Controversies about the value of the third cosmic velocity - Biblioteka UMCS

A N N A L E S

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ABSTRACT

The purpose of writing this article was to derivate the formula for the WKLUGFRVPLFYHORFLW\

XVLQJRQO\WKHODZVRIFRQVHUYDWLRQLQWKHKHOLRFHQWULFUHIHUHQFHV\VWHP,WWXUQVRXWWKDWLWFDQEH

GRQHE\XVLQJHOHPHQWDU\PDWKHPDWLFVWKDQNVWRZKLFKWKHUHDVRQLQJDQGFDOFXODWLRQVDUHDIIRUGD- EOHIRURQHLQWHUHVWHGLQWKLVVXEMHFW%\WKHZD\ZHZDQWHGWRGLVFXVVWKHHUURUVWKDWDSSHDUHYHQLQ

ZHOONQRZQWH[WERRNVDQGSURIHVVLRQDODUWLFOHVZKRVHFRPPLWPHQWOHDGVWRLQFRUUHFWUHVXOWV7KH

PDJQLWXGHVRIWKHWKLUGFRVPLFYHORFLW\REWDLQHGE\XVDUHY_3average NPVY3perihelion 

NPVDQGY_3aphelion NPV

,1752'8&7,21

The third cosmic velocityLVGHILQHGDVWKHPLQLPXPLQLWLDOYHORFLW\ZKLFK

WKH REMHFW RQ WKH (DUWK¶V VXUIDFH UHTXLUHV WR HVFDSH QRW RQO\ IURP WKH (DUWK¶V

JUDYLWDWLRQDOILHOGEXWIURPWKHVRODUJUDYLWDWLRQDOILHOGDVZHOO,QRWKHUZRUGVLW

LVWKHLQLWLDOYHORFLW\RIVXFKPDJQLWXGHWKDWWKHREMHFWODXQFKHGIURPWKH(DUWK¶V

VXUIDFHDWWKLVVSHHGFDQHVFDSHIURPWKH6RODU6\VWHPDQGPRYHRQWRWKHLQILQLW\

The WKLUGFRVPLFYHORFLW\LVGHILQHGZLWKUHVSHFWWRWKH(DUWK

7KH PLQLPXP YDOXH RI FRXUVH DSSOLHV WR WKH VLWXDWLRQ ZKHQ WKH YHORFLW\

YHFWRURIWKHREMHFWODXQFKHGIURPWKH(DUWK¶VVXUIDFHLVSDUDOOHOWRWKHLQVWDQWDQH- RXVRUELWDOYHORFLW\RIWKH(DUWK)LJ,QWKHHVWLPDWLQJWKHYDOXHRIWKHWKLUG

FRVPLFYHORFLW\VRPHDSSUR[LPDWLRQVDUHPDGHWKHURWDWLRQRIWKH(DUWKDQGWKH

Controversies about the value of the third cosmic velocity

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3ODF0&XULH6NáRGRZVNLHM/XEOLQ3RODQG

1&RUUHVSRQGLQJDXWKRU(PDLODGGUHVVVWDQLVODZKDODV#XPFVSO

LQWHUDFWLRQRIWKH(DUWK¶VDWPRVSKHUHZLWKWKHODXQFKHGREMHFWDUHQHJOHFWHG7KH

JUDYLWDWLRQDOLQWHUDFWLRQEHWZHHQWKHODXQFKHGREMHFWDQGRWKHUREMHFWVLQWKH6R- ODU6\VWHPH[FHSWWKH(DUWKDQGWKH6XQLWVHOIDUHQHJOHFWHGDVZHOO

)LJ/DXQFKLQJDQREMHFWIURPWKH(DUWKZLWKWKHWKLUGFRVPLFYHORFLW\

Around the WKLUGFRVPLFYHORFLW\DURVHVRPHFRQWURYHUVLHVPLVXQGHUVWDQG- LQJVDQGHYHQGLVSXWHVFRQFHUQLQJLWVYDOXH,WVHHPVWKDWLQWKHWZHQW\ILUVWFHQ- WXU\LQWKHWLPHZKHQFRVPLFIOLJKWVGRQRWFDXVHDQ\VHQVDWLRQDQGZKHQSUREHV

ODXQFKHGIURPWKH(DUWKUHDFKQRWRQO\WKHVXUIDFHRIRXUQDWXUDOVDWHOOLWHEXWDOVR

UHDFK0DUV-XSLWHUDQG3OXWRDQGVRPHRIWKHPOHDYHWKH6RODU6\VWHPIRUHYHU

the knowledge of the cosmic velocities should have been already well established DQGDYDLODEOHDWOHDVWIRUWKRVHZKRDUHLQWHUHVWHG

:KLOHLWVHHPVDOOWKHSK\VLFVWH[WERRNVIRUVHFRQGDU\VFKRROVDQGDQXPEHU

RIZHEVLWHVIRUSK\VLFVDVWURQRP\RUDVWURSK\VLFVHYHQDQDUWLFOHRQ:LNLSHGLD allow the reader to thoroughly acquaint with the issue of the first and second FRVPLFYHORFLW\JLYLQJDSUHFLVHGHILQLWLRQDQGGHULYDWLRQRIWKHZKROHIRUPXOD

