi laina e hoʻohui ana i kahi honua a me ka Lā e hoʻopau i nā wahi like i nā wā like, ʻo ia hoʻi ka neʻe wikiwiki o ka honua ke kokoke aku i ka Lā a lohi ke mamao aku mai ka Lā.
3. Ke Kānāwai ʻEkolu o Kepler (Kānāwai Harmonic): Ua like pololei ka huinahā o ka wā orbital o ka honua me ka cube o ke axis semi-major o kona orbit.
Ke Kānāwai ʻEkolu o Kepler
ʻŌlelo ke Kānāwai 3 o Kepler no nā hōkūhele āpau e kaapuni ana i ka Lā, ʻo ka lakio o ka huinahā o ka wā orbital (T) i ka cube o ke axis semi-major (a) he paʻa. Ma ke ʻano makemakika, hiki ke kākau ʻia kēia kānāwai penei:
\[ \frac{T^2}{a^3} = \text{cons} \]
Ma hea:
– ʻO T ka wā orbital o ka honua, ʻo ia ka manawa e pono ai no ka honua e hoʻopau i hoʻokahi orbit a puni ka Lā (ma nā makahiki inā aia ʻo a ma AU).
– ʻo a ka axis semi-major o ka orbit elliptical o ka honua (ma nā ʻāpana astronomical, AU).
- pili kēia mau i nā hōkūhele āpau o ka ʻōnaehana lā.
Ka Hoʻokumu ʻana o ke Kānāwai 3 o Kepler
No kā mākou ʻōnaehana lā, ua like nā mea mau e hoʻomalu ana i ke kānāwai ʻekolu o Kepler no nā hōkūhele āpau e kaapuni ana i ka Lā. Hōʻike ke ʻano ma luna i ka pilina makemakika ma waena o ka wā orbital a me ka nui o ka orbital, hiki ke hoʻohana ʻia e wānana i kekahi loli inā ʻike ʻia kekahi.
Ma nā ʻāpana astronomical (AU) no ke axis semi-major a me nā makahiki no ka wā orbital, ʻo kēia mau mea paʻa he 1:
\[ \frac{T^2}{a^3} = 1 \]
Eia nō naʻe, inā mākou e hoʻohana i nā anakahi ʻē aʻe a i ʻole e ana i nā kaapuni o nā mea a puni nā hōkū ʻē aʻe ma mua o ka Lā, pono mākou e noʻonoʻo i ke kūpaʻa umekaumaha honua (G) a me ka nuipa o ka mea kikowaena (M).
ʻO ke ʻano maʻamau no ke kānāwai 3 o Kepler ma kēia hihia:
\[ \frac{T^2}{a^3} = \frac{4 \pi^2}{G (M+m)} \]
Ma hea:
– ʻO G ke kūpaʻa umekaumaha honua (\( 6,674 \times 10^{-11} \, \text{Nm}^2 \text{kg}^{-2} \)).
– ʻO M ka nuipa o ka mea kikowaena (e.g., ka nuipa o ka Lā).
– ʻo m ka nuipa o ka honua, akā i ka hapanui o nā hihia, ʻoi aku ka liʻiliʻi o \( m \) ma mua o \( M \) a hiki ke hoʻowahāwahā ʻia.
Laʻana o ka Helu ʻana me ka hoʻohana ʻana i ke kānāwai 3 o Kepler
No ka wehewehe ʻana i ka hoʻohana ʻia ʻana o kēia kānāwai, eia kekahi mau laʻana o ka helu ʻana:
Laʻana 1: Ka Wā Hoʻohele o ka Honua
Ua ʻike ʻia he axis semi-major ko ka Honua ma kahi o 1 AU. Ke hoʻohana nei i ke kānāwai 3 o Kepler:
\[ \frac{T^2}{1^3} = 1 \]
No laila, ʻo ka wā orbital o ka Honua (T) he 1 makahiki, kahi e kūlike me ka ʻike ʻana he hoʻokahi makahiki ka Honua e kaapuni ai i ka Lā.
Laʻana 2: Ka Wā Hoʻopāpā o Mars
ʻO kahi axis semi-major ko Mars ma kahi o 1,524 AU. No ka helu ʻana i ka wā orbital o Mars:
\[ \frac{T^2}{(1,524)^3} = 1 \]
\[ T^2 = (1,524)^3 \]
\[ T^2 \approx 3,54 \]
\[ T \approx \sqrt{3,54} \]
\[ T \approx 1,88 \text{ makahiki} \]
No laila, ʻo ka wā o ke kaʻapuni o Mars ma kahi o 1,88 mau makahiki.
Ka Hoʻopili ʻana o ke Kānāwai 3 o Kepler
He nui nā noi kūpono a me nā noi kumumanaʻo o ke Kānāwai 3 o Kepler i ka ʻike hōkū a me ka ʻepekema lewa:
1. Ke hoʻoholo nei i ka nuipa o kahi hōkū: Ma ka nānā ʻana i ka wā a me ke axis semi-major o kahi honua e kaapuni ana i kahi hōkū, hiki i nā kilo hōkū ke hoʻoholo i ka nuipa o ka hōkū me ka hoʻohana ʻana i kahi mana laulā o ke kānāwai 3 o Kepler.
2. Wānana o nā kaʻapuni Satellite: Pili pū kēia kānāwai i nā satelite hana e kaʻapuni ana i nā hōkūhele, e kōkua ana i ka hoʻolālā a me ka nānā ʻana i nā satelite.
3. Ka ʻImi ʻana i ka ʻŌnaehana Lā: Ma nā misionari interplanetary, kōkua kēia kānāwai i ka hoʻolālā ʻana i ke ala orbital o kahi mokulele e hiki ai a kaʻapuni i kahi honua a mahina paha.
4. Ke aʻo ʻana i nā ʻōnaehana hōkū binary: Ma nā ʻōnaehana hōkū binary, hoʻohana ʻia ke kānāwai 3 o Kepler e hoʻoholo i nā palena orbital a me nā nuipa o nā hōkū ʻelua ma muli o nā nānā ʻana o kā lākou neʻe ʻana.
Ka hopena
ʻO ke Kānāwai ʻEkolu o Kepler, a i ʻole ke Kānāwai o Harmonics, he kia koʻikoʻi ia i ka hoʻomaopopo ʻana i nā dinamika orbital o nā hōkūhele a me nā kino lani ʻē aʻe. Ma ka hōʻike ʻana ua kūlike pololei ka huinahā o ka wā orbital i ka cube o ke axis semi-major o ka orbit, hāʻawi kēia kānāwai i kahi mea hana makemakika ikaika no ka nānā ʻana a me ka wānana ʻana i ka neʻe ʻana orbital.
Ma o nā hiʻohiʻona helu a me nā noi kūpono, ʻike mākou i ke ʻano o kēia kānāwai i nā ʻano like ʻole o ka ʻike hōkū a me ka ʻimi ʻana i ka lewa. Mai ka hoʻoholo ʻana i nā nui o nā hōkū a hiki i ka hoʻolālā ʻana i nā misionari interplanetary, ke hoʻomau nei ke Kānāwai ʻEkolu o Kepler i kumu koʻikoʻi no nā kānaka ʻepekema a me nā ʻenekinia e ʻimi nei a hoʻomaopopo i ke ao holoʻokoʻa.
Me kēia ʻike, ʻaʻole hiki iā mākou ke mahalo wale i ka nani makemakika a me ke kūlike o ka neʻe ʻana o ka honua, akā e hoʻohana pū i kēia mau kānāwai e wehe i nā ʻaoʻao hou i ka ʻimi ʻana a me ka ʻike ʻana i nā hōkū. ʻO ke Kānāwai ʻEkolu o Kepler, ʻoiai he maʻalahi ke ʻano, ua loaʻa iā ia kahi hopena hohonu a ākea i ka ʻepekema.