ZKHUHDVWKHPDMRULW\RIVRXUFHVERRNVDQGZHEVLWHVLQ3ROLVKDQG(QJOLVKUDWKHU

ignore the WKLUGFRVPLFYHORFLW\LVVXH7KRVHVRXUFHVZKLFKGRPHQWLRQRQWKLVLV- VXHXVXDOO\SURYLGHDPRUHRUOHVVFOHDUGHILQLWLRQHVSHFLDOO\WKH,QWHUQHWFRQWDLQV

PDQ\LQDFFXUDFLHVDQGDOPRVWDOZD\VWKHYDOXHWKDWLVH[SHFWHGWREHY₃ 

NPV,WLVYHU\GLIILFXOWWRILQGDFRPSOHWHGHULYDWLRQRIWKHIRUPXOD7KHLVVXHLV

UHDOO\QRWWULYLDOEHFDXVHLWFRQFHUQVLQWKHPRVWVLPSOHWHUPVWKHWKUHHERGLHV

%XWRQHVKRXOGEHDZDUHWKDWE\DFFHSWLQJVRPHDSSUR[LPDWLRQVEXWQRWIDOVLI\- LQJWKHSK\VLFDOVHQVHDQGQRWEHLQJRYHUVLPSOLILHGWKLVSUREOHPFDQEHVROYHG

ZLWKRXWWKHXVHRIQXPHULFDOPHWKRGV

However it appears that the value of the WKLUGFRVPLFYHORFLW\ reported by WKHDXWKRUVRIJHQHUDOSK\VLFVDFDGHPLFWH[WERRNVLVGLIIHUHQW7KH³3K\VLFV´E\

2UHDUJLYHVNPV>@ZKHUHDVWKH³,QWURGXFWLRQWR3K\VLFV´E\:UyEOHZVNL

DQG=DNU]HZVNLJLYHVNPV>@7KHGLIIHUHQFHLVQRWGXHWRDSSUR[LPDWLRQV

LQSK\VLFDOFRQVWDQWVRUDSSUR[LPDWLRQVLQVRPHVWDJHVRIFDOFXODWLRQVEXWGXHWR

GLIIHUHQWZD\RIGHULYDWLRQZKLFKOHDGVWRGLIIHUHQWILQDOIRUPXOD

,Q  DQ DUWLFOH ³$ 6WXG\ RQ WKH7KHRUHWLFDO9DOXHRI WKH7KLUG &RVPLF

9HORFLW\´ ZDV SXEOLVKHG LQ WKH RQOLQH VFLHQWLILF MRXUQDO >@ 7KH DXWKRUV DIWHU

FRQGXFWLQJ WKHLU RZQ FRQVLGHUDWLRQV DFFRUGLQJ WR XV ZURQJ DQG FDOFXODWLRQV

TXHVWLRQHGWKHPRVWIUHTXHQWO\SXEOLVKHGYDOXHRINPV7KH\FODLPWKDWWKH

FRUUHFWYDOXHLVNPVZKLFKLVH[DFWO\WKHVDPHDVLQWKH2UHDU¶VWH[WERRN>@

In this short paper we present a derivation of the WKLUGFRVPLFYHORFLW\ for- PXODEDVHGRQO\RQWKHWKUHHODZVRIFRQVHUYDWLRQPRPHQWXPDQJXODUPRPHQ- WXPDQGHQHUJ\LQWKHUHIHUHQFHIUDPHDWWDFKHGWRWKH6XQ7KHQZHFRPSDUHRXU

UHDVRQLQJZLWKWKRVHE\2UHDUDQGE\:UyEOHZVNLDQG=DNU]HZVNLDQGE\WKDW

GHVFULEHGLQWKHDUWLFOHPHQWLRQHGDERYH:HEHOLHYHWKDWIRUDFRPSOHWHXQGHU- standing of our considerations the knowledge of physics and mathematics at the VHFRQGDU\VFKRROOHYHOLVVXIILFLHQW

$1$/<6,62)7+(352%/(0

:HFRQVLGHUWKHODXQFKIURPWKH(DUWK¶VVXUIDFHRIDQREMHFWKDYLQJDPDVV

RIPDQ\RUGHUVRIPDJQLWXGHVPDOOHUWKDQWKHPDVVRIWKH(DUWK7KLVREMHFWZLOO

be called the EXOOHW6LQFHWKH(DUWK¶VPDVVLV0 Â^NJDVVXPLQJWKHEXOOHW mass of P = 10⁶NJWKHPDVVUDWLRLVRIRUGHU

0

P 10^-18

Let us introduce the term gravitational interaction sphere>@7KHJUDYLWD

WLRQDOLQWHUDFWLRQ VSKHUHwill be called the area of the spherical surface centered DWWKHFHQWHURIWKHSODQHWLQZKLFKWKHPRWLRQRIDQ\REMHFWVXFKDVDEXOOHWLV

better described as a motion in the gravitational field of the planet with the per- turbations from the Sun than the motion in the gravitational field of the Sun with SHUWXUEDWLRQVIURPWKHSODQHW7KHUDGLXVRIWKH(DUWK¶VJUDYLWDWLRQDOLQWHUDFWLRQ VSKHUHis U_§NP¹

1 :HUHIHUWKHFDUHIXOUHDGHUWRWKHVHFWLRQ,,RIWKHWH[WERRN>@KRZHYHUZHQRWHWKDW

a precise determination of the U_UDGLXVLVQRWQHFHVVDU\LQRXUFDVH

)LJ7KHWUDMHFWRU\RIWKHEXOOHW after launch from the Earth with the WKLUGFRVPLFYHORFLW\

The lowest escape velocity ZLWKUHVSHFWWRWKH6XQRIDQREMHFWPRYLQJRQ

the Earth’s orbit will be called the parabolic velocity7KHQDPHFRPHVIURPWKH

IDFWWKDWDWUDMHFWRU\VXFKDQHVFDSLQJIURPWKH(DUWK¶VRUELWREMHFWLVDSDUDEROD

ZKLFKLQLWLDOO\FRQYHUJHVZLWKWKHRUELWRIWKH(DUWKDQGWKHQLVPRUHDQGPRUH

GHIOHFWLQJIURPLW)LJ-XVWDVWKHYDOXHRIWKHVHFRQGFRVPLFYHORFLW\LV 2 WLPHVJUHDWHUWKDQWKHYDOXHRIWKHILUVWFRVPLFYHORFLW\WKHYDOXHRIWKHSDUDEROLF

YHORFLW\ is 2 times greater than the value of the orbital velocity of the Earth and is v_p NPV

In our considerations we take into account the recoil effect of the Earth at the moment of launching of the EXOOHW from the Earth’s surface and so-called JUDYL

WDWLRQDOSXOOLQJHIIHFWLHHQWDLOLQJWKH(DUWKE\WKHODXQFKHGEXOOHW as a result of the gravitational interaction between the EXOOHWDQGWKH(DUWK7RVLPSOLI\WKHGLV- FXVVLRQZHDVVXPHWKDWWKH(DUWK¶VRUELWLVDFLUFOHRIUDGLXVU NP

:HZLOOXVHWKHIROORZLQJQRWDWLRQV

0₆LVWKHPDVVRIWKH6XQ

0LVWKHPDVVRIWKH(DUWK

P is the mass of the EXOOHWP0 ULVWKH(DUWK¶VRUELWUDGLXV

RLVWKH(DUWK¶VUDGLXV

U_ is the radius of the Earth’s JUDYLWDWLRQDOLQWHUDFWLRQ VSKHUH

Y_ is the orbital velocity of the Earth relative to the sun Y_ NPV

Y_’ is the Earth’s velocity relativHWRWKHVXQMXVWDIWHUWKHODXQFKRIDEXOOHW IURP(DUWK¶VVXUIDFH

Y_´LVWKH³ILQDO´(DUWK¶VYHORFLW\UHODWLYHWRWKHVXQLH(DUWK¶VYHORFLW\LQ

the moment when the receding EXOOHW³VWRSV´LQWHUDFWZLWKWKH(DUWKVRZKHQLW

crosses the Earth’s JUDYLWDWLRQDOLQWHUDFWLRQ VSKHUH

Y_p is the pDUDEROLFYHORFLW\

Y_ is the the second cosmic velocity Y_ =

R

*0

2 NPV

Y₃ is the WKLUGFRVPLFYHORFLW\

7KHODXQFKRIWKHbulletIURPWKH(DUWK¶VVXUIDFH

'XHWRWKHIDFWWKDWZHFRQVLGHUDVKRUWLQWHUYDORIWLPHIURPMXVWEHIRUHWLOO

MXVWDIWHUWKHODXQFKRIWKHEXOOHWIURPWKH(DUWKVXUIDFHLWFDQEHDVVXPHGWKDWDW

this time both the Earth and the EXOOHW move along the straight line locally tangen- WLDOWRWKHRUELWRIWKHSODQHW,QDGGLWLRQLIZHWUHDWWKH(DUWKDQGWKHEXOOHW as the WZRERGLHVLQWHUDFWLQJRQO\ZLWKHDFKRWKHURULJQRUHWKHLQWHUDFWLRQRIWKH6XQ

then the momentum of the Earth – EXOOHWVV\VWHPKDVWREHFRQVHUYHG7KLVOHDGV

LQWKHUHIHUHQFHIUDPHDWWDFKHGWRWKH6XQWRWKHIROORZLQJHTXDWLRQ



0 P



Y₀ Y0 ₀'P



Y₀'Y₃



 2QWKHOHIWVLGHRIWKHHTXDWLRQWKHUHLVZULWWHQDQH[SUHVVLRQIRUWKHPRPHQWXP

RIWKHV\VWHPMXVWEHIRUHWKHODXQFKRIWKHEXOOHWIURPWKH(DUWKDQGRQWKHULJKW

±MXVWDIWHUODXQFKLWIURPWKH(DUWKZLWKWKHWKLUGFRVPLFYHORFLW\

*UDYLWDWLRQDOLQWHUDFWLRQODXQFKHGbulletZLWKWKH(DUWKDQGWKH6XQ Moving away from the Earth the EXOOHW eventually crosses theEarth’sJUDYL

WDWLRQDOLQWHUDFWLRQ VpKHUH)URPWKLVSRLQWZHEHJLQWRWUHDWEXOOHW¶V motion as DPRWLRQLQWKHJUDYLWDWLRQDOILHOGRIWKH6XQ,QRUGHUWRHVFDSHIURPWKH6RODU

6\VWHPDQGPRYHWRLQILQLW\WKHEXOOHW has to move at least with the pDUDEROLF

YHORFLW\ Y_p)LJ

)LJ7KHEXOOHW crossing the Earth’s JUDYLWDWLRQDOLQWHUDFWLRQ VpKHUH7KHEXOOHWVWUDMHFWRU\

FRQYHUJHVZLWK(DUWK¶VRUELWLQVLGHWKHVSKHUH

M

v

Here we assume that the EXOOHW launched from the Earth with the WKLUGFRVPLF

YHORFLW\PRYHVDORQJDSDUDERODZKLFKZLWKLQWKH(DUWK¶VJUDYLWDWLRQDOLQWHUDF

WLRQ VpKHUHFRQYHUJHVZLWKWKH(DUWK¶VRUELW)LJ7KLVLVQRWWRRFRDUVHDS- proximation because the diameter of the Earth’s JUDYLWDWLRQDOLQWHUDFWLRQ VpKHUH LVOHVVWKDQRIWKHOHQJWKRILWVRUELWZKLFKFRUUHVSRQGVWRDQDQJXODUDUF

OHQJWKRI(DUWK¶VRUELWDERXWƒ$QGDFFRUGLQJWRVLPSOHFDOFXODWLRQVWKHGLV- tance between the points of intersection of the JUDYLWDWLRQDOLQWHUDFWLRQ VpKHUH ZLWKWKH(DUWK¶VRUELWDQGZLWKWKHSDUDERODLVDERXWNPZKLFKLVOHVVWKDQ

10^-5$87KHUHIRUH ZH FODLP WKDW LQ WKH UHIHUHQFH IUDPH DWWDFKHGWR WKH 6XQ

according to the law of conservation of angular momentum of the Earth-EXOOHW V\VWHPDQJXODUPRPHQWXPDWWKHWLPHMXVWDIWHUWKHODXQFKRIWKHEXOOHW from the surface of the Earth is equal to the angular momentum at a time when the EXOOHW leaves the Earth’s JUDYLWDWLRQDOLQWHUDFWLRQ VpKHUH



Y Y



U 0Y U PY U P

U

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HQHUJ\

 

0 2 2

0 2

0 3 2

0 2

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U 0

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Y P

0Y     _p 

 ,Q WKH ODVW HTXDWLRQ ZH KDYH QHJOHFWHG RQ ERWK VLGHV WKH SRWHQWLDO HQHUJ\ RI the Earth and the EXOOHWLQWKH6XQ¶VJUDYLW\ILHOG

U P 0

*0⁶  

  as it is iden-

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0 ²



²

2 0 0

2 0

3 1 1 " '

 2

' Y Y Y_p

P 0 U

*0 R Y

Y     



7KHQZHVHHWKDWEHFDXVH

7DNLQJLQWRDFFRXQWLQHTXDWLRQWKLVDSSUR[LPDWLRQVXEVWLWXWLQJWKH

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007

 0

0

U | R

M

v

M

v

mv

Mv Mm Mv mv Mm

GM

R

*0 U R R

*0 U

*0 R1 1 2 1 2 2

0 0

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¹

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*0 2



8VLQJWKHDSSUR[LPDWLRQZHREWDLQ²



_{3 0}'



² ₂² 2



Y₀" Y₀'



Y₀' Y_p² P

Y 0 Y

Y     

&DOFXODWLQJ IURP HTXDWLRQ 



Y Y



Y Y Y_p P

0 ₀" ₀' ₃  ₀' and substituting to HTXDWLRQZHJHW

  

3 0



0 ²

2 2 2 0

3 Y ' Y 2Y Y ' Y_p Y ' Y_p

Y       ,QHTXDWLRQZHSHUIRUPPXOWLSOLFDWLRQDQGUHGXFHVLPLODUWHUPV$IWHUWKHVH

operations we obtain the following expression for the WKLUGFRVPLFYHORFLW\



0



²

2 2 2

3 Y Y Y '

Y  _p  

$IWHUVXEVWLWXWLRQLQHTXDWLRQY_p 2Y₀ and making approximation ³Y₀' Y| ₀ we take the square root of both sides and obtain the final formula for the WKLUG

FRVPLFYHORFLW\

 

0²

2 2 2

3 Y 2 1 Y

Y    If we substitute Y_   NPV DQG Y_   NPV ZH JHW WKH YDOXH RI

a WKLUGFRVPLFYHORFLW\ZKLFKWXUQVRXWWREHY₃ NPV

',6&866,21

The interaction of the Earth with the launched EXOOHW may be considered in WZRVWDJHV,QLWLDOO\GXHWRWKHUHFRLOWKHRUELWDO(DUWKYHORFLW\GHFUHDVHVDQG

then increases as a result of gravitational attraction between these two bodies JUDYLWDWLRQDOpXOOLQJ7KLVODWWHUHIIHFWKLJKO\FRPSHQVDWHVIRUUHFRLOVRWKDWLQ

2 (TXDWLRQOHDGVWR

$IWHUVXEVWLWXWLQJ

10^ we get

+RZ- HYHUZHHPSKDVL]HWKDWWKHUHLVDOLWWOHGLIIHUHQFHEHWZHHQWKHVHYHORFLWLHVZKLFKZHGRQRWQHJOHFW

LQHTXDWLRQ

3 (TXDWLRQOHDGVWR

6XEVWLWXWLQJ^-18ZHJHW

3 0 3 0 0

1

' Y

0 P 0 P Y PY 0 Y P Y



 



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" 0 3 0

0 Y Y Y Y

0

Y P   p 0

P Y₀"|Y₀'

0

P Y₀"|Y₀'

  

0 0



0 0



²

2 2 2 0

3 ' Y " Y ' Y " Y ' Y_p P

Y 0 Y

Y     

' 2 '

" _{0 0}

0 Y Y

Y  |

the final result the orbital Earth speed is only insignificantly lower than the speed MXVWEHIRUHWKHODXQFKRIWKHEXOOHW^

The value of the WKLUGFRVPLFYHORFLW\ we obtained Y₃ NPVLVHTXDO

WRWKHYDOXHUHSRUWHGLQPRVWRIWKHWH[WERRNLQFOXGLQJWKDWE\:UREOHZVNLDQG

=DNU]HZVNL>@DVZHOODVWKHDUWLFOHV>@DQG>@,QWKHODVWDUWLFOHWKHDXWKRUV

evaluated the WKLUG FRVPLF YHORFLW\ QXPHULFDOO\ ZLWKRXW UHIHUHQFH WR WKH FRQ- VHUYDWLRQODZVEXWRQO\RQWKHEDVLVRIJHQHUDOIRUPRI1HZWRQ¶VVHFRQGODZ

dt ) p d& & 

The values Y₃   NPV DQG Y₃   NPV SUHVHQWHG LQ >@ DQG LQ WKH

WH[WERRN>@UHVSHFWLYHO\DUHREWDLQHGZKHQWKHtKLUdFRVPLFYHORFLt\ formula LVGHULYHGRQO\RQWKHEDVLVRIWKHODZRIFRQVHUYDWLRQRIHQHUJ\DVVXPLQJLQ- H[SOLFLWO\WKDWQRWRQO\WKH6XQEXWDOVRWKH(DUWKLVVWDWLRQDU\)ROORZLQJ

WKLVDSSURDFKWKHHQHUJ\RIWKHEXOOHtMXVWDIWHULWVODXQFKIURPWKHVXUIDFHRIWKH

VWDWLRQDU\(DUWKLV

2 0

1 2

3  

U P

*0 R 0

*P

PY ⁶ 

According to the definition of the tKLUdFRVPLFYHORFLt\WRWDOEXOOHt energy in LQILQLW\LVRIFRXUVH]HUR7KHYDOXHRIY₃ FDOFXODWHGIURPWKHHTXDWLRQLV

NPV/HWXVHPSKDVL]HWKDWLQWKLVUHDVRQLQJWKHYDOXHRINPVLVUHODWLYHWR

WKH6XQDQGWRWKH(DUWKDVZHOO+RZHYHUWKHDXWKRUVRIWKHDUWLFOH>@DQG2UHDU

>@VXEWUDFWIURPWKLVYDOXHWKHRUELWDOYHORFLW\RIWKH(DUWKDVWKHYHORFLW\DOUHDG\

acquired by a EXOOHtDWWKHWLPHRILWVODXQFKDQGWKXVJHWDYDOXHRIUHVSHFWLYHO\

NPVDQGNPV7KHVHYDOXHVDUHORZHUWKDQWKHYDOXHREWDLQHGE\XV

7KHH[SODQDWLRQRIWKLVGLIIHUHQFHLVVLPSOHEXWQRWREYLRXV7KXVLQWKHDERYH

UHDVRQLQJWKHLQFUHDVHRIWKH(DUWK¶VHQHUJ\LQWKHUHIHUHQFHIUDPHDWWDFKHGWR

WKH6XQFRUUHVSRQGLQJWRDQLQFUHDVHRI(DUWK¶VVSHHGGXHWRWKHJUDYLtDtLRQDO

pXOOLQJLVFRPSOHWHO\QHJOHFWHG7KLVLVXQDFFHSWDEOHVLPSOLILFDWLRQEHFDXVHWKLV

energy is turning out to be more than twice higher than the energy of the EXOOHt es- FDSLQJIURPWKH(DUWK>@2QO\LQWKHUHIHUHQFHIUDPHDWWDFKHGWRWKH(DUWKWKHUH

is no effect of JUDYLtDtLRQDOpXOOLQJEXWWKHQWKHPRWLRQRIWKH(DUWKZLWKUHVSHFW

WRWKH6XQFDQQRWEHLJQRUHG/HW¶VFOHDUO\SRLQWRXWKHUHWKH HQHUJ\RIWKHbullet ODXQFKHGIURPWKH(DUWKKDVWREHVXIILFLHQWQRWRQO\IRULWVHVFDSHIURPWKH

6RODU6\VWHPEXWDOVRIRUDFFHOHUDWLQJWKH(DUWK

It is worth noting that the derivation of the tKLUd FRVPLF YHORFLt\ formula EHFRPHVYHU\HDV\LQDUHIHUHQFHIUDPHDWWDFKHGWRWKH(DUWK6RWKDWLQDQDWX- ral way one can avoid the need to take into account the recoil of the Earth and

 The decrease of the Earth’s orbital velocity due to the recoil is and the following increase of the orbital velocity as a result of the JUDYLtDtLRQDOpXOOLQJLV

7KHUHIRUHWKHRYHUDOOFKDQJHRIWKH(DUWK¶VRUELWDOYHORFLW\LVQHJDWLYH

3 0

0 ' Y

P 0 Y P

Y  

Y Yp

0 Y P Y₀"₀| ₀'

Y Y Yp

0 Y P

Y0"0' 0'3

mv₃2 Mm

dp Fdt.

the JUDYLtDtLRQDO pXOOLQJ 6XFK DSSURDFK KDV EHHQ DGRSWHG E\:UyEOHZVNL DQG

=DNU]HZVNL>@/HW¶VUHFDOOWKDWWKHpDUDEROLFYHORFLt\LVWKHHVFDSHYHORFLW\ZLWK

UHVSHFWWRWKH6XQIURPWKH6RODU6\VWHPRIDQREMHFWPRYLQJRQWKH(DUWK¶VRUELW

IDUDZD\IURPWKH(DUWKVRWKDWWKHJUDYLWDWLRQDOLQWHUDFWLRQHQHUJ\ZLWKWKH(DUWK

FDQEHQHJOHFWHG6RWKHZKROHUHDVRQLQJFDQEHSUHVHQWHGDVIROORZV

In order to escape from the Solar System the EXOOHt has to be launched from WKH(DUWK¶VVXUIDFHZLWKVXFKDVSHHGZKLFKHQDEOHVWRFURVVWKH(DUWK¶VJUDYLtD

tLRQDOLQtHUDFtLRQ VpKHUH with at least the pDUDEROLFYHORFLt\)LJ7KLVVSHHG

with respect to the Earth is Y_p^Y0 2Y0^Y0



2^1



Y0   NPV ,I ZH

assume that the potential energy of the EXOOHt at the Earth’sJUDYLtDtLRQDOLQtHUDFtLRQ VpKHUHLV]HURWKHQWKHODZRIFRQVHUYDWLRQRIHQHUJ\FDQEHZULWWHQDVIROORZV

         

Since the potential energy of the EXOOHt in the Sun gravity field at the moment of launching from the Earth is the same as at the moment of crossing the Earth’s JUDYLtDtLRQDOLQtHUDFtLRQ VpKHUHLQWKHDERYHHTXDWLRQWKLVHQHUJ\LVQHJOHFWHG

MXVWOLNHZHGLGLWLQHTXDWLRQ$IWHUUHGXFWLRQRIWKHEXOOHtPDVVPLQHTXDWLRQ

PXOWLSOLFDWLRQE\IDFWRUDQGVXEVWLWXWLRQRIWKHVTXDUHRIWKHVHFRQGFRVPLF

YHORFLW\Y₂²IRUH[SUHVVLRQ

R

*0

2 ZHJHWWKHILQDOIRUPXODIRUWKHtKLUdFRVPLF YHORFLt\

ZKLFKLVLGHQWLFDODVHTXDWLRQ

:HVHHWKDWWKHFDOFXODWLRQVLQWKHUHIHUHQFHIUDPHDWWDFKHGWRWKH(DUWKDUH

VLPSOHUEXWUHTXLUHKDYLQJWKHNQRZOHGJHDERXWWKHPRYHPHQWVRIREMHFWVLQWKH

heliocentric system and a mental transition from one system to the other in the GHULYDWLRQLVQHFHVVDU\,QFRQWUDVWLQWKHGHULYDWLRQGHPRQVWUDWHGDERYHZHXVHG

FRQVLVWHQWO\WKHKHOLRFHQWULFV\VWHPRQO\

The tKLUdFRVPLFYHORFLt\ formula can be generalized by considering the an- JOHijEHWZHHQWKHYHORFLW\YHFWRURIWKHEXOOHt and the instantaneous Earth’s or- ELWDOYHORFLW\>@

 

0²

2 2

3  Y 3 2 2cos Y

Y M   M

         

For ij ƒZKHQWKHEXOOHt velocity is consistent with the instantaneous Earth RUELWDOYHORFLW\WKHHTXDWLRQWXUQVLQWRHTXDWLRQVRY₃ƒ NPV

For ij ƒODXQFKLQJWKHEXOOHtWRZDUGVWKH(DUWK6XQGLUHFWLRQY₃ƒ 

NPV)RUij ƒODXQFKLQJWKHEXOOHt in reverse to the instantaneous Earth or- ELWDOYHORFLW\Y₃ƒ NPV

 

2 1 2 2

2 0 2 2

3 P Y

R 0

*P

PY 



 

0²

2 2 2

3 Y 2 1 Y

Y  

mv₃² Mm

It is worth noting that the numerical calculations performed by the authors RI WKH DUWLFOH ³&RPSDULQJ VROXWLRQV IRU WKH VRODU HVFDSH WKH SUREOHP´ >@ JLYH

DQLQWHUHVWLQJQRQLQWXLWLYHUHVXOWV$FFRUGLQJWRWKHPWKHRSWLPDOGLUHFWLRQIRU

launching the EXOOHtLVQRWLQVWDQWDQHRXVGLUHFWLRQRIWKH(DUWK¶VRUELWDOYHORFLW\

EXWWKHGLUHFWLRQGHIOHFWHGE\ƒRXWZDUGVRILW)RUVXFKDGLUHFWLRQQXPHULFDOFDO- culations of the tKLUdFRVPLFYHORFLt\OHDGWRDYDOXHORZHUE\LQFRPSDULVRQ

WRWKDWDFKLHYHGDOVRQXPHULFDOO\IRUWKHGLUHFWLRQDORQJWKH(DUWK¶VRUELW7KH

DXWKRUVGRQRWSURYLGHDSK\VLFDOH[SODQDWLRQIRUWKLVXQH[SHFWHGUHVXOWDFWXDOO\

WKH\GRQRWJLYHDQ\H[SODQDWLRQRUFRPPHQWDWDOO2QHPXVWWDNHLQWRDFFRXQW

that their numerical calculations were based only on the general form of New- ton’s second law and the assumption that the motionless Sun is at the center of WKH(DUWK¶VRUELWZKLFKZDVDUWLILFLDOO\VHWDVDFLUFOH,QRWKHUZRUGVWKHDXWKRUV

DVWKHFHQWHURIWKHPDVVRIWKHV\VWHPWUHDWHGWKHFHQWHURIWKH6XQWKH\DOVR

neglected the constant movement of the Earth due to the gravitational interaction with the EXOOHtJUDYLtDtLRQDOpXOOLQJ6XFKDQLWHUDWLYH³IROORZLQJ´DORQJWKHLQ- stantaneous growth of d&pYHFWRUPD\OHDGWRREWDLQLQJDWUDMHFWRU\YHU\FORVHWR

the actual EXOOHtWUDMHFWRU\+RZHYHUQRWWDNLQJLQWRDFFRXQWWKHODZVRIFRQVHUYD- WLRQDOWKRXJKWKH\DULVHIURPWKH1HZWRQ¶VODZVDQGXVLQJLQHDFKVWHSRILWHUD- WLRQDSSUR[LPDWHGYDOXHVZKLFKLVLQKHUHQWLQWKHQXPHULFDOFDOFXODWLRQVPD\

OHDGWRQRQSK\VLFDOHIIHFWVOLNHWKHGHIOHFWLRQRIHVFDSHYHORFLW\YHFWRURXWZDUGV

RIWDQJHQWLDOWRWKH(DUWK¶VRUELWDOYHORFLW\7KHUHLVQRHPSLULFDOFRQFOXVLYHGDWD

LQWKDWPDWWHU7KHDXWKRUVRIPHQWLRQHGSDSHU>@VWDWHWKDWWKH\FDOFXODWHGWKH

YHORFLW\ZLWKDFFXUDF\FORVHUWKDQ

:HJHWHTXDWLRQXQGHUDVVXPSWLRQWKDWWKH(DUWK¶VRUELWLVDFLUFOHVRWKDW

WKH(DUWK¶VRUELWDOYHORFLW\LVFRQVWDQW7KXVWKLVIRUPXODLVRQO\DQH[SUHVVLRQ

for the average magnitude of the tKLUdFRVPLFYHORFLt\+RZHYHURQHFDQEHOLHYH

WKDWEHFDXVHWKH(DUWK¶VRUELWLVDQHOOLSVHWKHtKLUdFRVPLFYHORFLt\ has a different PDJQLWXGHGHSHQGLQJRQWKHSRVLWLRQRIWKH(DUWKRQWKHRUELWDQGWKHH[WUHPH

YDOXHVVKRXOGRFFXUIRUH[WUHPH(DUWK6XQGLVWDQFHZKLFKLVWKHSHULKHOLRQDQG

DSKHOLRQ,QGHHGWKHSUHFLVHFDOFXODWLRQVOHDGWRVXFKFRQFOXVLRQ+RZHYHULQ

order to calculate the tKLUdFRVPLFYHORFLt\LQWKHSHULKHOLRQDQGDSKHOLRQLWLV

QRWHQRXJKWRVXEVWLWXWHLQWRWKHIRUPXODUHVSHFWLYH(DUWK¶VYHORFLWLHV7KHHO- liptical shape of the orbit causes not only different Earth’s orbital velocities in the SHULKHOLRQDQGDSKHOLRQEXWDOVRGLIIHUHQW(DUWK6XQGLVWDQFHVDQGFRQVHTXHQWO\

different pDUDEROLFYHORFLtLHVLQWKHVHH[WUHPHSRVLWLRQVRIWKH(DUWK$OORIWKHVH

concerns have to be taken into account already in derivation of the formula for the tKLUdFRVPLFYHORFLt\:KHQZHWDNHDOOWKLVLQWRFRQVLGHUDWLRQDQGWKDWWKHUDGLXV

RIFXUYDWXUHRIWKHHOOLSVHDWDSKHOLRQDQGRUSHULKHOLRQLV



²



2

1 e a a

E 

U 

we receive the following formula for the Earth’s orbital velocity at perihelion and DSKHOLRQUHVSHFWLYHO\

e Y e

Y _peU\KeOLRQ



 1 1

0

0    

e Y e

Y _apKeOLRQ



 1 1

0 0

         

Then the respective formulas for the paUaEROLFYeORFLt\ in perihelion and aphelion DUHDVIROORZV

Y e Y_ppeU\KeOLRQ

 1

2

0   

Y e Y_papKeOLRQ

 1

2

0



         

7KH V\PEROV XVHG LQ DERYH H[SUHVVLRQV PHDQ a LV WKH PDMRU VHPLD[LV RI WKH

(DUWK¶VRUELWELVDPLQRUVHPLD[LVRI(DUWK¶VRUELWe is eccentricity of the el- OLSWLFDO(DUWK¶VRUELW$VDUHVXOWRIUHDVRQLQJOHDGLQJWRHTXDWLRQDQGXVLQJ

HTXDWLRQVZHJHWWKHILQDOIRUPXODVIRUWKHtKLUdFRVPLFYeORFLt\ at SHULKHOLRQDQGDSKHOLRQUHVSHFWLYHO\

 

²

0 2 2

2

3 1

1

2 Y

e Y e

Y _peU\KeOLRQ ˜





   

 

²

0 2 2

2

3 1

1

2 Y

e Y e

Y_apKeOLRQ ˜





 

         

$IWHUVXEVWLWXWLQJLQWRWKHPe >@Y_ NPVDQGY_ NPV

we get the tKLUdFRVPLFYeORFLt\PDJQLWXGHVLQWKHSHULKHOLRQDQGDSKHOLRQZKLFK

DUHDVIROORZVv3SHULKHOLRQ NPV and v_3DSKHOLRQ NPV

6RLWWXUQVRXWSHUKDSVDELWVXUSULVLQJO\WKDWODXQFKLQJWKHEXOOet from the (DUWKLQRUGHUWRHVFDSHIURPWKH6RODU6\VWHPLVHDVLHULQWKHSHULKHOLRQZKHUH

WKHJUDYLWDWLRQDOIRUFHRIWKH6XQLVVWURQJHUWKDQWKDWLQDSKHOLRQ1RWHWKDWIRU

e LHIRUWKHFLUFXODU(DUWK¶VRUELWVWKHIRUPXODVDQGERWKWXUQLQWR

WKHIRUPXOD:HGLGQRWILQGWKHIRUPXODVDQGLQWKHOLWHUDWXUHEXW

PDJQLWXGHVWRZKLFKWKH\OHDGDUHFRQILUPHGE\WKHDUWLFOH³3DUDGR[HVRIFRVPLF

IOLJKWV´E\$U\6]WHUQIHOG>@

,QWKHFRQWH[WRIHTXDWLRQVDQGRQHFDQFRQVLGHUWKHPRUHJHQHUDO

SUREOHPQDPHO\ZKHWKHUWKHGLIIHUHQWPDJQLWXGHRIWKHtKLUdFRVPLFYeORFLt\ at different positions of the Earth on its orbit is not contradictory with the law of con- VHUYDWLRQRIHQHUJ\$UHZHQRWGHDOLQJZLWKDSDUDGR[FRQVLVWLQJLQWKHIDFWWKDW

GHVSLWHWKHFRQVWDQWWRWDOHQHUJ\LQWKH(DUWK¶VRUELWDOPRWLRQDGLIIHUHQWLQLWLDO

kinetic energy is necessary in order to launch a bullet beyond the Solar System Y3peryhelionY_3aphelion"7RH[SODLQWKLVLVVXHRQHKDVWREHDZDUHWKDWLQIDFWWKH(DUWK

GRHVQRWPRYHDURXQGWKH6XQKDYLQJDIL[HGSRVLWLRQLQVSDFH:HEHOLHYHWKDW

accurate calculations carried out in the reference frame attached to the center of the mass of the Earth-Sun system would remove the paradox and would not lead WRLQFRQVLVWHQF\ZLWKWKHODZRIFRQVHUYDWLRQRIHQHUJ\+RZHYHUVXFKFRQVLGHUD- WLRQVH[FHHGWKHIUDPHVRIWKLVDUWLFOH

Let us finally note that the Earth orbits around the Sun and it additionally URWDWHVDURXQGLWVD[LV7KHOLQHDUVSHHGRIDSRLQWRQWKH(DUWK¶VHTXDWRUGXHWR

WKH(DUWK¶VURWDWLRQLVFORVHWRNPV7KLVIDFWDOVRFRXOGEHXVHGWRIXUWKHU

optimize the initial conditions in which the bullet is launched in order to leave the 6RODU6\VWHPMXVWOLNHLWLVPDGHLQUHJXODUIOLJKWVWRVSDFH

6800$5<

The formulas and magnitudes of the tKLUdFRVPLFYeORFLt\REWDLQHGLQWKLVSDSHUDUH

Third cosmic

velocity Formula Magnitude

>NPV@

Average

 

0²

2 2 2

3 Y 2 1 Y

Y   ˜ ^16

In peryhelion 

In aphelion 

 

²

0 2 2

2

3 1

1

2 Y

e Y e

Y _peU\KeOLRQ ˜





 

 

²

0 2 2

2

3 1

1

2 Y

e Y e

Y_apKeOLRQ ˜





 

REFERENCES

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VFLHQWLILFQHW$00

>@$=+HQGHO6ROaUeVFape$PHULFDQ-RXUQDORI3K\VLFVYRO51, ±

>@$=+HQGHO0-/RQJR&RPpaULQJVROXtLRQVIRUtKeVROaUeVFapepUREOeP$PHULFDQ-RXUQDO

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